Step | Hyp | Ref
| Expression |
1 | | mulsuniflem.3 |
. . . 4
⊢ (𝜑 → 𝐴 = (𝐿 |s 𝑅)) |
2 | | mulsuniflem.1 |
. . . . 5
⊢ (𝜑 → 𝐿 <<s 𝑅) |
3 | 2 | scutcld 27233 |
. . . 4
⊢ (𝜑 → (𝐿 |s 𝑅) ∈ No
) |
4 | 1, 3 | eqeltrd 2833 |
. . 3
⊢ (𝜑 → 𝐴 ∈ No
) |
5 | | mulsuniflem.4 |
. . . 4
⊢ (𝜑 → 𝐵 = (𝑀 |s 𝑆)) |
6 | | mulsuniflem.2 |
. . . . 5
⊢ (𝜑 → 𝑀 <<s 𝑆) |
7 | 6 | scutcld 27233 |
. . . 4
⊢ (𝜑 → (𝑀 |s 𝑆) ∈ No
) |
8 | 5, 7 | eqeltrd 2833 |
. . 3
⊢ (𝜑 → 𝐵 ∈ No
) |
9 | | mulsval 27494 |
. . 3
⊢ ((𝐴 ∈
No ∧ 𝐵 ∈
No ) → (𝐴 ·s 𝐵) = (({𝑒 ∣ ∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑒 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔))} ∪ {ℎ ∣ ∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵)ℎ = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗))}) |s ({𝑘 ∣ ∃𝑙 ∈ ( L ‘𝐴)∃𝑚 ∈ ( R ‘𝐵)𝑘 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))} ∪ {𝑛 ∣ ∃𝑥 ∈ ( R ‘𝐴)∃𝑦 ∈ ( L ‘𝐵)𝑛 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))}))) |
10 | 4, 8, 9 | syl2anc 584 |
. 2
⊢ (𝜑 → (𝐴 ·s 𝐵) = (({𝑒 ∣ ∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑒 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔))} ∪ {ℎ ∣ ∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵)ℎ = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗))}) |s ({𝑘 ∣ ∃𝑙 ∈ ( L ‘𝐴)∃𝑚 ∈ ( R ‘𝐵)𝑘 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))} ∪ {𝑛 ∣ ∃𝑥 ∈ ( R ‘𝐴)∃𝑦 ∈ ( L ‘𝐵)𝑛 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))}))) |
11 | 4, 8 | mulscut2 27518 |
. . 3
⊢ (𝜑 → ({𝑒 ∣ ∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑒 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔))} ∪ {ℎ ∣ ∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵)ℎ = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗))}) <<s ({𝑘 ∣ ∃𝑙 ∈ ( L ‘𝐴)∃𝑚 ∈ ( R ‘𝐵)𝑘 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))} ∪ {𝑛 ∣ ∃𝑥 ∈ ( R ‘𝐴)∃𝑦 ∈ ( L ‘𝐵)𝑛 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))})) |
12 | 2, 1 | cofcutr1d 27341 |
. . . . . 6
⊢ (𝜑 → ∀𝑓 ∈ ( L ‘𝐴)∃𝑝 ∈ 𝐿 𝑓 ≤s 𝑝) |
13 | 6, 5 | cofcutr1d 27341 |
. . . . . . . . . 10
⊢ (𝜑 → ∀𝑔 ∈ ( L ‘𝐵)∃𝑞 ∈ 𝑀 𝑔 ≤s 𝑞) |
14 | 13 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ ∃𝑝 ∈ 𝐿 𝑓 ≤s 𝑝)) → ∀𝑔 ∈ ( L ‘𝐵)∃𝑞 ∈ 𝑀 𝑔 ≤s 𝑞) |
15 | | reeanv 3226 |
. . . . . . . . . . . . . . 15
⊢
(∃𝑝 ∈
𝐿 ∃𝑞 ∈ 𝑀 (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞) ↔ (∃𝑝 ∈ 𝐿 𝑓 ≤s 𝑝 ∧ ∃𝑞 ∈ 𝑀 𝑔 ≤s 𝑞)) |
16 | | leftssno 27304 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ( L
‘𝐴) ⊆ No |
17 | | simprl 769 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵))) → 𝑓 ∈ ( L ‘𝐴)) |
18 | 16, 17 | sselid 3977 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵))) → 𝑓 ∈ No
) |
19 | 18 | adantrr 715 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → 𝑓 ∈ No
) |
20 | 8 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → 𝐵 ∈ No
) |
21 | 19, 20 | mulscld 27520 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (𝑓 ·s 𝐵) ∈ No
) |
22 | 4 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → 𝐴 ∈ No
) |
23 | | leftssno 27304 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ( L
‘𝐵) ⊆ No |
24 | | simprr 771 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵))) → 𝑔 ∈ ( L ‘𝐵)) |
25 | 23, 24 | sselid 3977 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵))) → 𝑔 ∈ No
) |
26 | 25 | adantrr 715 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → 𝑔 ∈ No
) |
27 | 22, 26 | mulscld 27520 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (𝐴 ·s 𝑔) ∈ No
) |
28 | 21, 27 | addscld 27393 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → ((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) ∈ No
) |
29 | 19, 26 | mulscld 27520 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (𝑓 ·s 𝑔) ∈ No
) |
30 | 28, 29 | subscld 27464 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ∈ No
) |
31 | 30 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ∈ No
) |
32 | | ssltss1 27219 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝐿 <<s 𝑅 → 𝐿 ⊆ No
) |
33 | 2, 32 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝜑 → 𝐿 ⊆ No
) |
34 | 33 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → 𝐿 ⊆ No
) |
35 | | simprl 769 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → 𝑝 ∈ 𝐿) |
36 | 34, 35 | sseldd 3980 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → 𝑝 ∈ No
) |
37 | 36 | adantrl 714 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → 𝑝 ∈ No
) |
38 | 37, 20 | mulscld 27520 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (𝑝 ·s 𝐵) ∈ No
) |
39 | 38, 27 | addscld 27393 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → ((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑔)) ∈ No
) |
40 | 37, 26 | mulscld 27520 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (𝑝 ·s 𝑔) ∈ No
) |
41 | 39, 40 | subscld 27464 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑝 ·s 𝑔)) ∈ No
) |
42 | 41 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑝 ·s 𝑔)) ∈ No
) |
43 | | ssltss1 27219 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑀 <<s 𝑆 → 𝑀 ⊆ No
) |
44 | 6, 43 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝜑 → 𝑀 ⊆ No
) |
45 | 44 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → 𝑀 ⊆ No
) |
46 | | simprr 771 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → 𝑞 ∈ 𝑀) |
47 | 45, 46 | sseldd 3980 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → 𝑞 ∈ No
) |
48 | 47 | adantrl 714 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → 𝑞 ∈ No
) |
49 | 22, 48 | mulscld 27520 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (𝐴 ·s 𝑞) ∈ No
) |
50 | 38, 49 | addscld 27393 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → ((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) ∈ No
) |
51 | 37, 48 | mulscld 27520 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (𝑝 ·s 𝑞) ∈ No
) |
52 | 50, 51 | subscld 27464 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ∈ No
) |
53 | 52 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ∈ No
) |
54 | 18 | adantrr 715 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → 𝑓 ∈ No
) |
55 | 37 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → 𝑝 ∈ No
) |
56 | 25 | adantrr 715 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → 𝑔 ∈ No
) |
57 | 8 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → 𝐵 ∈ No
) |
58 | | simprrl 779 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞))) → 𝑓 ≤s 𝑝) |
59 | 58 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → 𝑓 ≤s 𝑝) |
60 | 8 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵))) → 𝐵 ∈ No
) |
61 | | ssltleft 27294 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝐵 ∈
No → ( L ‘𝐵) <<s {𝐵}) |
62 | 8, 61 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝜑 → ( L ‘𝐵) <<s {𝐵}) |
63 | 62 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵))) → ( L ‘𝐵) <<s {𝐵}) |
64 | | snidg 4657 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝐵 ∈
No → 𝐵 ∈
{𝐵}) |
65 | 8, 64 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝜑 → 𝐵 ∈ {𝐵}) |
66 | 65 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵))) → 𝐵 ∈ {𝐵}) |
67 | 63, 24, 66 | ssltsepcd 27224 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵))) → 𝑔 <s 𝐵) |
68 | 25, 60, 67 | sltled 27201 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵))) → 𝑔 ≤s 𝐵) |
69 | 68 | adantrr 715 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → 𝑔 ≤s 𝐵) |
70 | 54, 55, 56, 57, 59, 69 | slemuld 27523 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → ((𝑓 ·s 𝐵) -s (𝑓 ·s 𝑔)) ≤s ((𝑝 ·s 𝐵) -s (𝑝 ·s 𝑔))) |
71 | 21, 29 | subscld 27464 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → ((𝑓 ·s 𝐵) -s (𝑓 ·s 𝑔)) ∈ No
) |
72 | 38, 40 | subscld 27464 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → ((𝑝 ·s 𝐵) -s (𝑝 ·s 𝑔)) ∈ No
) |
73 | 71, 72, 27 | sleadd1d 27407 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (((𝑓 ·s 𝐵) -s (𝑓 ·s 𝑔)) ≤s ((𝑝 ·s 𝐵) -s (𝑝 ·s 𝑔)) ↔ (((𝑓 ·s 𝐵) -s (𝑓 ·s 𝑔)) +s (𝐴 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) -s (𝑝 ·s 𝑔)) +s (𝐴 ·s 𝑔)))) |
74 | 73 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → (((𝑓 ·s 𝐵) -s (𝑓 ·s 𝑔)) ≤s ((𝑝 ·s 𝐵) -s (𝑝 ·s 𝑔)) ↔ (((𝑓 ·s 𝐵) -s (𝑓 ·s 𝑔)) +s (𝐴 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) -s (𝑝 ·s 𝑔)) +s (𝐴 ·s 𝑔)))) |
75 | 70, 74 | mpbid 231 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → (((𝑓 ·s 𝐵) -s (𝑓 ·s 𝑔)) +s (𝐴 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) -s (𝑝 ·s 𝑔)) +s (𝐴 ·s 𝑔))) |
76 | 21, 27, 29 | addsubsd 27478 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) = (((𝑓 ·s 𝐵) -s (𝑓 ·s 𝑔)) +s (𝐴 ·s 𝑔))) |
77 | 76 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) = (((𝑓 ·s 𝐵) -s (𝑓 ·s 𝑔)) +s (𝐴 ·s 𝑔))) |
78 | 38, 27, 40 | addsubsd 27478 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑝 ·s 𝑔)) = (((𝑝 ·s 𝐵) -s (𝑝 ·s 𝑔)) +s (𝐴 ·s 𝑔))) |
79 | 78 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑝 ·s 𝑔)) = (((𝑝 ·s 𝐵) -s (𝑝 ·s 𝑔)) +s (𝐴 ·s 𝑔))) |
80 | 75, 77, 79 | 3brtr4d 5174 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑝 ·s 𝑔))) |
81 | 4 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → 𝐴 ∈ No
) |
82 | 48 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → 𝑞 ∈ No
) |
83 | 4 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → 𝐴 ∈ No
) |
84 | | scutcut 27231 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝐿 <<s 𝑅 → ((𝐿 |s 𝑅) ∈ No
∧ 𝐿 <<s {(𝐿 |s 𝑅)} ∧ {(𝐿 |s 𝑅)} <<s 𝑅)) |
85 | 2, 84 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝜑 → ((𝐿 |s 𝑅) ∈ No
∧ 𝐿 <<s {(𝐿 |s 𝑅)} ∧ {(𝐿 |s 𝑅)} <<s 𝑅)) |
86 | 85 | simp2d 1143 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝜑 → 𝐿 <<s {(𝐿 |s 𝑅)}) |
87 | 86 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → 𝐿 <<s {(𝐿 |s 𝑅)}) |
88 | | ovex 7427 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝐿 |s 𝑅) ∈ V |
89 | 88 | snid 4659 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝐿 |s 𝑅) ∈ {(𝐿 |s 𝑅)} |
90 | 1, 89 | eqeltrdi 2841 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝜑 → 𝐴 ∈ {(𝐿 |s 𝑅)}) |
91 | 90 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → 𝐴 ∈ {(𝐿 |s 𝑅)}) |
92 | 87, 35, 91 | ssltsepcd 27224 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → 𝑝 <s 𝐴) |
93 | 36, 83, 92 | sltled 27201 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → 𝑝 ≤s 𝐴) |
94 | 93 | adantrl 714 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → 𝑝 ≤s 𝐴) |
95 | 94 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → 𝑝 ≤s 𝐴) |
96 | | simprrr 780 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞))) → 𝑔 ≤s 𝑞) |
97 | 96 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → 𝑔 ≤s 𝑞) |
98 | 55, 81, 56, 82, 95, 97 | slemuld 27523 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → ((𝑝 ·s 𝑞) -s (𝑝 ·s 𝑔)) ≤s ((𝐴 ·s 𝑞) -s (𝐴 ·s 𝑔))) |
99 | 51, 49, 40, 27 | slesubsub3bd 27484 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (((𝑝 ·s 𝑞) -s (𝑝 ·s 𝑔)) ≤s ((𝐴 ·s 𝑞) -s (𝐴 ·s 𝑔)) ↔ ((𝐴 ·s 𝑔) -s (𝑝 ·s 𝑔)) ≤s ((𝐴 ·s 𝑞) -s (𝑝 ·s 𝑞)))) |
100 | 27, 40 | subscld 27464 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → ((𝐴 ·s 𝑔) -s (𝑝 ·s 𝑔)) ∈ No
) |
101 | 49, 51 | subscld 27464 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → ((𝐴 ·s 𝑞) -s (𝑝 ·s 𝑞)) ∈ No
) |
102 | 100, 101,
38 | sleadd2d 27408 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (((𝐴 ·s 𝑔) -s (𝑝 ·s 𝑔)) ≤s ((𝐴 ·s 𝑞) -s (𝑝 ·s 𝑞)) ↔ ((𝑝 ·s 𝐵) +s ((𝐴 ·s 𝑔) -s (𝑝 ·s 𝑔))) ≤s ((𝑝 ·s 𝐵) +s ((𝐴 ·s 𝑞) -s (𝑝 ·s 𝑞))))) |
103 | 99, 102 | bitrd 278 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (((𝑝 ·s 𝑞) -s (𝑝 ·s 𝑔)) ≤s ((𝐴 ·s 𝑞) -s (𝐴 ·s 𝑔)) ↔ ((𝑝 ·s 𝐵) +s ((𝐴 ·s 𝑔) -s (𝑝 ·s 𝑔))) ≤s ((𝑝 ·s 𝐵) +s ((𝐴 ·s 𝑞) -s (𝑝 ·s 𝑞))))) |
104 | 103 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → (((𝑝 ·s 𝑞) -s (𝑝 ·s 𝑔)) ≤s ((𝐴 ·s 𝑞) -s (𝐴 ·s 𝑔)) ↔ ((𝑝 ·s 𝐵) +s ((𝐴 ·s 𝑔) -s (𝑝 ·s 𝑔))) ≤s ((𝑝 ·s 𝐵) +s ((𝐴 ·s 𝑞) -s (𝑝 ·s 𝑞))))) |
105 | 98, 104 | mpbid 231 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → ((𝑝 ·s 𝐵) +s ((𝐴 ·s 𝑔) -s (𝑝 ·s 𝑔))) ≤s ((𝑝 ·s 𝐵) +s ((𝐴 ·s 𝑞) -s (𝑝 ·s 𝑞)))) |
106 | 38, 27, 40 | addsubsassd 27477 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑝 ·s 𝑔)) = ((𝑝 ·s 𝐵) +s ((𝐴 ·s 𝑔) -s (𝑝 ·s 𝑔)))) |
107 | 106 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑝 ·s 𝑔)) = ((𝑝 ·s 𝐵) +s ((𝐴 ·s 𝑔) -s (𝑝 ·s 𝑔)))) |
108 | 38, 49, 51 | addsubsassd 27477 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) = ((𝑝 ·s 𝐵) +s ((𝐴 ·s 𝑞) -s (𝑝 ·s 𝑞)))) |
109 | 108 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) = ((𝑝 ·s 𝐵) +s ((𝐴 ·s 𝑞) -s (𝑝 ·s 𝑞)))) |
110 | 105, 107,
109 | 3brtr4d 5174 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑝 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))) |
111 | 31, 42, 53, 80, 110 | sletrd 27194 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))) |
112 | 111 | anassrs 468 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵))) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞))) → (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))) |
113 | 112 | expr 457 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵))) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → ((𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞) → (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)))) |
114 | 113 | reximdvva 3205 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵))) → (∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞) → ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)))) |
115 | 114 | expcom 414 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) → (𝜑 → (∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞) → ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))))) |
116 | 115 | com23 86 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) → (∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞) → (𝜑 → ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))))) |
117 | 116 | imp 407 |
. . . . . . . . . . . . . . 15
⊢ (((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)) → (𝜑 → ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)))) |
118 | 15, 117 | sylan2br 595 |
. . . . . . . . . . . . . 14
⊢ (((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (∃𝑝 ∈ 𝐿 𝑓 ≤s 𝑝 ∧ ∃𝑞 ∈ 𝑀 𝑔 ≤s 𝑞)) → (𝜑 → ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)))) |
119 | 118 | an4s 658 |
. . . . . . . . . . . . 13
⊢ (((𝑓 ∈ ( L ‘𝐴) ∧ ∃𝑝 ∈ 𝐿 𝑓 ≤s 𝑝) ∧ (𝑔 ∈ ( L ‘𝐵) ∧ ∃𝑞 ∈ 𝑀 𝑔 ≤s 𝑞)) → (𝜑 → ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)))) |
120 | 119 | impcom 408 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ ∃𝑝 ∈ 𝐿 𝑓 ≤s 𝑝) ∧ (𝑔 ∈ ( L ‘𝐵) ∧ ∃𝑞 ∈ 𝑀 𝑔 ≤s 𝑞))) → ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))) |
121 | 120 | anassrs 468 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ ∃𝑝 ∈ 𝐿 𝑓 ≤s 𝑝)) ∧ (𝑔 ∈ ( L ‘𝐵) ∧ ∃𝑞 ∈ 𝑀 𝑔 ≤s 𝑞)) → ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))) |
122 | 121 | expr 457 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ ∃𝑝 ∈ 𝐿 𝑓 ≤s 𝑝)) ∧ 𝑔 ∈ ( L ‘𝐵)) → (∃𝑞 ∈ 𝑀 𝑔 ≤s 𝑞 → ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)))) |
123 | 122 | ralimdva 3167 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ ∃𝑝 ∈ 𝐿 𝑓 ≤s 𝑝)) → (∀𝑔 ∈ ( L ‘𝐵)∃𝑞 ∈ 𝑀 𝑔 ≤s 𝑞 → ∀𝑔 ∈ ( L ‘𝐵)∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)))) |
124 | 14, 123 | mpd 15 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ ∃𝑝 ∈ 𝐿 𝑓 ≤s 𝑝)) → ∀𝑔 ∈ ( L ‘𝐵)∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))) |
125 | 124 | expr 457 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑓 ∈ ( L ‘𝐴)) → (∃𝑝 ∈ 𝐿 𝑓 ≤s 𝑝 → ∀𝑔 ∈ ( L ‘𝐵)∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)))) |
126 | 125 | ralimdva 3167 |
. . . . . 6
⊢ (𝜑 → (∀𝑓 ∈ ( L ‘𝐴)∃𝑝 ∈ 𝐿 𝑓 ≤s 𝑝 → ∀𝑓 ∈ ( L ‘𝐴)∀𝑔 ∈ ( L ‘𝐵)∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)))) |
127 | 12, 126 | mpd 15 |
. . . . 5
⊢ (𝜑 → ∀𝑓 ∈ ( L ‘𝐴)∀𝑔 ∈ ( L ‘𝐵)∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))) |
128 | | eqeq1 2736 |
. . . . . . . . . 10
⊢ (𝑎 = 𝑧 → (𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ↔ 𝑧 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)))) |
129 | 128 | 2rexbidv 3219 |
. . . . . . . . 9
⊢ (𝑎 = 𝑧 → (∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ↔ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑧 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)))) |
130 | 129 | rexab 3687 |
. . . . . . . 8
⊢
(∃𝑧 ∈
{𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s 𝑧 ↔ ∃𝑧(∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑧 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ∧ (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s 𝑧)) |
131 | | r19.41vv 3224 |
. . . . . . . . 9
⊢
(∃𝑝 ∈
𝐿 ∃𝑞 ∈ 𝑀 (𝑧 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ∧ (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s 𝑧) ↔ (∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑧 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ∧ (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s 𝑧)) |
132 | 131 | exbii 1850 |
. . . . . . . 8
⊢
(∃𝑧∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (𝑧 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ∧ (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s 𝑧) ↔ ∃𝑧(∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑧 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ∧ (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s 𝑧)) |
133 | | rexcom4 3285 |
. . . . . . . . 9
⊢
(∃𝑝 ∈
𝐿 ∃𝑧∃𝑞 ∈ 𝑀 (𝑧 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ∧ (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s 𝑧) ↔ ∃𝑧∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (𝑧 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ∧ (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s 𝑧)) |
134 | | rexcom4 3285 |
. . . . . . . . . . 11
⊢
(∃𝑞 ∈
𝑀 ∃𝑧(𝑧 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ∧ (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s 𝑧) ↔ ∃𝑧∃𝑞 ∈ 𝑀 (𝑧 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ∧ (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s 𝑧)) |
135 | | ovex 7427 |
. . . . . . . . . . . . 13
⊢ (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ∈ V |
136 | | breq2 5146 |
. . . . . . . . . . . . 13
⊢ (𝑧 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) → ((((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s 𝑧 ↔ (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)))) |
137 | 135, 136 | ceqsexv 3523 |
. . . . . . . . . . . 12
⊢
(∃𝑧(𝑧 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ∧ (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s 𝑧) ↔ (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))) |
138 | 137 | rexbii 3094 |
. . . . . . . . . . 11
⊢
(∃𝑞 ∈
𝑀 ∃𝑧(𝑧 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ∧ (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s 𝑧) ↔ ∃𝑞 ∈ 𝑀 (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))) |
139 | 134, 138 | bitr3i 276 |
. . . . . . . . . 10
⊢
(∃𝑧∃𝑞 ∈ 𝑀 (𝑧 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ∧ (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s 𝑧) ↔ ∃𝑞 ∈ 𝑀 (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))) |
140 | 139 | rexbii 3094 |
. . . . . . . . 9
⊢
(∃𝑝 ∈
𝐿 ∃𝑧∃𝑞 ∈ 𝑀 (𝑧 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ∧ (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s 𝑧) ↔ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))) |
141 | 133, 140 | bitr3i 276 |
. . . . . . . 8
⊢
(∃𝑧∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (𝑧 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ∧ (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s 𝑧) ↔ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))) |
142 | 130, 132,
141 | 3bitr2i 298 |
. . . . . . 7
⊢
(∃𝑧 ∈
{𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s 𝑧 ↔ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))) |
143 | | ssun1 4169 |
. . . . . . . 8
⊢ {𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ⊆ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) |
144 | | ssrexv 4048 |
. . . . . . . 8
⊢ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ⊆ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) → (∃𝑧 ∈ {𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s 𝑧 → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})(((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s 𝑧)) |
145 | 143, 144 | ax-mp 5 |
. . . . . . 7
⊢
(∃𝑧 ∈
{𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s 𝑧 → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})(((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s 𝑧) |
146 | 142, 145 | sylbir 234 |
. . . . . 6
⊢
(∃𝑝 ∈
𝐿 ∃𝑞 ∈ 𝑀 (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})(((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s 𝑧) |
147 | 146 | 2ralimi 3123 |
. . . . 5
⊢
(∀𝑓 ∈ (
L ‘𝐴)∀𝑔 ∈ ( L ‘𝐵)∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) → ∀𝑓 ∈ ( L ‘𝐴)∀𝑔 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})(((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s 𝑧) |
148 | 127, 147 | syl 17 |
. . . 4
⊢ (𝜑 → ∀𝑓 ∈ ( L ‘𝐴)∀𝑔 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})(((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s 𝑧) |
149 | 2, 1 | cofcutr2d 27342 |
. . . . . 6
⊢ (𝜑 → ∀𝑖 ∈ ( R ‘𝐴)∃𝑟 ∈ 𝑅 𝑟 ≤s 𝑖) |
150 | 6, 5 | cofcutr2d 27342 |
. . . . . . . . . 10
⊢ (𝜑 → ∀𝑗 ∈ ( R ‘𝐵)∃𝑠 ∈ 𝑆 𝑠 ≤s 𝑗) |
151 | 150 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ ∃𝑟 ∈ 𝑅 𝑟 ≤s 𝑖)) → ∀𝑗 ∈ ( R ‘𝐵)∃𝑠 ∈ 𝑆 𝑠 ≤s 𝑗) |
152 | | reeanv 3226 |
. . . . . . . . . . . . . . 15
⊢
(∃𝑟 ∈
𝑅 ∃𝑠 ∈ 𝑆 (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗) ↔ (∃𝑟 ∈ 𝑅 𝑟 ≤s 𝑖 ∧ ∃𝑠 ∈ 𝑆 𝑠 ≤s 𝑗)) |
153 | | rightssno 27305 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ( R
‘𝐴) ⊆ No |
154 | | simprl 769 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵))) → 𝑖 ∈ ( R ‘𝐴)) |
155 | 153, 154 | sselid 3977 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵))) → 𝑖 ∈ No
) |
156 | 155 | adantrr 715 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → 𝑖 ∈ No
) |
157 | 8 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → 𝐵 ∈ No
) |
158 | 156, 157 | mulscld 27520 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (𝑖 ·s 𝐵) ∈ No
) |
159 | 4 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → 𝐴 ∈ No
) |
160 | | rightssno 27305 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ( R
‘𝐵) ⊆ No |
161 | | simprr 771 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵))) → 𝑗 ∈ ( R ‘𝐵)) |
162 | 160, 161 | sselid 3977 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵))) → 𝑗 ∈ No
) |
163 | 162 | adantrr 715 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → 𝑗 ∈ No
) |
164 | 159, 163 | mulscld 27520 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (𝐴 ·s 𝑗) ∈ No
) |
165 | 158, 164 | addscld 27393 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → ((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) ∈ No
) |
166 | 156, 163 | mulscld 27520 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (𝑖 ·s 𝑗) ∈ No
) |
167 | 165, 166 | subscld 27464 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ∈ No
) |
168 | 167 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ∈ No
) |
169 | | ssltss2 27220 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝐿 <<s 𝑅 → 𝑅 ⊆ No
) |
170 | 2, 169 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝜑 → 𝑅 ⊆ No
) |
171 | 170 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝑅 ⊆ No
) |
172 | | simprl 769 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝑟 ∈ 𝑅) |
173 | 171, 172 | sseldd 3980 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝑟 ∈ No
) |
174 | 173 | adantrl 714 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → 𝑟 ∈ No
) |
175 | 174, 157 | mulscld 27520 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (𝑟 ·s 𝐵) ∈ No
) |
176 | 175, 164 | addscld 27393 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → ((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑗)) ∈ No
) |
177 | 174, 163 | mulscld 27520 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (𝑟 ·s 𝑗) ∈ No
) |
178 | 176, 177 | subscld 27464 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑟 ·s 𝑗)) ∈ No
) |
179 | 178 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑟 ·s 𝑗)) ∈ No
) |
180 | | ssltss2 27220 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑀 <<s 𝑆 → 𝑆 ⊆ No
) |
181 | 6, 180 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝜑 → 𝑆 ⊆ No
) |
182 | 181 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝑆 ⊆ No
) |
183 | | simprr 771 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝑠 ∈ 𝑆) |
184 | 182, 183 | sseldd 3980 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝑠 ∈ No
) |
185 | 184 | adantrl 714 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → 𝑠 ∈ No
) |
186 | 159, 185 | mulscld 27520 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (𝐴 ·s 𝑠) ∈ No
) |
187 | 175, 186 | addscld 27393 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → ((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) ∈ No
) |
188 | 173, 184 | mulscld 27520 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (𝑟 ·s 𝑠) ∈ No
) |
189 | 188 | adantrl 714 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (𝑟 ·s 𝑠) ∈ No
) |
190 | 187, 189 | subscld 27464 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ∈ No
) |
191 | 190 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ∈ No
) |
192 | 174 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → 𝑟 ∈ No
) |
193 | 155 | adantrr 715 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → 𝑖 ∈ No
) |
194 | 8 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → 𝐵 ∈ No
) |
195 | 162 | adantrr 715 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → 𝑗 ∈ No
) |
196 | | simprrl 779 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗))) → 𝑟 ≤s 𝑖) |
197 | 196 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → 𝑟 ≤s 𝑖) |
198 | 8 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵))) → 𝐵 ∈ No
) |
199 | | ssltright 27295 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝐵 ∈
No → {𝐵}
<<s ( R ‘𝐵)) |
200 | 8, 199 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝜑 → {𝐵} <<s ( R ‘𝐵)) |
201 | 200 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵))) → {𝐵} <<s ( R ‘𝐵)) |
202 | 65 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵))) → 𝐵 ∈ {𝐵}) |
203 | 201, 202,
161 | ssltsepcd 27224 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵))) → 𝐵 <s 𝑗) |
204 | 198, 162,
203 | sltled 27201 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵))) → 𝐵 ≤s 𝑗) |
205 | 204 | adantrr 715 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → 𝐵 ≤s 𝑗) |
206 | 192, 193,
194, 195, 197, 205 | slemuld 27523 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → ((𝑟 ·s 𝑗) -s (𝑟 ·s 𝐵)) ≤s ((𝑖 ·s 𝑗) -s (𝑖 ·s 𝐵))) |
207 | 177, 175,
166, 158 | slesubsub2bd 27483 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (((𝑟 ·s 𝑗) -s (𝑟 ·s 𝐵)) ≤s ((𝑖 ·s 𝑗) -s (𝑖 ·s 𝐵)) ↔ ((𝑖 ·s 𝐵) -s (𝑖 ·s 𝑗)) ≤s ((𝑟 ·s 𝐵) -s (𝑟 ·s 𝑗)))) |
208 | 158, 166 | subscld 27464 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → ((𝑖 ·s 𝐵) -s (𝑖 ·s 𝑗)) ∈ No
) |
209 | 175, 177 | subscld 27464 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → ((𝑟 ·s 𝐵) -s (𝑟 ·s 𝑗)) ∈ No
) |
210 | 208, 209,
164 | sleadd1d 27407 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (((𝑖 ·s 𝐵) -s (𝑖 ·s 𝑗)) ≤s ((𝑟 ·s 𝐵) -s (𝑟 ·s 𝑗)) ↔ (((𝑖 ·s 𝐵) -s (𝑖 ·s 𝑗)) +s (𝐴 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) -s (𝑟 ·s 𝑗)) +s (𝐴 ·s 𝑗)))) |
211 | 207, 210 | bitrd 278 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (((𝑟 ·s 𝑗) -s (𝑟 ·s 𝐵)) ≤s ((𝑖 ·s 𝑗) -s (𝑖 ·s 𝐵)) ↔ (((𝑖 ·s 𝐵) -s (𝑖 ·s 𝑗)) +s (𝐴 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) -s (𝑟 ·s 𝑗)) +s (𝐴 ·s 𝑗)))) |
212 | 211 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → (((𝑟 ·s 𝑗) -s (𝑟 ·s 𝐵)) ≤s ((𝑖 ·s 𝑗) -s (𝑖 ·s 𝐵)) ↔ (((𝑖 ·s 𝐵) -s (𝑖 ·s 𝑗)) +s (𝐴 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) -s (𝑟 ·s 𝑗)) +s (𝐴 ·s 𝑗)))) |
213 | 206, 212 | mpbid 231 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → (((𝑖 ·s 𝐵) -s (𝑖 ·s 𝑗)) +s (𝐴 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) -s (𝑟 ·s 𝑗)) +s (𝐴 ·s 𝑗))) |
214 | 158, 164,
166 | addsubsd 27478 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) = (((𝑖 ·s 𝐵) -s (𝑖 ·s 𝑗)) +s (𝐴 ·s 𝑗))) |
215 | 214 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) = (((𝑖 ·s 𝐵) -s (𝑖 ·s 𝑗)) +s (𝐴 ·s 𝑗))) |
216 | 175, 164,
177 | addsubsd 27478 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑟 ·s 𝑗)) = (((𝑟 ·s 𝐵) -s (𝑟 ·s 𝑗)) +s (𝐴 ·s 𝑗))) |
217 | 216 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑟 ·s 𝑗)) = (((𝑟 ·s 𝐵) -s (𝑟 ·s 𝑗)) +s (𝐴 ·s 𝑗))) |
218 | 213, 215,
217 | 3brtr4d 5174 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑟 ·s 𝑗))) |
219 | 4 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → 𝐴 ∈ No
) |
220 | 185 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → 𝑠 ∈ No
) |
221 | 4 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝐴 ∈ No
) |
222 | 85 | simp3d 1144 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝜑 → {(𝐿 |s 𝑅)} <<s 𝑅) |
223 | 222 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → {(𝐿 |s 𝑅)} <<s 𝑅) |
224 | 90 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝐴 ∈ {(𝐿 |s 𝑅)}) |
225 | 223, 224,
172 | ssltsepcd 27224 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝐴 <s 𝑟) |
226 | 221, 173,
225 | sltled 27201 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝐴 ≤s 𝑟) |
227 | 226 | adantrl 714 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → 𝐴 ≤s 𝑟) |
228 | 227 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → 𝐴 ≤s 𝑟) |
229 | | simprrr 780 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗))) → 𝑠 ≤s 𝑗) |
230 | 229 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → 𝑠 ≤s 𝑗) |
231 | 219, 192,
220, 195, 228, 230 | slemuld 27523 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → ((𝐴 ·s 𝑗) -s (𝐴 ·s 𝑠)) ≤s ((𝑟 ·s 𝑗) -s (𝑟 ·s 𝑠))) |
232 | 164, 177,
186, 189 | slesubsubbd 27482 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (((𝐴 ·s 𝑗) -s (𝐴 ·s 𝑠)) ≤s ((𝑟 ·s 𝑗) -s (𝑟 ·s 𝑠)) ↔ ((𝐴 ·s 𝑗) -s (𝑟 ·s 𝑗)) ≤s ((𝐴 ·s 𝑠) -s (𝑟 ·s 𝑠)))) |
233 | 164, 177 | subscld 27464 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → ((𝐴 ·s 𝑗) -s (𝑟 ·s 𝑗)) ∈ No
) |
234 | 186, 189 | subscld 27464 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → ((𝐴 ·s 𝑠) -s (𝑟 ·s 𝑠)) ∈ No
) |
235 | 233, 234,
175 | sleadd2d 27408 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (((𝐴 ·s 𝑗) -s (𝑟 ·s 𝑗)) ≤s ((𝐴 ·s 𝑠) -s (𝑟 ·s 𝑠)) ↔ ((𝑟 ·s 𝐵) +s ((𝐴 ·s 𝑗) -s (𝑟 ·s 𝑗))) ≤s ((𝑟 ·s 𝐵) +s ((𝐴 ·s 𝑠) -s (𝑟 ·s 𝑠))))) |
236 | 232, 235 | bitrd 278 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (((𝐴 ·s 𝑗) -s (𝐴 ·s 𝑠)) ≤s ((𝑟 ·s 𝑗) -s (𝑟 ·s 𝑠)) ↔ ((𝑟 ·s 𝐵) +s ((𝐴 ·s 𝑗) -s (𝑟 ·s 𝑗))) ≤s ((𝑟 ·s 𝐵) +s ((𝐴 ·s 𝑠) -s (𝑟 ·s 𝑠))))) |
237 | 236 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → (((𝐴 ·s 𝑗) -s (𝐴 ·s 𝑠)) ≤s ((𝑟 ·s 𝑗) -s (𝑟 ·s 𝑠)) ↔ ((𝑟 ·s 𝐵) +s ((𝐴 ·s 𝑗) -s (𝑟 ·s 𝑗))) ≤s ((𝑟 ·s 𝐵) +s ((𝐴 ·s 𝑠) -s (𝑟 ·s 𝑠))))) |
238 | 231, 237 | mpbid 231 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → ((𝑟 ·s 𝐵) +s ((𝐴 ·s 𝑗) -s (𝑟 ·s 𝑗))) ≤s ((𝑟 ·s 𝐵) +s ((𝐴 ·s 𝑠) -s (𝑟 ·s 𝑠)))) |
239 | 175, 164,
177 | addsubsassd 27477 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑟 ·s 𝑗)) = ((𝑟 ·s 𝐵) +s ((𝐴 ·s 𝑗) -s (𝑟 ·s 𝑗)))) |
240 | 239 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑟 ·s 𝑗)) = ((𝑟 ·s 𝐵) +s ((𝐴 ·s 𝑗) -s (𝑟 ·s 𝑗)))) |
241 | 175, 186,
189 | addsubsassd 27477 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) = ((𝑟 ·s 𝐵) +s ((𝐴 ·s 𝑠) -s (𝑟 ·s 𝑠)))) |
242 | 241 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) = ((𝑟 ·s 𝐵) +s ((𝐴 ·s 𝑠) -s (𝑟 ·s 𝑠)))) |
243 | 238, 240,
242 | 3brtr4d 5174 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑟 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))) |
244 | 168, 179,
191, 218, 243 | sletrd 27194 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))) |
245 | 244 | anassrs 468 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗))) → (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))) |
246 | 245 | expr 457 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵))) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → ((𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗) → (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)))) |
247 | 246 | reximdvva 3205 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵))) → (∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗) → ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)))) |
248 | 247 | expcom 414 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) → (𝜑 → (∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗) → ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))))) |
249 | 248 | com23 86 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) → (∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗) → (𝜑 → ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))))) |
250 | 249 | imp 407 |
. . . . . . . . . . . . . . 15
⊢ (((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)) → (𝜑 → ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)))) |
251 | 152, 250 | sylan2br 595 |
. . . . . . . . . . . . . 14
⊢ (((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (∃𝑟 ∈ 𝑅 𝑟 ≤s 𝑖 ∧ ∃𝑠 ∈ 𝑆 𝑠 ≤s 𝑗)) → (𝜑 → ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)))) |
252 | 251 | an4s 658 |
. . . . . . . . . . . . 13
⊢ (((𝑖 ∈ ( R ‘𝐴) ∧ ∃𝑟 ∈ 𝑅 𝑟 ≤s 𝑖) ∧ (𝑗 ∈ ( R ‘𝐵) ∧ ∃𝑠 ∈ 𝑆 𝑠 ≤s 𝑗)) → (𝜑 → ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)))) |
253 | 252 | impcom 408 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ ∃𝑟 ∈ 𝑅 𝑟 ≤s 𝑖) ∧ (𝑗 ∈ ( R ‘𝐵) ∧ ∃𝑠 ∈ 𝑆 𝑠 ≤s 𝑗))) → ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))) |
254 | 253 | anassrs 468 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ ∃𝑟 ∈ 𝑅 𝑟 ≤s 𝑖)) ∧ (𝑗 ∈ ( R ‘𝐵) ∧ ∃𝑠 ∈ 𝑆 𝑠 ≤s 𝑗)) → ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))) |
255 | 254 | expr 457 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ ∃𝑟 ∈ 𝑅 𝑟 ≤s 𝑖)) ∧ 𝑗 ∈ ( R ‘𝐵)) → (∃𝑠 ∈ 𝑆 𝑠 ≤s 𝑗 → ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)))) |
256 | 255 | ralimdva 3167 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ ∃𝑟 ∈ 𝑅 𝑟 ≤s 𝑖)) → (∀𝑗 ∈ ( R ‘𝐵)∃𝑠 ∈ 𝑆 𝑠 ≤s 𝑗 → ∀𝑗 ∈ ( R ‘𝐵)∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)))) |
257 | 151, 256 | mpd 15 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ ∃𝑟 ∈ 𝑅 𝑟 ≤s 𝑖)) → ∀𝑗 ∈ ( R ‘𝐵)∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))) |
258 | 257 | expr 457 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ ( R ‘𝐴)) → (∃𝑟 ∈ 𝑅 𝑟 ≤s 𝑖 → ∀𝑗 ∈ ( R ‘𝐵)∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)))) |
259 | 258 | ralimdva 3167 |
. . . . . 6
⊢ (𝜑 → (∀𝑖 ∈ ( R ‘𝐴)∃𝑟 ∈ 𝑅 𝑟 ≤s 𝑖 → ∀𝑖 ∈ ( R ‘𝐴)∀𝑗 ∈ ( R ‘𝐵)∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)))) |
260 | 149, 259 | mpd 15 |
. . . . 5
⊢ (𝜑 → ∀𝑖 ∈ ( R ‘𝐴)∀𝑗 ∈ ( R ‘𝐵)∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))) |
261 | | eqeq1 2736 |
. . . . . . . . . 10
⊢ (𝑏 = 𝑧 → (𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ↔ 𝑧 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)))) |
262 | 261 | 2rexbidv 3219 |
. . . . . . . . 9
⊢ (𝑏 = 𝑧 → (∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ↔ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑧 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)))) |
263 | 262 | rexab 3687 |
. . . . . . . 8
⊢
(∃𝑧 ∈
{𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))} (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s 𝑧 ↔ ∃𝑧(∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑧 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ∧ (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s 𝑧)) |
264 | | r19.41vv 3224 |
. . . . . . . . 9
⊢
(∃𝑟 ∈
𝑅 ∃𝑠 ∈ 𝑆 (𝑧 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ∧ (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s 𝑧) ↔ (∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑧 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ∧ (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s 𝑧)) |
265 | 264 | exbii 1850 |
. . . . . . . 8
⊢
(∃𝑧∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (𝑧 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ∧ (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s 𝑧) ↔ ∃𝑧(∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑧 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ∧ (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s 𝑧)) |
266 | | rexcom4 3285 |
. . . . . . . . 9
⊢
(∃𝑟 ∈
𝑅 ∃𝑧∃𝑠 ∈ 𝑆 (𝑧 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ∧ (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s 𝑧) ↔ ∃𝑧∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (𝑧 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ∧ (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s 𝑧)) |
267 | | rexcom4 3285 |
. . . . . . . . . . 11
⊢
(∃𝑠 ∈
𝑆 ∃𝑧(𝑧 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ∧ (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s 𝑧) ↔ ∃𝑧∃𝑠 ∈ 𝑆 (𝑧 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ∧ (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s 𝑧)) |
268 | | ovex 7427 |
. . . . . . . . . . . . 13
⊢ (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ∈ V |
269 | | breq2 5146 |
. . . . . . . . . . . . 13
⊢ (𝑧 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) → ((((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s 𝑧 ↔ (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)))) |
270 | 268, 269 | ceqsexv 3523 |
. . . . . . . . . . . 12
⊢
(∃𝑧(𝑧 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ∧ (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s 𝑧) ↔ (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))) |
271 | 270 | rexbii 3094 |
. . . . . . . . . . 11
⊢
(∃𝑠 ∈
𝑆 ∃𝑧(𝑧 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ∧ (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s 𝑧) ↔ ∃𝑠 ∈ 𝑆 (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))) |
272 | 267, 271 | bitr3i 276 |
. . . . . . . . . 10
⊢
(∃𝑧∃𝑠 ∈ 𝑆 (𝑧 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ∧ (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s 𝑧) ↔ ∃𝑠 ∈ 𝑆 (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))) |
273 | 272 | rexbii 3094 |
. . . . . . . . 9
⊢
(∃𝑟 ∈
𝑅 ∃𝑧∃𝑠 ∈ 𝑆 (𝑧 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ∧ (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s 𝑧) ↔ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))) |
274 | 266, 273 | bitr3i 276 |
. . . . . . . 8
⊢
(∃𝑧∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (𝑧 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ∧ (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s 𝑧) ↔ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))) |
275 | 263, 265,
274 | 3bitr2i 298 |
. . . . . . 7
⊢
(∃𝑧 ∈
{𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))} (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s 𝑧 ↔ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))) |
276 | | ssun2 4170 |
. . . . . . . 8
⊢ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))} ⊆ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) |
277 | | ssrexv 4048 |
. . . . . . . 8
⊢ ({𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))} ⊆ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) → (∃𝑧 ∈ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))} (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s 𝑧 → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})(((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s 𝑧)) |
278 | 276, 277 | ax-mp 5 |
. . . . . . 7
⊢
(∃𝑧 ∈
{𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))} (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s 𝑧 → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})(((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s 𝑧) |
279 | 275, 278 | sylbir 234 |
. . . . . 6
⊢
(∃𝑟 ∈
𝑅 ∃𝑠 ∈ 𝑆 (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})(((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s 𝑧) |
280 | 279 | 2ralimi 3123 |
. . . . 5
⊢
(∀𝑖 ∈ (
R ‘𝐴)∀𝑗 ∈ ( R ‘𝐵)∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) → ∀𝑖 ∈ ( R ‘𝐴)∀𝑗 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})(((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s 𝑧) |
281 | 260, 280 | syl 17 |
. . . 4
⊢ (𝜑 → ∀𝑖 ∈ ( R ‘𝐴)∀𝑗 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})(((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s 𝑧) |
282 | | ralunb 4188 |
. . . . 5
⊢
(∀𝑥𝑂 ∈ ({𝑒 ∣ ∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑒 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔))} ∪ {ℎ ∣ ∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵)ℎ = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗))})∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧 ↔ (∀𝑥𝑂 ∈ {𝑒 ∣ ∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑒 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔))}∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧 ∧ ∀𝑥𝑂 ∈ {ℎ ∣ ∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵)ℎ = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗))}∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧)) |
283 | | eqeq1 2736 |
. . . . . . . . 9
⊢ (𝑒 = 𝑥𝑂 → (𝑒 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ↔ 𝑥𝑂 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)))) |
284 | 283 | 2rexbidv 3219 |
. . . . . . . 8
⊢ (𝑒 = 𝑥𝑂 → (∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑒 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ↔ ∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑥𝑂 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)))) |
285 | 284 | ralab 3684 |
. . . . . . 7
⊢
(∀𝑥𝑂 ∈ {𝑒 ∣ ∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑒 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔))}∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧 ↔ ∀𝑥𝑂(∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑥𝑂 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧)) |
286 | | r19.23v 3182 |
. . . . . . . . . 10
⊢
(∀𝑔 ∈ (
L ‘𝐵)(𝑥𝑂 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧) ↔ (∃𝑔 ∈ ( L ‘𝐵)𝑥𝑂 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧)) |
287 | 286 | ralbii 3093 |
. . . . . . . . 9
⊢
(∀𝑓 ∈ (
L ‘𝐴)∀𝑔 ∈ ( L ‘𝐵)(𝑥𝑂 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧) ↔ ∀𝑓 ∈ ( L ‘𝐴)(∃𝑔 ∈ ( L ‘𝐵)𝑥𝑂 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧)) |
288 | | r19.23v 3182 |
. . . . . . . . 9
⊢
(∀𝑓 ∈ (
L ‘𝐴)(∃𝑔 ∈ ( L ‘𝐵)𝑥𝑂 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧) ↔ (∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑥𝑂 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧)) |
289 | 287, 288 | bitri 274 |
. . . . . . . 8
⊢
(∀𝑓 ∈ (
L ‘𝐴)∀𝑔 ∈ ( L ‘𝐵)(𝑥𝑂 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧) ↔ (∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑥𝑂 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧)) |
290 | 289 | albii 1821 |
. . . . . . 7
⊢
(∀𝑥𝑂∀𝑓 ∈ ( L ‘𝐴)∀𝑔 ∈ ( L ‘𝐵)(𝑥𝑂 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧) ↔ ∀𝑥𝑂(∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑥𝑂 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧)) |
291 | | ralcom4 3283 |
. . . . . . . 8
⊢
(∀𝑓 ∈ (
L ‘𝐴)∀𝑥𝑂∀𝑔 ∈ ( L ‘𝐵)(𝑥𝑂 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧) ↔ ∀𝑥𝑂∀𝑓 ∈ ( L ‘𝐴)∀𝑔 ∈ ( L ‘𝐵)(𝑥𝑂 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧)) |
292 | | ralcom4 3283 |
. . . . . . . . . 10
⊢
(∀𝑔 ∈ (
L ‘𝐵)∀𝑥𝑂(𝑥𝑂 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧) ↔ ∀𝑥𝑂∀𝑔 ∈ ( L ‘𝐵)(𝑥𝑂 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧)) |
293 | | ovex 7427 |
. . . . . . . . . . . 12
⊢ (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ∈ V |
294 | | breq1 5145 |
. . . . . . . . . . . . 13
⊢ (𝑥𝑂 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) → (𝑥𝑂 ≤s 𝑧 ↔ (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s 𝑧)) |
295 | 294 | rexbidv 3178 |
. . . . . . . . . . . 12
⊢ (𝑥𝑂 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) → (∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧 ↔ ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})(((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s 𝑧)) |
296 | 293, 295 | ceqsalv 3510 |
. . . . . . . . . . 11
⊢
(∀𝑥𝑂(𝑥𝑂 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧) ↔ ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})(((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s 𝑧) |
297 | 296 | ralbii 3093 |
. . . . . . . . . 10
⊢
(∀𝑔 ∈ (
L ‘𝐵)∀𝑥𝑂(𝑥𝑂 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧) ↔ ∀𝑔 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})(((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s 𝑧) |
298 | 292, 297 | bitr3i 276 |
. . . . . . . . 9
⊢
(∀𝑥𝑂∀𝑔 ∈ ( L ‘𝐵)(𝑥𝑂 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧) ↔ ∀𝑔 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})(((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s 𝑧) |
299 | 298 | ralbii 3093 |
. . . . . . . 8
⊢
(∀𝑓 ∈ (
L ‘𝐴)∀𝑥𝑂∀𝑔 ∈ ( L ‘𝐵)(𝑥𝑂 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧) ↔ ∀𝑓 ∈ ( L ‘𝐴)∀𝑔 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})(((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s 𝑧) |
300 | 291, 299 | bitr3i 276 |
. . . . . . 7
⊢
(∀𝑥𝑂∀𝑓 ∈ ( L ‘𝐴)∀𝑔 ∈ ( L ‘𝐵)(𝑥𝑂 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧) ↔ ∀𝑓 ∈ ( L ‘𝐴)∀𝑔 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})(((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s 𝑧) |
301 | 285, 290,
300 | 3bitr2i 298 |
. . . . . 6
⊢
(∀𝑥𝑂 ∈ {𝑒 ∣ ∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑒 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔))}∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧 ↔ ∀𝑓 ∈ ( L ‘𝐴)∀𝑔 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})(((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s 𝑧) |
302 | | eqeq1 2736 |
. . . . . . . . 9
⊢ (ℎ = 𝑥𝑂 → (ℎ = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ↔ 𝑥𝑂 = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)))) |
303 | 302 | 2rexbidv 3219 |
. . . . . . . 8
⊢ (ℎ = 𝑥𝑂 → (∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵)ℎ = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ↔ ∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵)𝑥𝑂 = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)))) |
304 | 303 | ralab 3684 |
. . . . . . 7
⊢
(∀𝑥𝑂 ∈ {ℎ ∣ ∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵)ℎ = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗))}∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧 ↔ ∀𝑥𝑂(∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵)𝑥𝑂 = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧)) |
305 | | r19.23v 3182 |
. . . . . . . . . 10
⊢
(∀𝑗 ∈ (
R ‘𝐵)(𝑥𝑂 = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧) ↔ (∃𝑗 ∈ ( R ‘𝐵)𝑥𝑂 = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧)) |
306 | 305 | ralbii 3093 |
. . . . . . . . 9
⊢
(∀𝑖 ∈ (
R ‘𝐴)∀𝑗 ∈ ( R ‘𝐵)(𝑥𝑂 = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧) ↔ ∀𝑖 ∈ ( R ‘𝐴)(∃𝑗 ∈ ( R ‘𝐵)𝑥𝑂 = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧)) |
307 | | r19.23v 3182 |
. . . . . . . . 9
⊢
(∀𝑖 ∈ (
R ‘𝐴)(∃𝑗 ∈ ( R ‘𝐵)𝑥𝑂 = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧) ↔ (∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵)𝑥𝑂 = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧)) |
308 | 306, 307 | bitri 274 |
. . . . . . . 8
⊢
(∀𝑖 ∈ (
R ‘𝐴)∀𝑗 ∈ ( R ‘𝐵)(𝑥𝑂 = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧) ↔ (∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵)𝑥𝑂 = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧)) |
309 | 308 | albii 1821 |
. . . . . . 7
⊢
(∀𝑥𝑂∀𝑖 ∈ ( R ‘𝐴)∀𝑗 ∈ ( R ‘𝐵)(𝑥𝑂 = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧) ↔ ∀𝑥𝑂(∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵)𝑥𝑂 = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧)) |
310 | | ralcom4 3283 |
. . . . . . . 8
⊢
(∀𝑖 ∈ (
R ‘𝐴)∀𝑥𝑂∀𝑗 ∈ ( R ‘𝐵)(𝑥𝑂 = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧) ↔ ∀𝑥𝑂∀𝑖 ∈ ( R ‘𝐴)∀𝑗 ∈ ( R ‘𝐵)(𝑥𝑂 = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧)) |
311 | | ralcom4 3283 |
. . . . . . . . . 10
⊢
(∀𝑗 ∈ (
R ‘𝐵)∀𝑥𝑂(𝑥𝑂 = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧) ↔ ∀𝑥𝑂∀𝑗 ∈ ( R ‘𝐵)(𝑥𝑂 = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧)) |
312 | | ovex 7427 |
. . . . . . . . . . . 12
⊢ (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ∈ V |
313 | | breq1 5145 |
. . . . . . . . . . . . 13
⊢ (𝑥𝑂 = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) → (𝑥𝑂 ≤s 𝑧 ↔ (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s 𝑧)) |
314 | 313 | rexbidv 3178 |
. . . . . . . . . . . 12
⊢ (𝑥𝑂 = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) → (∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧 ↔ ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})(((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s 𝑧)) |
315 | 312, 314 | ceqsalv 3510 |
. . . . . . . . . . 11
⊢
(∀𝑥𝑂(𝑥𝑂 = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧) ↔ ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})(((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s 𝑧) |
316 | 315 | ralbii 3093 |
. . . . . . . . . 10
⊢
(∀𝑗 ∈ (
R ‘𝐵)∀𝑥𝑂(𝑥𝑂 = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧) ↔ ∀𝑗 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})(((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s 𝑧) |
317 | 311, 316 | bitr3i 276 |
. . . . . . . . 9
⊢
(∀𝑥𝑂∀𝑗 ∈ ( R ‘𝐵)(𝑥𝑂 = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧) ↔ ∀𝑗 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})(((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s 𝑧) |
318 | 317 | ralbii 3093 |
. . . . . . . 8
⊢
(∀𝑖 ∈ (
R ‘𝐴)∀𝑥𝑂∀𝑗 ∈ ( R ‘𝐵)(𝑥𝑂 = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧) ↔ ∀𝑖 ∈ ( R ‘𝐴)∀𝑗 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})(((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s 𝑧) |
319 | 310, 318 | bitr3i 276 |
. . . . . . 7
⊢
(∀𝑥𝑂∀𝑖 ∈ ( R ‘𝐴)∀𝑗 ∈ ( R ‘𝐵)(𝑥𝑂 = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧) ↔ ∀𝑖 ∈ ( R ‘𝐴)∀𝑗 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})(((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s 𝑧) |
320 | 304, 309,
319 | 3bitr2i 298 |
. . . . . 6
⊢
(∀𝑥𝑂 ∈ {ℎ ∣ ∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵)ℎ = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗))}∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧 ↔ ∀𝑖 ∈ ( R ‘𝐴)∀𝑗 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})(((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s 𝑧) |
321 | 301, 320 | anbi12i 627 |
. . . . 5
⊢
((∀𝑥𝑂 ∈ {𝑒 ∣ ∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑒 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔))}∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧 ∧ ∀𝑥𝑂 ∈ {ℎ ∣ ∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵)ℎ = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗))}∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧) ↔ (∀𝑓 ∈ ( L ‘𝐴)∀𝑔 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})(((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s 𝑧 ∧ ∀𝑖 ∈ ( R ‘𝐴)∀𝑗 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})(((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s 𝑧)) |
322 | 282, 321 | bitri 274 |
. . . 4
⊢
(∀𝑥𝑂 ∈ ({𝑒 ∣ ∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑒 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔))} ∪ {ℎ ∣ ∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵)ℎ = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗))})∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧 ↔ (∀𝑓 ∈ ( L ‘𝐴)∀𝑔 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})(((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s 𝑧 ∧ ∀𝑖 ∈ ( R ‘𝐴)∀𝑗 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})(((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s 𝑧)) |
323 | 148, 281,
322 | sylanbrc 583 |
. . 3
⊢ (𝜑 → ∀𝑥𝑂 ∈ ({𝑒 ∣ ∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑒 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔))} ∪ {ℎ ∣ ∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵)ℎ = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗))})∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧) |
324 | 2, 1 | cofcutr1d 27341 |
. . . . . 6
⊢ (𝜑 → ∀𝑙 ∈ ( L ‘𝐴)∃𝑡 ∈ 𝐿 𝑙 ≤s 𝑡) |
325 | 6, 5 | cofcutr2d 27342 |
. . . . . . . . . 10
⊢ (𝜑 → ∀𝑚 ∈ ( R ‘𝐵)∃𝑢 ∈ 𝑆 𝑢 ≤s 𝑚) |
326 | 325 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ ∃𝑡 ∈ 𝐿 𝑙 ≤s 𝑡)) → ∀𝑚 ∈ ( R ‘𝐵)∃𝑢 ∈ 𝑆 𝑢 ≤s 𝑚) |
327 | | reeanv 3226 |
. . . . . . . . . . . . . . 15
⊢
(∃𝑡 ∈
𝐿 ∃𝑢 ∈ 𝑆 (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚) ↔ (∃𝑡 ∈ 𝐿 𝑙 ≤s 𝑡 ∧ ∃𝑢 ∈ 𝑆 𝑢 ≤s 𝑚)) |
328 | 33 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝐿 ⊆ No
) |
329 | | simprl 769 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝑡 ∈ 𝐿) |
330 | 328, 329 | sseldd 3980 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝑡 ∈ No
) |
331 | 330 | adantrl 714 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → 𝑡 ∈ No
) |
332 | 8 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → 𝐵 ∈ No
) |
333 | 331, 332 | mulscld 27520 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (𝑡 ·s 𝐵) ∈ No
) |
334 | 4 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → 𝐴 ∈ No
) |
335 | 181 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝑆 ⊆ No
) |
336 | | simprr 771 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝑢 ∈ 𝑆) |
337 | 335, 336 | sseldd 3980 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝑢 ∈ No
) |
338 | 337 | adantrl 714 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → 𝑢 ∈ No
) |
339 | 334, 338 | mulscld 27520 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (𝐴 ·s 𝑢) ∈ No
) |
340 | 333, 339 | addscld 27393 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → ((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) ∈ No
) |
341 | 331, 338 | mulscld 27520 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (𝑡 ·s 𝑢) ∈ No
) |
342 | 340, 341 | subscld 27464 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ∈ No
) |
343 | 342 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ∈ No
) |
344 | | simprl 769 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → 𝑙 ∈ ( L ‘𝐴)) |
345 | 16, 344 | sselid 3977 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → 𝑙 ∈ No
) |
346 | 8 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → 𝐵 ∈ No
) |
347 | 345, 346 | mulscld 27520 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → (𝑙 ·s 𝐵) ∈ No
) |
348 | 347 | adantrr 715 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (𝑙 ·s 𝐵) ∈ No
) |
349 | 348, 339 | addscld 27393 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → ((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑢)) ∈ No
) |
350 | 345 | adantrr 715 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → 𝑙 ∈ No
) |
351 | 350, 338 | mulscld 27520 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (𝑙 ·s 𝑢) ∈ No
) |
352 | 349, 351 | subscld 27464 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑙 ·s 𝑢)) ∈ No
) |
353 | 352 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑙 ·s 𝑢)) ∈ No
) |
354 | 4 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → 𝐴 ∈ No
) |
355 | | simprr 771 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → 𝑚 ∈ ( R ‘𝐵)) |
356 | 160, 355 | sselid 3977 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → 𝑚 ∈ No
) |
357 | 354, 356 | mulscld 27520 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → (𝐴 ·s 𝑚) ∈ No
) |
358 | 357 | adantrr 715 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (𝐴 ·s 𝑚) ∈ No
) |
359 | 348, 358 | addscld 27393 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → ((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) ∈ No
) |
360 | 345, 356 | mulscld 27520 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → (𝑙 ·s 𝑚) ∈ No
) |
361 | 360 | adantrr 715 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (𝑙 ·s 𝑚) ∈ No
) |
362 | 359, 361 | subscld 27464 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) ∈ No
) |
363 | 362 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) ∈ No
) |
364 | 345 | adantrr 715 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → 𝑙 ∈ No
) |
365 | 331 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → 𝑡 ∈ No
) |
366 | 8 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → 𝐵 ∈ No
) |
367 | 338 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → 𝑢 ∈ No
) |
368 | | simprrl 779 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚))) → 𝑙 ≤s 𝑡) |
369 | 368 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → 𝑙 ≤s 𝑡) |
370 | 8 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝐵 ∈ No
) |
371 | | scutcut 27231 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑀 <<s 𝑆 → ((𝑀 |s 𝑆) ∈ No
∧ 𝑀 <<s {(𝑀 |s 𝑆)} ∧ {(𝑀 |s 𝑆)} <<s 𝑆)) |
372 | 6, 371 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝜑 → ((𝑀 |s 𝑆) ∈ No
∧ 𝑀 <<s {(𝑀 |s 𝑆)} ∧ {(𝑀 |s 𝑆)} <<s 𝑆)) |
373 | 372 | simp3d 1144 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝜑 → {(𝑀 |s 𝑆)} <<s 𝑆) |
374 | 373 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → {(𝑀 |s 𝑆)} <<s 𝑆) |
375 | | ovex 7427 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑀 |s 𝑆) ∈ V |
376 | 375 | snid 4659 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑀 |s 𝑆) ∈ {(𝑀 |s 𝑆)} |
377 | 5, 376 | eqeltrdi 2841 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝜑 → 𝐵 ∈ {(𝑀 |s 𝑆)}) |
378 | 377 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝐵 ∈ {(𝑀 |s 𝑆)}) |
379 | 374, 378,
336 | ssltsepcd 27224 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝐵 <s 𝑢) |
380 | 370, 337,
379 | sltled 27201 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝐵 ≤s 𝑢) |
381 | 380 | adantrl 714 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → 𝐵 ≤s 𝑢) |
382 | 381 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → 𝐵 ≤s 𝑢) |
383 | 364, 365,
366, 367, 369, 382 | slemuld 27523 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → ((𝑙 ·s 𝑢) -s (𝑙 ·s 𝐵)) ≤s ((𝑡 ·s 𝑢) -s (𝑡 ·s 𝐵))) |
384 | 351, 348,
341, 333 | slesubsub2bd 27483 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (((𝑙 ·s 𝑢) -s (𝑙 ·s 𝐵)) ≤s ((𝑡 ·s 𝑢) -s (𝑡 ·s 𝐵)) ↔ ((𝑡 ·s 𝐵) -s (𝑡 ·s 𝑢)) ≤s ((𝑙 ·s 𝐵) -s (𝑙 ·s 𝑢)))) |
385 | 333, 341 | subscld 27464 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → ((𝑡 ·s 𝐵) -s (𝑡 ·s 𝑢)) ∈ No
) |
386 | 348, 351 | subscld 27464 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → ((𝑙 ·s 𝐵) -s (𝑙 ·s 𝑢)) ∈ No
) |
387 | 385, 386,
339 | sleadd1d 27407 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (((𝑡 ·s 𝐵) -s (𝑡 ·s 𝑢)) ≤s ((𝑙 ·s 𝐵) -s (𝑙 ·s 𝑢)) ↔ (((𝑡 ·s 𝐵) -s (𝑡 ·s 𝑢)) +s (𝐴 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) -s (𝑙 ·s 𝑢)) +s (𝐴 ·s 𝑢)))) |
388 | 384, 387 | bitrd 278 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (((𝑙 ·s 𝑢) -s (𝑙 ·s 𝐵)) ≤s ((𝑡 ·s 𝑢) -s (𝑡 ·s 𝐵)) ↔ (((𝑡 ·s 𝐵) -s (𝑡 ·s 𝑢)) +s (𝐴 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) -s (𝑙 ·s 𝑢)) +s (𝐴 ·s 𝑢)))) |
389 | 388 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → (((𝑙 ·s 𝑢) -s (𝑙 ·s 𝐵)) ≤s ((𝑡 ·s 𝑢) -s (𝑡 ·s 𝐵)) ↔ (((𝑡 ·s 𝐵) -s (𝑡 ·s 𝑢)) +s (𝐴 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) -s (𝑙 ·s 𝑢)) +s (𝐴 ·s 𝑢)))) |
390 | 383, 389 | mpbid 231 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → (((𝑡 ·s 𝐵) -s (𝑡 ·s 𝑢)) +s (𝐴 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) -s (𝑙 ·s 𝑢)) +s (𝐴 ·s 𝑢))) |
391 | 333, 339,
341 | addsubsd 27478 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) = (((𝑡 ·s 𝐵) -s (𝑡 ·s 𝑢)) +s (𝐴 ·s 𝑢))) |
392 | 391 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) = (((𝑡 ·s 𝐵) -s (𝑡 ·s 𝑢)) +s (𝐴 ·s 𝑢))) |
393 | 348, 339,
351 | addsubsd 27478 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑙 ·s 𝑢)) = (((𝑙 ·s 𝐵) -s (𝑙 ·s 𝑢)) +s (𝐴 ·s 𝑢))) |
394 | 393 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑙 ·s 𝑢)) = (((𝑙 ·s 𝐵) -s (𝑙 ·s 𝑢)) +s (𝐴 ·s 𝑢))) |
395 | 390, 392,
394 | 3brtr4d 5174 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑙 ·s 𝑢))) |
396 | 4 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → 𝐴 ∈ No
) |
397 | 356 | adantrr 715 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → 𝑚 ∈ No
) |
398 | | ssltleft 27294 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝐴 ∈
No → ( L ‘𝐴) <<s {𝐴}) |
399 | 4, 398 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝜑 → ( L ‘𝐴) <<s {𝐴}) |
400 | 399 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → ( L ‘𝐴) <<s {𝐴}) |
401 | | snidg 4657 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝐴 ∈
No → 𝐴 ∈
{𝐴}) |
402 | 4, 401 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝜑 → 𝐴 ∈ {𝐴}) |
403 | 402 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → 𝐴 ∈ {𝐴}) |
404 | 400, 344,
403 | ssltsepcd 27224 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → 𝑙 <s 𝐴) |
405 | 345, 354,
404 | sltled 27201 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → 𝑙 ≤s 𝐴) |
406 | 405 | adantrr 715 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → 𝑙 ≤s 𝐴) |
407 | | simprrr 780 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚))) → 𝑢 ≤s 𝑚) |
408 | 407 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → 𝑢 ≤s 𝑚) |
409 | 364, 396,
367, 397, 406, 408 | slemuld 27523 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → ((𝑙 ·s 𝑚) -s (𝑙 ·s 𝑢)) ≤s ((𝐴 ·s 𝑚) -s (𝐴 ·s 𝑢))) |
410 | 361, 358,
351, 339 | slesubsub3bd 27484 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (((𝑙 ·s 𝑚) -s (𝑙 ·s 𝑢)) ≤s ((𝐴 ·s 𝑚) -s (𝐴 ·s 𝑢)) ↔ ((𝐴 ·s 𝑢) -s (𝑙 ·s 𝑢)) ≤s ((𝐴 ·s 𝑚) -s (𝑙 ·s 𝑚)))) |
411 | 339, 351 | subscld 27464 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → ((𝐴 ·s 𝑢) -s (𝑙 ·s 𝑢)) ∈ No
) |
412 | 358, 361 | subscld 27464 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → ((𝐴 ·s 𝑚) -s (𝑙 ·s 𝑚)) ∈ No
) |
413 | 411, 412,
348 | sleadd2d 27408 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (((𝐴 ·s 𝑢) -s (𝑙 ·s 𝑢)) ≤s ((𝐴 ·s 𝑚) -s (𝑙 ·s 𝑚)) ↔ ((𝑙 ·s 𝐵) +s ((𝐴 ·s 𝑢) -s (𝑙 ·s 𝑢))) ≤s ((𝑙 ·s 𝐵) +s ((𝐴 ·s 𝑚) -s (𝑙 ·s 𝑚))))) |
414 | 410, 413 | bitrd 278 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (((𝑙 ·s 𝑚) -s (𝑙 ·s 𝑢)) ≤s ((𝐴 ·s 𝑚) -s (𝐴 ·s 𝑢)) ↔ ((𝑙 ·s 𝐵) +s ((𝐴 ·s 𝑢) -s (𝑙 ·s 𝑢))) ≤s ((𝑙 ·s 𝐵) +s ((𝐴 ·s 𝑚) -s (𝑙 ·s 𝑚))))) |
415 | 414 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → (((𝑙 ·s 𝑚) -s (𝑙 ·s 𝑢)) ≤s ((𝐴 ·s 𝑚) -s (𝐴 ·s 𝑢)) ↔ ((𝑙 ·s 𝐵) +s ((𝐴 ·s 𝑢) -s (𝑙 ·s 𝑢))) ≤s ((𝑙 ·s 𝐵) +s ((𝐴 ·s 𝑚) -s (𝑙 ·s 𝑚))))) |
416 | 409, 415 | mpbid 231 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → ((𝑙 ·s 𝐵) +s ((𝐴 ·s 𝑢) -s (𝑙 ·s 𝑢))) ≤s ((𝑙 ·s 𝐵) +s ((𝐴 ·s 𝑚) -s (𝑙 ·s 𝑚)))) |
417 | 348, 339,
351 | addsubsassd 27477 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑙 ·s 𝑢)) = ((𝑙 ·s 𝐵) +s ((𝐴 ·s 𝑢) -s (𝑙 ·s 𝑢)))) |
418 | 417 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑙 ·s 𝑢)) = ((𝑙 ·s 𝐵) +s ((𝐴 ·s 𝑢) -s (𝑙 ·s 𝑢)))) |
419 | 348, 358,
361 | addsubsassd 27477 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) = ((𝑙 ·s 𝐵) +s ((𝐴 ·s 𝑚) -s (𝑙 ·s 𝑚)))) |
420 | 419 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) = ((𝑙 ·s 𝐵) +s ((𝐴 ·s 𝑚) -s (𝑙 ·s 𝑚)))) |
421 | 416, 418,
420 | 3brtr4d 5174 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑙 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))) |
422 | 343, 353,
363, 395, 421 | sletrd 27194 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))) |
423 | 422 | anassrs 468 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚))) → (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))) |
424 | 423 | expr 457 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → ((𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚) → (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)))) |
425 | 424 | reximdvva 3205 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → (∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚) → ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)))) |
426 | 425 | expcom 414 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) → (𝜑 → (∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚) → ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))))) |
427 | 426 | com23 86 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) → (∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚) → (𝜑 → ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))))) |
428 | 427 | imp 407 |
. . . . . . . . . . . . . . 15
⊢ (((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)) → (𝜑 → ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)))) |
429 | 327, 428 | sylan2br 595 |
. . . . . . . . . . . . . 14
⊢ (((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (∃𝑡 ∈ 𝐿 𝑙 ≤s 𝑡 ∧ ∃𝑢 ∈ 𝑆 𝑢 ≤s 𝑚)) → (𝜑 → ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)))) |
430 | 429 | an4s 658 |
. . . . . . . . . . . . 13
⊢ (((𝑙 ∈ ( L ‘𝐴) ∧ ∃𝑡 ∈ 𝐿 𝑙 ≤s 𝑡) ∧ (𝑚 ∈ ( R ‘𝐵) ∧ ∃𝑢 ∈ 𝑆 𝑢 ≤s 𝑚)) → (𝜑 → ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)))) |
431 | 430 | impcom 408 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ ∃𝑡 ∈ 𝐿 𝑙 ≤s 𝑡) ∧ (𝑚 ∈ ( R ‘𝐵) ∧ ∃𝑢 ∈ 𝑆 𝑢 ≤s 𝑚))) → ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))) |
432 | 431 | anassrs 468 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ ∃𝑡 ∈ 𝐿 𝑙 ≤s 𝑡)) ∧ (𝑚 ∈ ( R ‘𝐵) ∧ ∃𝑢 ∈ 𝑆 𝑢 ≤s 𝑚)) → ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))) |
433 | 432 | expr 457 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ ∃𝑡 ∈ 𝐿 𝑙 ≤s 𝑡)) ∧ 𝑚 ∈ ( R ‘𝐵)) → (∃𝑢 ∈ 𝑆 𝑢 ≤s 𝑚 → ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)))) |
434 | 433 | ralimdva 3167 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ ∃𝑡 ∈ 𝐿 𝑙 ≤s 𝑡)) → (∀𝑚 ∈ ( R ‘𝐵)∃𝑢 ∈ 𝑆 𝑢 ≤s 𝑚 → ∀𝑚 ∈ ( R ‘𝐵)∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)))) |
435 | 326, 434 | mpd 15 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ ∃𝑡 ∈ 𝐿 𝑙 ≤s 𝑡)) → ∀𝑚 ∈ ( R ‘𝐵)∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))) |
436 | 435 | expr 457 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝐴)) → (∃𝑡 ∈ 𝐿 𝑙 ≤s 𝑡 → ∀𝑚 ∈ ( R ‘𝐵)∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)))) |
437 | 436 | ralimdva 3167 |
. . . . . 6
⊢ (𝜑 → (∀𝑙 ∈ ( L ‘𝐴)∃𝑡 ∈ 𝐿 𝑙 ≤s 𝑡 → ∀𝑙 ∈ ( L ‘𝐴)∀𝑚 ∈ ( R ‘𝐵)∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)))) |
438 | 324, 437 | mpd 15 |
. . . . 5
⊢ (𝜑 → ∀𝑙 ∈ ( L ‘𝐴)∀𝑚 ∈ ( R ‘𝐵)∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))) |
439 | | eqeq1 2736 |
. . . . . . . . . 10
⊢ (𝑐 = 𝑧 → (𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ↔ 𝑧 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)))) |
440 | 439 | 2rexbidv 3219 |
. . . . . . . . 9
⊢ (𝑐 = 𝑧 → (∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ↔ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑧 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)))) |
441 | 440 | rexab 3687 |
. . . . . . . 8
⊢
(∃𝑧 ∈
{𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))}𝑧 ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) ↔ ∃𝑧(∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑧 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ∧ 𝑧 ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)))) |
442 | | r19.41vv 3224 |
. . . . . . . . 9
⊢
(∃𝑡 ∈
𝐿 ∃𝑢 ∈ 𝑆 (𝑧 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ∧ 𝑧 ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))) ↔ (∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑧 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ∧ 𝑧 ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)))) |
443 | 442 | exbii 1850 |
. . . . . . . 8
⊢
(∃𝑧∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (𝑧 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ∧ 𝑧 ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))) ↔ ∃𝑧(∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑧 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ∧ 𝑧 ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)))) |
444 | | rexcom4 3285 |
. . . . . . . . 9
⊢
(∃𝑡 ∈
𝐿 ∃𝑧∃𝑢 ∈ 𝑆 (𝑧 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ∧ 𝑧 ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))) ↔ ∃𝑧∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (𝑧 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ∧ 𝑧 ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)))) |
445 | | rexcom4 3285 |
. . . . . . . . . . 11
⊢
(∃𝑢 ∈
𝑆 ∃𝑧(𝑧 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ∧ 𝑧 ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))) ↔ ∃𝑧∃𝑢 ∈ 𝑆 (𝑧 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ∧ 𝑧 ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)))) |
446 | | ovex 7427 |
. . . . . . . . . . . . 13
⊢ (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ∈ V |
447 | | breq1 5145 |
. . . . . . . . . . . . 13
⊢ (𝑧 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) → (𝑧 ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) ↔ (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)))) |
448 | 446, 447 | ceqsexv 3523 |
. . . . . . . . . . . 12
⊢
(∃𝑧(𝑧 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ∧ 𝑧 ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))) ↔ (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))) |
449 | 448 | rexbii 3094 |
. . . . . . . . . . 11
⊢
(∃𝑢 ∈
𝑆 ∃𝑧(𝑧 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ∧ 𝑧 ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))) ↔ ∃𝑢 ∈ 𝑆 (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))) |
450 | 445, 449 | bitr3i 276 |
. . . . . . . . . 10
⊢
(∃𝑧∃𝑢 ∈ 𝑆 (𝑧 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ∧ 𝑧 ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))) ↔ ∃𝑢 ∈ 𝑆 (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))) |
451 | 450 | rexbii 3094 |
. . . . . . . . 9
⊢
(∃𝑡 ∈
𝐿 ∃𝑧∃𝑢 ∈ 𝑆 (𝑧 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ∧ 𝑧 ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))) ↔ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))) |
452 | 444, 451 | bitr3i 276 |
. . . . . . . 8
⊢
(∃𝑧∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (𝑧 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ∧ 𝑧 ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))) ↔ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))) |
453 | 441, 443,
452 | 3bitr2i 298 |
. . . . . . 7
⊢
(∃𝑧 ∈
{𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))}𝑧 ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) ↔ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))) |
454 | | ssun1 4169 |
. . . . . . . 8
⊢ {𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ⊆ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}) |
455 | | ssrexv 4048 |
. . . . . . . 8
⊢ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ⊆ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}) → (∃𝑧 ∈ {𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))}𝑧 ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)))) |
456 | 454, 455 | ax-mp 5 |
. . . . . . 7
⊢
(∃𝑧 ∈
{𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))}𝑧 ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))) |
457 | 453, 456 | sylbir 234 |
. . . . . 6
⊢
(∃𝑡 ∈
𝐿 ∃𝑢 ∈ 𝑆 (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))) |
458 | 457 | 2ralimi 3123 |
. . . . 5
⊢
(∀𝑙 ∈ (
L ‘𝐴)∀𝑚 ∈ ( R ‘𝐵)∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) → ∀𝑙 ∈ ( L ‘𝐴)∀𝑚 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))) |
459 | 438, 458 | syl 17 |
. . . 4
⊢ (𝜑 → ∀𝑙 ∈ ( L ‘𝐴)∀𝑚 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))) |
460 | 2, 1 | cofcutr2d 27342 |
. . . . . 6
⊢ (𝜑 → ∀𝑥 ∈ ( R ‘𝐴)∃𝑣 ∈ 𝑅 𝑣 ≤s 𝑥) |
461 | 6, 5 | cofcutr1d 27341 |
. . . . . . . . . 10
⊢ (𝜑 → ∀𝑦 ∈ ( L ‘𝐵)∃𝑤 ∈ 𝑀 𝑦 ≤s 𝑤) |
462 | 461 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ ∃𝑣 ∈ 𝑅 𝑣 ≤s 𝑥)) → ∀𝑦 ∈ ( L ‘𝐵)∃𝑤 ∈ 𝑀 𝑦 ≤s 𝑤) |
463 | | reeanv 3226 |
. . . . . . . . . . . . . . 15
⊢
(∃𝑣 ∈
𝑅 ∃𝑤 ∈ 𝑀 (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤) ↔ (∃𝑣 ∈ 𝑅 𝑣 ≤s 𝑥 ∧ ∃𝑤 ∈ 𝑀 𝑦 ≤s 𝑤)) |
464 | 170 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑅 ⊆ No
) |
465 | | simprl 769 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑣 ∈ 𝑅) |
466 | 464, 465 | sseldd 3980 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑣 ∈ No
) |
467 | 8 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝐵 ∈ No
) |
468 | 466, 467 | mulscld 27520 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (𝑣 ·s 𝐵) ∈ No
) |
469 | 468 | adantrl 714 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → (𝑣 ·s 𝐵) ∈ No
) |
470 | 4 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝐴 ∈ No
) |
471 | 44 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑀 ⊆ No
) |
472 | | simprr 771 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑤 ∈ 𝑀) |
473 | 471, 472 | sseldd 3980 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑤 ∈ No
) |
474 | 470, 473 | mulscld 27520 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (𝐴 ·s 𝑤) ∈ No
) |
475 | 474 | adantrl 714 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → (𝐴 ·s 𝑤) ∈ No
) |
476 | 469, 475 | addscld 27393 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → ((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) ∈ No
) |
477 | 466, 473 | mulscld 27520 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (𝑣 ·s 𝑤) ∈ No
) |
478 | 477 | adantrl 714 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → (𝑣 ·s 𝑤) ∈ No
) |
479 | 476, 478 | subscld 27464 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ∈ No
) |
480 | 479 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ∈ No
) |
481 | 4 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → 𝐴 ∈ No
) |
482 | | simprr 771 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → 𝑦 ∈ ( L ‘𝐵)) |
483 | 23, 482 | sselid 3977 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → 𝑦 ∈ No
) |
484 | 483 | adantrr 715 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → 𝑦 ∈ No
) |
485 | 481, 484 | mulscld 27520 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → (𝐴 ·s 𝑦) ∈ No
) |
486 | 469, 485 | addscld 27393 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → ((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑦)) ∈ No
) |
487 | 466 | adantrl 714 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → 𝑣 ∈ No
) |
488 | 487, 484 | mulscld 27520 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → (𝑣 ·s 𝑦) ∈ No
) |
489 | 486, 488 | subscld 27464 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑣 ·s 𝑦)) ∈ No
) |
490 | 489 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑣 ·s 𝑦)) ∈ No
) |
491 | | simprl 769 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → 𝑥 ∈ ( R ‘𝐴)) |
492 | 153, 491 | sselid 3977 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → 𝑥 ∈ No
) |
493 | 8 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → 𝐵 ∈ No
) |
494 | 492, 493 | mulscld 27520 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → (𝑥 ·s 𝐵) ∈ No
) |
495 | 494 | adantrr 715 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → (𝑥 ·s 𝐵) ∈ No
) |
496 | 495, 485 | addscld 27393 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → ((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) ∈ No
) |
497 | 492, 483 | mulscld 27520 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → (𝑥 ·s 𝑦) ∈ No
) |
498 | 497 | adantrr 715 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → (𝑥 ·s 𝑦) ∈ No
) |
499 | 496, 498 | subscld 27464 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) ∈ No
) |
500 | 499 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) ∈ No
) |
501 | 4 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → 𝐴 ∈ No
) |
502 | 487 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → 𝑣 ∈ No
) |
503 | 483 | adantrr 715 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → 𝑦 ∈ No
) |
504 | 473 | adantrl 714 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → 𝑤 ∈ No
) |
505 | 504 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → 𝑤 ∈ No
) |
506 | 1 | sneqd 4635 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝜑 → {𝐴} = {(𝐿 |s 𝑅)}) |
507 | 506, 222 | eqbrtrd 5164 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝜑 → {𝐴} <<s 𝑅) |
508 | 507 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → {𝐴} <<s 𝑅) |
509 | 481, 401 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → 𝐴 ∈ {𝐴}) |
510 | | simprrl 779 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → 𝑣 ∈ 𝑅) |
511 | 508, 509,
510 | ssltsepcd 27224 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → 𝐴 <s 𝑣) |
512 | 481, 487,
511 | sltled 27201 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → 𝐴 ≤s 𝑣) |
513 | 512 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → 𝐴 ≤s 𝑣) |
514 | | simprrr 780 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤))) → 𝑦 ≤s 𝑤) |
515 | 514 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → 𝑦 ≤s 𝑤) |
516 | 501, 502,
503, 505, 513, 515 | slemuld 27523 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → ((𝐴 ·s 𝑤) -s (𝐴 ·s 𝑦)) ≤s ((𝑣 ·s 𝑤) -s (𝑣 ·s 𝑦))) |
517 | 475, 478,
485, 488 | slesubsubbd 27482 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → (((𝐴 ·s 𝑤) -s (𝐴 ·s 𝑦)) ≤s ((𝑣 ·s 𝑤) -s (𝑣 ·s 𝑦)) ↔ ((𝐴 ·s 𝑤) -s (𝑣 ·s 𝑤)) ≤s ((𝐴 ·s 𝑦) -s (𝑣 ·s 𝑦)))) |
518 | 475, 478 | subscld 27464 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → ((𝐴 ·s 𝑤) -s (𝑣 ·s 𝑤)) ∈ No
) |
519 | 485, 488 | subscld 27464 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → ((𝐴 ·s 𝑦) -s (𝑣 ·s 𝑦)) ∈ No
) |
520 | 518, 519,
469 | sleadd2d 27408 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → (((𝐴 ·s 𝑤) -s (𝑣 ·s 𝑤)) ≤s ((𝐴 ·s 𝑦) -s (𝑣 ·s 𝑦)) ↔ ((𝑣 ·s 𝐵) +s ((𝐴 ·s 𝑤) -s (𝑣 ·s 𝑤))) ≤s ((𝑣 ·s 𝐵) +s ((𝐴 ·s 𝑦) -s (𝑣 ·s 𝑦))))) |
521 | 517, 520 | bitrd 278 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → (((𝐴 ·s 𝑤) -s (𝐴 ·s 𝑦)) ≤s ((𝑣 ·s 𝑤) -s (𝑣 ·s 𝑦)) ↔ ((𝑣 ·s 𝐵) +s ((𝐴 ·s 𝑤) -s (𝑣 ·s 𝑤))) ≤s ((𝑣 ·s 𝐵) +s ((𝐴 ·s 𝑦) -s (𝑣 ·s 𝑦))))) |
522 | 521 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → (((𝐴 ·s 𝑤) -s (𝐴 ·s 𝑦)) ≤s ((𝑣 ·s 𝑤) -s (𝑣 ·s 𝑦)) ↔ ((𝑣 ·s 𝐵) +s ((𝐴 ·s 𝑤) -s (𝑣 ·s 𝑤))) ≤s ((𝑣 ·s 𝐵) +s ((𝐴 ·s 𝑦) -s (𝑣 ·s 𝑦))))) |
523 | 516, 522 | mpbid 231 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → ((𝑣 ·s 𝐵) +s ((𝐴 ·s 𝑤) -s (𝑣 ·s 𝑤))) ≤s ((𝑣 ·s 𝐵) +s ((𝐴 ·s 𝑦) -s (𝑣 ·s 𝑦)))) |
524 | 469, 475,
478 | addsubsassd 27477 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) = ((𝑣 ·s 𝐵) +s ((𝐴 ·s 𝑤) -s (𝑣 ·s 𝑤)))) |
525 | 524 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) = ((𝑣 ·s 𝐵) +s ((𝐴 ·s 𝑤) -s (𝑣 ·s 𝑤)))) |
526 | 469, 485,
488 | addsubsassd 27477 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑣 ·s 𝑦)) = ((𝑣 ·s 𝐵) +s ((𝐴 ·s 𝑦) -s (𝑣 ·s 𝑦)))) |
527 | 526 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑣 ·s 𝑦)) = ((𝑣 ·s 𝐵) +s ((𝐴 ·s 𝑦) -s (𝑣 ·s 𝑦)))) |
528 | 523, 525,
527 | 3brtr4d 5174 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ≤s (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑣 ·s 𝑦))) |
529 | 492 | adantrr 715 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → 𝑥 ∈ No
) |
530 | 8 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → 𝐵 ∈ No
) |
531 | | simprrl 779 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤))) → 𝑣 ≤s 𝑥) |
532 | 531 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → 𝑣 ≤s 𝑥) |
533 | 493, 61 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → ( L ‘𝐵) <<s {𝐵}) |
534 | 65 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → 𝐵 ∈ {𝐵}) |
535 | 533, 482,
534 | ssltsepcd 27224 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → 𝑦 <s 𝐵) |
536 | 483, 493,
535 | sltled 27201 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → 𝑦 ≤s 𝐵) |
537 | 536 | adantrr 715 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → 𝑦 ≤s 𝐵) |
538 | 502, 529,
503, 530, 532, 537 | slemuld 27523 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → ((𝑣 ·s 𝐵) -s (𝑣 ·s 𝑦)) ≤s ((𝑥 ·s 𝐵) -s (𝑥 ·s 𝑦))) |
539 | 469, 488 | subscld 27464 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → ((𝑣 ·s 𝐵) -s (𝑣 ·s 𝑦)) ∈ No
) |
540 | 539 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → ((𝑣 ·s 𝐵) -s (𝑣 ·s 𝑦)) ∈ No
) |
541 | 495, 498 | subscld 27464 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → ((𝑥 ·s 𝐵) -s (𝑥 ·s 𝑦)) ∈ No
) |
542 | 541 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → ((𝑥 ·s 𝐵) -s (𝑥 ·s 𝑦)) ∈ No
) |
543 | 485 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → (𝐴 ·s 𝑦) ∈ No
) |
544 | 540, 542,
543 | sleadd1d 27407 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → (((𝑣 ·s 𝐵) -s (𝑣 ·s 𝑦)) ≤s ((𝑥 ·s 𝐵) -s (𝑥 ·s 𝑦)) ↔ (((𝑣 ·s 𝐵) -s (𝑣 ·s 𝑦)) +s (𝐴 ·s 𝑦)) ≤s (((𝑥 ·s 𝐵) -s (𝑥 ·s 𝑦)) +s (𝐴 ·s 𝑦)))) |
545 | 538, 544 | mpbid 231 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → (((𝑣 ·s 𝐵) -s (𝑣 ·s 𝑦)) +s (𝐴 ·s 𝑦)) ≤s (((𝑥 ·s 𝐵) -s (𝑥 ·s 𝑦)) +s (𝐴 ·s 𝑦))) |
546 | 469, 485,
488 | addsubsd 27478 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑣 ·s 𝑦)) = (((𝑣 ·s 𝐵) -s (𝑣 ·s 𝑦)) +s (𝐴 ·s 𝑦))) |
547 | 546 | adantrrr 723 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑣 ·s 𝑦)) = (((𝑣 ·s 𝐵) -s (𝑣 ·s 𝑦)) +s (𝐴 ·s 𝑦))) |
548 | 4 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → 𝐴 ∈ No
) |
549 | 548, 483 | mulscld 27520 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → (𝐴 ·s 𝑦) ∈ No
) |
550 | 494, 549,
497 | addsubsd 27478 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) = (((𝑥 ·s 𝐵) -s (𝑥 ·s 𝑦)) +s (𝐴 ·s 𝑦))) |
551 | 550 | adantrr 715 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) = (((𝑥 ·s 𝐵) -s (𝑥 ·s 𝑦)) +s (𝐴 ·s 𝑦))) |
552 | 545, 547,
551 | 3brtr4d 5174 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑣 ·s 𝑦)) ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))) |
553 | 480, 490,
500, 528, 552 | sletrd 27194 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))) |
554 | 553 | anassrs 468 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤))) → (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))) |
555 | 554 | expr 457 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → ((𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤) → (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)))) |
556 | 555 | reximdvva 3205 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → (∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤) → ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)))) |
557 | 556 | expcom 414 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) → (𝜑 → (∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤) → ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))))) |
558 | 557 | com23 86 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) → (∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤) → (𝜑 → ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))))) |
559 | 558 | imp 407 |
. . . . . . . . . . . . . . 15
⊢ (((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)) → (𝜑 → ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)))) |
560 | 463, 559 | sylan2br 595 |
. . . . . . . . . . . . . 14
⊢ (((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (∃𝑣 ∈ 𝑅 𝑣 ≤s 𝑥 ∧ ∃𝑤 ∈ 𝑀 𝑦 ≤s 𝑤)) → (𝜑 → ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)))) |
561 | 560 | an4s 658 |
. . . . . . . . . . . . 13
⊢ (((𝑥 ∈ ( R ‘𝐴) ∧ ∃𝑣 ∈ 𝑅 𝑣 ≤s 𝑥) ∧ (𝑦 ∈ ( L ‘𝐵) ∧ ∃𝑤 ∈ 𝑀 𝑦 ≤s 𝑤)) → (𝜑 → ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)))) |
562 | 561 | impcom 408 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ ∃𝑣 ∈ 𝑅 𝑣 ≤s 𝑥) ∧ (𝑦 ∈ ( L ‘𝐵) ∧ ∃𝑤 ∈ 𝑀 𝑦 ≤s 𝑤))) → ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))) |
563 | 562 | anassrs 468 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ ∃𝑣 ∈ 𝑅 𝑣 ≤s 𝑥)) ∧ (𝑦 ∈ ( L ‘𝐵) ∧ ∃𝑤 ∈ 𝑀 𝑦 ≤s 𝑤)) → ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))) |
564 | 563 | expr 457 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ ∃𝑣 ∈ 𝑅 𝑣 ≤s 𝑥)) ∧ 𝑦 ∈ ( L ‘𝐵)) → (∃𝑤 ∈ 𝑀 𝑦 ≤s 𝑤 → ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)))) |
565 | 564 | ralimdva 3167 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ ∃𝑣 ∈ 𝑅 𝑣 ≤s 𝑥)) → (∀𝑦 ∈ ( L ‘𝐵)∃𝑤 ∈ 𝑀 𝑦 ≤s 𝑤 → ∀𝑦 ∈ ( L ‘𝐵)∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)))) |
566 | 462, 565 | mpd 15 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ ∃𝑣 ∈ 𝑅 𝑣 ≤s 𝑥)) → ∀𝑦 ∈ ( L ‘𝐵)∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))) |
567 | 566 | expr 457 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ( R ‘𝐴)) → (∃𝑣 ∈ 𝑅 𝑣 ≤s 𝑥 → ∀𝑦 ∈ ( L ‘𝐵)∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)))) |
568 | 567 | ralimdva 3167 |
. . . . . 6
⊢ (𝜑 → (∀𝑥 ∈ ( R ‘𝐴)∃𝑣 ∈ 𝑅 𝑣 ≤s 𝑥 → ∀𝑥 ∈ ( R ‘𝐴)∀𝑦 ∈ ( L ‘𝐵)∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)))) |
569 | 460, 568 | mpd 15 |
. . . . 5
⊢ (𝜑 → ∀𝑥 ∈ ( R ‘𝐴)∀𝑦 ∈ ( L ‘𝐵)∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))) |
570 | | eqeq1 2736 |
. . . . . . . . . 10
⊢ (𝑑 = 𝑧 → (𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ↔ 𝑧 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)))) |
571 | 570 | 2rexbidv 3219 |
. . . . . . . . 9
⊢ (𝑑 = 𝑧 → (∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ↔ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑧 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)))) |
572 | 571 | rexab 3687 |
. . . . . . . 8
⊢
(∃𝑧 ∈
{𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}𝑧 ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) ↔ ∃𝑧(∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑧 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ∧ 𝑧 ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)))) |
573 | | r19.41vv 3224 |
. . . . . . . . 9
⊢
(∃𝑣 ∈
𝑅 ∃𝑤 ∈ 𝑀 (𝑧 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ∧ 𝑧 ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))) ↔ (∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑧 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ∧ 𝑧 ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)))) |
574 | 573 | exbii 1850 |
. . . . . . . 8
⊢
(∃𝑧∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (𝑧 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ∧ 𝑧 ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))) ↔ ∃𝑧(∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑧 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ∧ 𝑧 ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)))) |
575 | | rexcom4 3285 |
. . . . . . . . 9
⊢
(∃𝑣 ∈
𝑅 ∃𝑧∃𝑤 ∈ 𝑀 (𝑧 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ∧ 𝑧 ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))) ↔ ∃𝑧∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (𝑧 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ∧ 𝑧 ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)))) |
576 | | rexcom4 3285 |
. . . . . . . . . . 11
⊢
(∃𝑤 ∈
𝑀 ∃𝑧(𝑧 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ∧ 𝑧 ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))) ↔ ∃𝑧∃𝑤 ∈ 𝑀 (𝑧 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ∧ 𝑧 ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)))) |
577 | | ovex 7427 |
. . . . . . . . . . . . 13
⊢ (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ∈ V |
578 | | breq1 5145 |
. . . . . . . . . . . . 13
⊢ (𝑧 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) → (𝑧 ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) ↔ (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)))) |
579 | 577, 578 | ceqsexv 3523 |
. . . . . . . . . . . 12
⊢
(∃𝑧(𝑧 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ∧ 𝑧 ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))) ↔ (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))) |
580 | 579 | rexbii 3094 |
. . . . . . . . . . 11
⊢
(∃𝑤 ∈
𝑀 ∃𝑧(𝑧 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ∧ 𝑧 ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))) ↔ ∃𝑤 ∈ 𝑀 (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))) |
581 | 576, 580 | bitr3i 276 |
. . . . . . . . . 10
⊢
(∃𝑧∃𝑤 ∈ 𝑀 (𝑧 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ∧ 𝑧 ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))) ↔ ∃𝑤 ∈ 𝑀 (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))) |
582 | 581 | rexbii 3094 |
. . . . . . . . 9
⊢
(∃𝑣 ∈
𝑅 ∃𝑧∃𝑤 ∈ 𝑀 (𝑧 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ∧ 𝑧 ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))) ↔ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))) |
583 | 575, 582 | bitr3i 276 |
. . . . . . . 8
⊢
(∃𝑧∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (𝑧 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ∧ 𝑧 ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))) ↔ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))) |
584 | 572, 574,
583 | 3bitr2i 298 |
. . . . . . 7
⊢
(∃𝑧 ∈
{𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}𝑧 ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) ↔ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))) |
585 | | ssun2 4170 |
. . . . . . . 8
⊢ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))} ⊆ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}) |
586 | | ssrexv 4048 |
. . . . . . . 8
⊢ ({𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))} ⊆ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}) → (∃𝑧 ∈ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}𝑧 ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)))) |
587 | 585, 586 | ax-mp 5 |
. . . . . . 7
⊢
(∃𝑧 ∈
{𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}𝑧 ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))) |
588 | 584, 587 | sylbir 234 |
. . . . . 6
⊢
(∃𝑣 ∈
𝑅 ∃𝑤 ∈ 𝑀 (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))) |
589 | 588 | 2ralimi 3123 |
. . . . 5
⊢
(∀𝑥 ∈ (
R ‘𝐴)∀𝑦 ∈ ( L ‘𝐵)∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) → ∀𝑥 ∈ ( R ‘𝐴)∀𝑦 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))) |
590 | 569, 589 | syl 17 |
. . . 4
⊢ (𝜑 → ∀𝑥 ∈ ( R ‘𝐴)∀𝑦 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))) |
591 | | ralunb 4188 |
. . . . 5
⊢
(∀𝑥𝑂 ∈ ({𝑘 ∣ ∃𝑙 ∈ ( L ‘𝐴)∃𝑚 ∈ ( R ‘𝐵)𝑘 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))} ∪ {𝑛 ∣ ∃𝑥 ∈ ( R ‘𝐴)∃𝑦 ∈ ( L ‘𝐵)𝑛 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))})∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂 ↔ (∀𝑥𝑂 ∈
{𝑘 ∣ ∃𝑙 ∈ ( L ‘𝐴)∃𝑚 ∈ ( R ‘𝐵)𝑘 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))}∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂 ∧ ∀𝑥𝑂 ∈
{𝑛 ∣ ∃𝑥 ∈ ( R ‘𝐴)∃𝑦 ∈ ( L ‘𝐵)𝑛 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))}∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂)) |
592 | | eqeq1 2736 |
. . . . . . . . 9
⊢ (𝑘 = 𝑥𝑂 → (𝑘 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) ↔ 𝑥𝑂 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)))) |
593 | 592 | 2rexbidv 3219 |
. . . . . . . 8
⊢ (𝑘 = 𝑥𝑂 → (∃𝑙 ∈ ( L ‘𝐴)∃𝑚 ∈ ( R ‘𝐵)𝑘 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) ↔ ∃𝑙 ∈ ( L ‘𝐴)∃𝑚 ∈ ( R ‘𝐵)𝑥𝑂 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)))) |
594 | 593 | ralab 3684 |
. . . . . . 7
⊢
(∀𝑥𝑂 ∈ {𝑘 ∣ ∃𝑙 ∈ ( L ‘𝐴)∃𝑚 ∈ ( R ‘𝐵)𝑘 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))}∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂 ↔ ∀𝑥𝑂(∃𝑙 ∈ ( L ‘𝐴)∃𝑚 ∈ ( R ‘𝐵)𝑥𝑂 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂)) |
595 | | r19.23v 3182 |
. . . . . . . . . 10
⊢
(∀𝑚 ∈ (
R ‘𝐵)(𝑥𝑂 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂) ↔ (∃𝑚 ∈ ( R ‘𝐵)𝑥𝑂 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂)) |
596 | 595 | ralbii 3093 |
. . . . . . . . 9
⊢
(∀𝑙 ∈ (
L ‘𝐴)∀𝑚 ∈ ( R ‘𝐵)(𝑥𝑂 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂) ↔ ∀𝑙 ∈ ( L ‘𝐴)(∃𝑚 ∈ ( R ‘𝐵)𝑥𝑂 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂)) |
597 | | r19.23v 3182 |
. . . . . . . . 9
⊢
(∀𝑙 ∈ (
L ‘𝐴)(∃𝑚 ∈ ( R ‘𝐵)𝑥𝑂 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂) ↔ (∃𝑙 ∈ ( L ‘𝐴)∃𝑚 ∈ ( R ‘𝐵)𝑥𝑂 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂)) |
598 | 596, 597 | bitri 274 |
. . . . . . . 8
⊢
(∀𝑙 ∈ (
L ‘𝐴)∀𝑚 ∈ ( R ‘𝐵)(𝑥𝑂 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂) ↔ (∃𝑙 ∈ ( L ‘𝐴)∃𝑚 ∈ ( R ‘𝐵)𝑥𝑂 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂)) |
599 | 598 | albii 1821 |
. . . . . . 7
⊢
(∀𝑥𝑂∀𝑙 ∈ ( L ‘𝐴)∀𝑚 ∈ ( R ‘𝐵)(𝑥𝑂 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂) ↔ ∀𝑥𝑂(∃𝑙 ∈ ( L ‘𝐴)∃𝑚 ∈ ( R ‘𝐵)𝑥𝑂 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂)) |
600 | | ralcom4 3283 |
. . . . . . . 8
⊢
(∀𝑙 ∈ (
L ‘𝐴)∀𝑥𝑂∀𝑚 ∈ ( R ‘𝐵)(𝑥𝑂 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂) ↔ ∀𝑥𝑂∀𝑙 ∈ ( L ‘𝐴)∀𝑚 ∈ ( R ‘𝐵)(𝑥𝑂 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂)) |
601 | | ralcom4 3283 |
. . . . . . . . . 10
⊢
(∀𝑚 ∈ (
R ‘𝐵)∀𝑥𝑂(𝑥𝑂 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂) ↔ ∀𝑥𝑂∀𝑚 ∈ ( R ‘𝐵)(𝑥𝑂 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂)) |
602 | | ovex 7427 |
. . . . . . . . . . . 12
⊢ (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) ∈ V |
603 | | breq2 5146 |
. . . . . . . . . . . . 13
⊢ (𝑥𝑂 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) → (𝑧 ≤s 𝑥𝑂 ↔ 𝑧 ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)))) |
604 | 603 | rexbidv 3178 |
. . . . . . . . . . . 12
⊢ (𝑥𝑂 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) → (∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂 ↔ ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)))) |
605 | 602, 604 | ceqsalv 3510 |
. . . . . . . . . . 11
⊢
(∀𝑥𝑂(𝑥𝑂 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂) ↔ ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))) |
606 | 605 | ralbii 3093 |
. . . . . . . . . 10
⊢
(∀𝑚 ∈ (
R ‘𝐵)∀𝑥𝑂(𝑥𝑂 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂) ↔ ∀𝑚 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))) |
607 | 601, 606 | bitr3i 276 |
. . . . . . . . 9
⊢
(∀𝑥𝑂∀𝑚 ∈ ( R ‘𝐵)(𝑥𝑂 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂) ↔ ∀𝑚 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))) |
608 | 607 | ralbii 3093 |
. . . . . . . 8
⊢
(∀𝑙 ∈ (
L ‘𝐴)∀𝑥𝑂∀𝑚 ∈ ( R ‘𝐵)(𝑥𝑂 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂) ↔ ∀𝑙 ∈ ( L ‘𝐴)∀𝑚 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))) |
609 | 600, 608 | bitr3i 276 |
. . . . . . 7
⊢
(∀𝑥𝑂∀𝑙 ∈ ( L ‘𝐴)∀𝑚 ∈ ( R ‘𝐵)(𝑥𝑂 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂) ↔ ∀𝑙 ∈ ( L ‘𝐴)∀𝑚 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))) |
610 | 594, 599,
609 | 3bitr2i 298 |
. . . . . 6
⊢
(∀𝑥𝑂 ∈ {𝑘 ∣ ∃𝑙 ∈ ( L ‘𝐴)∃𝑚 ∈ ( R ‘𝐵)𝑘 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))}∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂 ↔ ∀𝑙 ∈ ( L ‘𝐴)∀𝑚 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))) |
611 | | eqeq1 2736 |
. . . . . . . . 9
⊢ (𝑛 = 𝑥𝑂 → (𝑛 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) ↔ 𝑥𝑂 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)))) |
612 | 611 | 2rexbidv 3219 |
. . . . . . . 8
⊢ (𝑛 = 𝑥𝑂 → (∃𝑥 ∈ ( R ‘𝐴)∃𝑦 ∈ ( L ‘𝐵)𝑛 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) ↔ ∃𝑥 ∈ ( R ‘𝐴)∃𝑦 ∈ ( L ‘𝐵)𝑥𝑂 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)))) |
613 | 612 | ralab 3684 |
. . . . . . 7
⊢
(∀𝑥𝑂 ∈ {𝑛 ∣ ∃𝑥 ∈ ( R ‘𝐴)∃𝑦 ∈ ( L ‘𝐵)𝑛 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))}∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂 ↔ ∀𝑥𝑂(∃𝑥 ∈ ( R ‘𝐴)∃𝑦 ∈ ( L ‘𝐵)𝑥𝑂 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂)) |
614 | | r19.23v 3182 |
. . . . . . . . . 10
⊢
(∀𝑦 ∈ (
L ‘𝐵)(𝑥𝑂 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂) ↔ (∃𝑦 ∈ ( L ‘𝐵)𝑥𝑂 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂)) |
615 | 614 | ralbii 3093 |
. . . . . . . . 9
⊢
(∀𝑥 ∈ (
R ‘𝐴)∀𝑦 ∈ ( L ‘𝐵)(𝑥𝑂 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂) ↔ ∀𝑥 ∈ ( R ‘𝐴)(∃𝑦 ∈ ( L ‘𝐵)𝑥𝑂 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂)) |
616 | | r19.23v 3182 |
. . . . . . . . 9
⊢
(∀𝑥 ∈ (
R ‘𝐴)(∃𝑦 ∈ ( L ‘𝐵)𝑥𝑂 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂) ↔ (∃𝑥 ∈ ( R ‘𝐴)∃𝑦 ∈ ( L ‘𝐵)𝑥𝑂 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂)) |
617 | 615, 616 | bitri 274 |
. . . . . . . 8
⊢
(∀𝑥 ∈ (
R ‘𝐴)∀𝑦 ∈ ( L ‘𝐵)(𝑥𝑂 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂) ↔ (∃𝑥 ∈ ( R ‘𝐴)∃𝑦 ∈ ( L ‘𝐵)𝑥𝑂 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂)) |
618 | 617 | albii 1821 |
. . . . . . 7
⊢
(∀𝑥𝑂∀𝑥 ∈ ( R ‘𝐴)∀𝑦 ∈ ( L ‘𝐵)(𝑥𝑂 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂) ↔ ∀𝑥𝑂(∃𝑥 ∈ ( R ‘𝐴)∃𝑦 ∈ ( L ‘𝐵)𝑥𝑂 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂)) |
619 | | ralcom4 3283 |
. . . . . . . 8
⊢
(∀𝑥 ∈ (
R ‘𝐴)∀𝑥𝑂∀𝑦 ∈ ( L ‘𝐵)(𝑥𝑂 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂) ↔ ∀𝑥𝑂∀𝑥 ∈ ( R ‘𝐴)∀𝑦 ∈ ( L ‘𝐵)(𝑥𝑂 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂)) |
620 | | ralcom4 3283 |
. . . . . . . . . 10
⊢
(∀𝑦 ∈ (
L ‘𝐵)∀𝑥𝑂(𝑥𝑂 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂) ↔ ∀𝑥𝑂∀𝑦 ∈ ( L ‘𝐵)(𝑥𝑂 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂)) |
621 | | ovex 7427 |
. . . . . . . . . . . 12
⊢ (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) ∈ V |
622 | | breq2 5146 |
. . . . . . . . . . . . 13
⊢ (𝑥𝑂 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) → (𝑧 ≤s 𝑥𝑂 ↔ 𝑧 ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)))) |
623 | 622 | rexbidv 3178 |
. . . . . . . . . . . 12
⊢ (𝑥𝑂 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) → (∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂 ↔ ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)))) |
624 | 621, 623 | ceqsalv 3510 |
. . . . . . . . . . 11
⊢
(∀𝑥𝑂(𝑥𝑂 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂) ↔ ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))) |
625 | 624 | ralbii 3093 |
. . . . . . . . . 10
⊢
(∀𝑦 ∈ (
L ‘𝐵)∀𝑥𝑂(𝑥𝑂 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂) ↔ ∀𝑦 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))) |
626 | 620, 625 | bitr3i 276 |
. . . . . . . . 9
⊢
(∀𝑥𝑂∀𝑦 ∈ ( L ‘𝐵)(𝑥𝑂 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂) ↔ ∀𝑦 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))) |
627 | 626 | ralbii 3093 |
. . . . . . . 8
⊢
(∀𝑥 ∈ (
R ‘𝐴)∀𝑥𝑂∀𝑦 ∈ ( L ‘𝐵)(𝑥𝑂 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂) ↔ ∀𝑥 ∈ ( R ‘𝐴)∀𝑦 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))) |
628 | 619, 627 | bitr3i 276 |
. . . . . . 7
⊢
(∀𝑥𝑂∀𝑥 ∈ ( R ‘𝐴)∀𝑦 ∈ ( L ‘𝐵)(𝑥𝑂 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂) ↔ ∀𝑥 ∈ ( R ‘𝐴)∀𝑦 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))) |
629 | 613, 618,
628 | 3bitr2i 298 |
. . . . . 6
⊢
(∀𝑥𝑂 ∈ {𝑛 ∣ ∃𝑥 ∈ ( R ‘𝐴)∃𝑦 ∈ ( L ‘𝐵)𝑛 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))}∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂 ↔ ∀𝑥 ∈ ( R ‘𝐴)∀𝑦 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))) |
630 | 610, 629 | anbi12i 627 |
. . . . 5
⊢
((∀𝑥𝑂 ∈ {𝑘 ∣ ∃𝑙 ∈ ( L ‘𝐴)∃𝑚 ∈ ( R ‘𝐵)𝑘 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))}∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂 ∧ ∀𝑥𝑂 ∈
{𝑛 ∣ ∃𝑥 ∈ ( R ‘𝐴)∃𝑦 ∈ ( L ‘𝐵)𝑛 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))}∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂) ↔ (∀𝑙 ∈ ( L ‘𝐴)∀𝑚 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) ∧ ∀𝑥 ∈ ( R ‘𝐴)∀𝑦 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)))) |
631 | 591, 630 | bitri 274 |
. . . 4
⊢
(∀𝑥𝑂 ∈ ({𝑘 ∣ ∃𝑙 ∈ ( L ‘𝐴)∃𝑚 ∈ ( R ‘𝐵)𝑘 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))} ∪ {𝑛 ∣ ∃𝑥 ∈ ( R ‘𝐴)∃𝑦 ∈ ( L ‘𝐵)𝑛 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))})∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂 ↔ (∀𝑙 ∈ ( L ‘𝐴)∀𝑚 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) ∧ ∀𝑥 ∈ ( R ‘𝐴)∀𝑦 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)))) |
632 | 459, 590,
631 | sylanbrc 583 |
. . 3
⊢ (𝜑 → ∀𝑥𝑂 ∈ ({𝑘 ∣ ∃𝑙 ∈ ( L ‘𝐴)∃𝑚 ∈ ( R ‘𝐵)𝑘 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))} ∪ {𝑛 ∣ ∃𝑥 ∈ ( R ‘𝐴)∃𝑦 ∈ ( L ‘𝐵)𝑛 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))})∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂) |
633 | 2, 6, 1, 5 | ssltmul1 27531 |
. . . 4
⊢ (𝜑 → ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) <<s {(𝐴 ·s 𝐵)}) |
634 | 10 | sneqd 4635 |
. . . 4
⊢ (𝜑 → {(𝐴 ·s 𝐵)} = {(({𝑒 ∣ ∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑒 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔))} ∪ {ℎ ∣ ∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵)ℎ = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗))}) |s ({𝑘 ∣ ∃𝑙 ∈ ( L ‘𝐴)∃𝑚 ∈ ( R ‘𝐵)𝑘 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))} ∪ {𝑛 ∣ ∃𝑥 ∈ ( R ‘𝐴)∃𝑦 ∈ ( L ‘𝐵)𝑛 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))}))}) |
635 | 633, 634 | breqtrd 5168 |
. . 3
⊢ (𝜑 → ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) <<s {(({𝑒 ∣ ∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑒 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔))} ∪ {ℎ ∣ ∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵)ℎ = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗))}) |s ({𝑘 ∣ ∃𝑙 ∈ ( L ‘𝐴)∃𝑚 ∈ ( R ‘𝐵)𝑘 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))} ∪ {𝑛 ∣ ∃𝑥 ∈ ( R ‘𝐴)∃𝑦 ∈ ( L ‘𝐵)𝑛 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))}))}) |
636 | 2, 6, 1, 5 | ssltmul2 27532 |
. . . 4
⊢ (𝜑 → {(𝐴 ·s 𝐵)} <<s ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})) |
637 | 634, 636 | eqbrtrrd 5166 |
. . 3
⊢ (𝜑 → {(({𝑒 ∣ ∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑒 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔))} ∪ {ℎ ∣ ∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵)ℎ = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗))}) |s ({𝑘 ∣ ∃𝑙 ∈ ( L ‘𝐴)∃𝑚 ∈ ( R ‘𝐵)𝑘 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))} ∪ {𝑛 ∣ ∃𝑥 ∈ ( R ‘𝐴)∃𝑦 ∈ ( L ‘𝐵)𝑛 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))}))} <<s ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})) |
638 | 11, 323, 632, 635, 637 | cofcut1d 27337 |
. 2
⊢ (𝜑 → (({𝑒 ∣ ∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑒 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔))} ∪ {ℎ ∣ ∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵)ℎ = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗))}) |s ({𝑘 ∣ ∃𝑙 ∈ ( L ‘𝐴)∃𝑚 ∈ ( R ‘𝐵)𝑘 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))} ∪ {𝑛 ∣ ∃𝑥 ∈ ( R ‘𝐴)∃𝑦 ∈ ( L ‘𝐵)𝑛 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))})) = (({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}))) |
639 | 10, 638 | eqtrd 2772 |
1
⊢ (𝜑 → (𝐴 ·s 𝐵) = (({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}))) |