Step | Hyp | Ref
| Expression |
1 | | addsunif.3 |
. . . 4
⊢ (𝜑 → 𝐴 = (𝐿 |s 𝑅)) |
2 | | addsunif.1 |
. . . . 5
⊢ (𝜑 → 𝐿 <<s 𝑅) |
3 | 2 | scutcld 27138 |
. . . 4
⊢ (𝜑 → (𝐿 |s 𝑅) ∈ No
) |
4 | 1, 3 | eqeltrd 2838 |
. . 3
⊢ (𝜑 → 𝐴 ∈ No
) |
5 | | addsunif.4 |
. . . 4
⊢ (𝜑 → 𝐵 = (𝑀 |s 𝑆)) |
6 | | addsunif.2 |
. . . . 5
⊢ (𝜑 → 𝑀 <<s 𝑆) |
7 | 6 | scutcld 27138 |
. . . 4
⊢ (𝜑 → (𝑀 |s 𝑆) ∈ No
) |
8 | 5, 7 | eqeltrd 2838 |
. . 3
⊢ (𝜑 → 𝐵 ∈ No
) |
9 | | addsval2 34301 |
. . 3
⊢ ((𝐴 ∈
No ∧ 𝐵 ∈
No ) → (𝐴 +s 𝐵) = (({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)𝑎 = (𝑝 +s 𝐵)} ∪ {𝑏 ∣ ∃𝑞 ∈ ( L ‘𝐵)𝑏 = (𝐴 +s 𝑞)}) |s ({𝑐 ∣ ∃𝑒 ∈ ( R ‘𝐴)𝑐 = (𝑒 +s 𝐵)} ∪ {𝑑 ∣ ∃𝑓 ∈ ( R ‘𝐵)𝑑 = (𝐴 +s 𝑓)}))) |
10 | 4, 8, 9 | syl2anc 584 |
. 2
⊢ (𝜑 → (𝐴 +s 𝐵) = (({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)𝑎 = (𝑝 +s 𝐵)} ∪ {𝑏 ∣ ∃𝑞 ∈ ( L ‘𝐵)𝑏 = (𝐴 +s 𝑞)}) |s ({𝑐 ∣ ∃𝑒 ∈ ( R ‘𝐴)𝑐 = (𝑒 +s 𝐵)} ∪ {𝑑 ∣ ∃𝑓 ∈ ( R ‘𝐵)𝑑 = (𝐴 +s 𝑓)}))) |
11 | 4, 8 | addscut 34315 |
. . . . 5
⊢ (𝜑 → ((𝐴 +s 𝐵) ∈ No
∧ ({𝑎 ∣
∃𝑝 ∈ ( L
‘𝐴)𝑎 = (𝑝 +s 𝐵)} ∪ {𝑏 ∣ ∃𝑞 ∈ ( L ‘𝐵)𝑏 = (𝐴 +s 𝑞)}) <<s {(𝐴 +s 𝐵)} ∧ {(𝐴 +s 𝐵)} <<s ({𝑐 ∣ ∃𝑒 ∈ ( R ‘𝐴)𝑐 = (𝑒 +s 𝐵)} ∪ {𝑑 ∣ ∃𝑓 ∈ ( R ‘𝐵)𝑑 = (𝐴 +s 𝑓)}))) |
12 | 11 | simp2d 1143 |
. . . 4
⊢ (𝜑 → ({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)𝑎 = (𝑝 +s 𝐵)} ∪ {𝑏 ∣ ∃𝑞 ∈ ( L ‘𝐵)𝑏 = (𝐴 +s 𝑞)}) <<s {(𝐴 +s 𝐵)}) |
13 | 11 | simp3d 1144 |
. . . 4
⊢ (𝜑 → {(𝐴 +s 𝐵)} <<s ({𝑐 ∣ ∃𝑒 ∈ ( R ‘𝐴)𝑐 = (𝑒 +s 𝐵)} ∪ {𝑑 ∣ ∃𝑓 ∈ ( R ‘𝐵)𝑑 = (𝐴 +s 𝑓)})) |
14 | | ovex 7387 |
. . . . . 6
⊢ (𝐴 +s 𝐵) ∈ V |
15 | 14 | snnz 4736 |
. . . . 5
⊢ {(𝐴 +s 𝐵)} ≠ ∅ |
16 | | sslttr 27142 |
. . . . 5
⊢ ((({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)𝑎 = (𝑝 +s 𝐵)} ∪ {𝑏 ∣ ∃𝑞 ∈ ( L ‘𝐵)𝑏 = (𝐴 +s 𝑞)}) <<s {(𝐴 +s 𝐵)} ∧ {(𝐴 +s 𝐵)} <<s ({𝑐 ∣ ∃𝑒 ∈ ( R ‘𝐴)𝑐 = (𝑒 +s 𝐵)} ∪ {𝑑 ∣ ∃𝑓 ∈ ( R ‘𝐵)𝑑 = (𝐴 +s 𝑓)}) ∧ {(𝐴 +s 𝐵)} ≠ ∅) → ({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)𝑎 = (𝑝 +s 𝐵)} ∪ {𝑏 ∣ ∃𝑞 ∈ ( L ‘𝐵)𝑏 = (𝐴 +s 𝑞)}) <<s ({𝑐 ∣ ∃𝑒 ∈ ( R ‘𝐴)𝑐 = (𝑒 +s 𝐵)} ∪ {𝑑 ∣ ∃𝑓 ∈ ( R ‘𝐵)𝑑 = (𝐴 +s 𝑓)})) |
17 | 15, 16 | mp3an3 1450 |
. . . 4
⊢ ((({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)𝑎 = (𝑝 +s 𝐵)} ∪ {𝑏 ∣ ∃𝑞 ∈ ( L ‘𝐵)𝑏 = (𝐴 +s 𝑞)}) <<s {(𝐴 +s 𝐵)} ∧ {(𝐴 +s 𝐵)} <<s ({𝑐 ∣ ∃𝑒 ∈ ( R ‘𝐴)𝑐 = (𝑒 +s 𝐵)} ∪ {𝑑 ∣ ∃𝑓 ∈ ( R ‘𝐵)𝑑 = (𝐴 +s 𝑓)})) → ({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)𝑎 = (𝑝 +s 𝐵)} ∪ {𝑏 ∣ ∃𝑞 ∈ ( L ‘𝐵)𝑏 = (𝐴 +s 𝑞)}) <<s ({𝑐 ∣ ∃𝑒 ∈ ( R ‘𝐴)𝑐 = (𝑒 +s 𝐵)} ∪ {𝑑 ∣ ∃𝑓 ∈ ( R ‘𝐵)𝑑 = (𝐴 +s 𝑓)})) |
18 | 12, 13, 17 | syl2anc 584 |
. . 3
⊢ (𝜑 → ({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)𝑎 = (𝑝 +s 𝐵)} ∪ {𝑏 ∣ ∃𝑞 ∈ ( L ‘𝐵)𝑏 = (𝐴 +s 𝑞)}) <<s ({𝑐 ∣ ∃𝑒 ∈ ( R ‘𝐴)𝑐 = (𝑒 +s 𝐵)} ∪ {𝑑 ∣ ∃𝑓 ∈ ( R ‘𝐵)𝑑 = (𝐴 +s 𝑓)})) |
19 | 2, 1 | cofcutr1d 27240 |
. . . . . 6
⊢ (𝜑 → ∀𝑝 ∈ ( L ‘𝐴)∃𝑙 ∈ 𝐿 𝑝 ≤s 𝑙) |
20 | | leftssno 27206 |
. . . . . . . . . . 11
⊢ ( L
‘𝐴) ⊆ No |
21 | 20 | sseli 3939 |
. . . . . . . . . 10
⊢ (𝑝 ∈ ( L ‘𝐴) → 𝑝 ∈ No
) |
22 | 21 | ad2antlr 725 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑝 ∈ ( L ‘𝐴)) ∧ 𝑙 ∈ 𝐿) → 𝑝 ∈ No
) |
23 | | ssltss1 27124 |
. . . . . . . . . . . 12
⊢ (𝐿 <<s 𝑅 → 𝐿 ⊆ No
) |
24 | 2, 23 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐿 ⊆ No
) |
25 | 24 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑝 ∈ ( L ‘𝐴)) → 𝐿 ⊆ No
) |
26 | 25 | sselda 3943 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑝 ∈ ( L ‘𝐴)) ∧ 𝑙 ∈ 𝐿) → 𝑙 ∈ No
) |
27 | 8 | ad2antrr 724 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑝 ∈ ( L ‘𝐴)) ∧ 𝑙 ∈ 𝐿) → 𝐵 ∈ No
) |
28 | 22, 26, 27 | sleadd1d 34330 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑝 ∈ ( L ‘𝐴)) ∧ 𝑙 ∈ 𝐿) → (𝑝 ≤s 𝑙 ↔ (𝑝 +s 𝐵) ≤s (𝑙 +s 𝐵))) |
29 | 28 | rexbidva 3172 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑝 ∈ ( L ‘𝐴)) → (∃𝑙 ∈ 𝐿 𝑝 ≤s 𝑙 ↔ ∃𝑙 ∈ 𝐿 (𝑝 +s 𝐵) ≤s (𝑙 +s 𝐵))) |
30 | 29 | ralbidva 3171 |
. . . . . 6
⊢ (𝜑 → (∀𝑝 ∈ ( L ‘𝐴)∃𝑙 ∈ 𝐿 𝑝 ≤s 𝑙 ↔ ∀𝑝 ∈ ( L ‘𝐴)∃𝑙 ∈ 𝐿 (𝑝 +s 𝐵) ≤s (𝑙 +s 𝐵))) |
31 | 19, 30 | mpbid 231 |
. . . . 5
⊢ (𝜑 → ∀𝑝 ∈ ( L ‘𝐴)∃𝑙 ∈ 𝐿 (𝑝 +s 𝐵) ≤s (𝑙 +s 𝐵)) |
32 | | eqeq1 2740 |
. . . . . . . . . 10
⊢ (𝑦 = 𝑠 → (𝑦 = (𝑙 +s 𝐵) ↔ 𝑠 = (𝑙 +s 𝐵))) |
33 | 32 | rexbidv 3174 |
. . . . . . . . 9
⊢ (𝑦 = 𝑠 → (∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵) ↔ ∃𝑙 ∈ 𝐿 𝑠 = (𝑙 +s 𝐵))) |
34 | 33 | rexab 3651 |
. . . . . . . 8
⊢
(∃𝑠 ∈
{𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} (𝑝 +s 𝐵) ≤s 𝑠 ↔ ∃𝑠(∃𝑙 ∈ 𝐿 𝑠 = (𝑙 +s 𝐵) ∧ (𝑝 +s 𝐵) ≤s 𝑠)) |
35 | | rexcom4 3270 |
. . . . . . . . 9
⊢
(∃𝑙 ∈
𝐿 ∃𝑠(𝑠 = (𝑙 +s 𝐵) ∧ (𝑝 +s 𝐵) ≤s 𝑠) ↔ ∃𝑠∃𝑙 ∈ 𝐿 (𝑠 = (𝑙 +s 𝐵) ∧ (𝑝 +s 𝐵) ≤s 𝑠)) |
36 | | ovex 7387 |
. . . . . . . . . . 11
⊢ (𝑙 +s 𝐵) ∈ V |
37 | | breq2 5108 |
. . . . . . . . . . 11
⊢ (𝑠 = (𝑙 +s 𝐵) → ((𝑝 +s 𝐵) ≤s 𝑠 ↔ (𝑝 +s 𝐵) ≤s (𝑙 +s 𝐵))) |
38 | 36, 37 | ceqsexv 3493 |
. . . . . . . . . 10
⊢
(∃𝑠(𝑠 = (𝑙 +s 𝐵) ∧ (𝑝 +s 𝐵) ≤s 𝑠) ↔ (𝑝 +s 𝐵) ≤s (𝑙 +s 𝐵)) |
39 | 38 | rexbii 3096 |
. . . . . . . . 9
⊢
(∃𝑙 ∈
𝐿 ∃𝑠(𝑠 = (𝑙 +s 𝐵) ∧ (𝑝 +s 𝐵) ≤s 𝑠) ↔ ∃𝑙 ∈ 𝐿 (𝑝 +s 𝐵) ≤s (𝑙 +s 𝐵)) |
40 | | r19.41v 3184 |
. . . . . . . . . 10
⊢
(∃𝑙 ∈
𝐿 (𝑠 = (𝑙 +s 𝐵) ∧ (𝑝 +s 𝐵) ≤s 𝑠) ↔ (∃𝑙 ∈ 𝐿 𝑠 = (𝑙 +s 𝐵) ∧ (𝑝 +s 𝐵) ≤s 𝑠)) |
41 | 40 | exbii 1850 |
. . . . . . . . 9
⊢
(∃𝑠∃𝑙 ∈ 𝐿 (𝑠 = (𝑙 +s 𝐵) ∧ (𝑝 +s 𝐵) ≤s 𝑠) ↔ ∃𝑠(∃𝑙 ∈ 𝐿 𝑠 = (𝑙 +s 𝐵) ∧ (𝑝 +s 𝐵) ≤s 𝑠)) |
42 | 35, 39, 41 | 3bitr3ri 301 |
. . . . . . . 8
⊢
(∃𝑠(∃𝑙 ∈ 𝐿 𝑠 = (𝑙 +s 𝐵) ∧ (𝑝 +s 𝐵) ≤s 𝑠) ↔ ∃𝑙 ∈ 𝐿 (𝑝 +s 𝐵) ≤s (𝑙 +s 𝐵)) |
43 | 34, 42 | bitri 274 |
. . . . . . 7
⊢
(∃𝑠 ∈
{𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} (𝑝 +s 𝐵) ≤s 𝑠 ↔ ∃𝑙 ∈ 𝐿 (𝑝 +s 𝐵) ≤s (𝑙 +s 𝐵)) |
44 | | ssun1 4131 |
. . . . . . . 8
⊢ {𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ⊆ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)}) |
45 | | ssrexv 4010 |
. . . . . . . 8
⊢ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ⊆ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)}) → (∃𝑠 ∈ {𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} (𝑝 +s 𝐵) ≤s 𝑠 → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝑝 +s 𝐵) ≤s 𝑠)) |
46 | 44, 45 | ax-mp 5 |
. . . . . . 7
⊢
(∃𝑠 ∈
{𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} (𝑝 +s 𝐵) ≤s 𝑠 → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝑝 +s 𝐵) ≤s 𝑠) |
47 | 43, 46 | sylbir 234 |
. . . . . 6
⊢
(∃𝑙 ∈
𝐿 (𝑝 +s 𝐵) ≤s (𝑙 +s 𝐵) → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝑝 +s 𝐵) ≤s 𝑠) |
48 | 47 | ralimi 3085 |
. . . . 5
⊢
(∀𝑝 ∈ (
L ‘𝐴)∃𝑙 ∈ 𝐿 (𝑝 +s 𝐵) ≤s (𝑙 +s 𝐵) → ∀𝑝 ∈ ( L ‘𝐴)∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝑝 +s 𝐵) ≤s 𝑠) |
49 | 31, 48 | syl 17 |
. . . 4
⊢ (𝜑 → ∀𝑝 ∈ ( L ‘𝐴)∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝑝 +s 𝐵) ≤s 𝑠) |
50 | 6, 5 | cofcutr1d 27240 |
. . . . . 6
⊢ (𝜑 → ∀𝑞 ∈ ( L ‘𝐵)∃𝑚 ∈ 𝑀 𝑞 ≤s 𝑚) |
51 | | leftssno 27206 |
. . . . . . . . . . 11
⊢ ( L
‘𝐵) ⊆ No |
52 | 51 | sseli 3939 |
. . . . . . . . . 10
⊢ (𝑞 ∈ ( L ‘𝐵) → 𝑞 ∈ No
) |
53 | 52 | ad2antlr 725 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑞 ∈ ( L ‘𝐵)) ∧ 𝑚 ∈ 𝑀) → 𝑞 ∈ No
) |
54 | | ssltss1 27124 |
. . . . . . . . . . . 12
⊢ (𝑀 <<s 𝑆 → 𝑀 ⊆ No
) |
55 | 6, 54 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑀 ⊆ No
) |
56 | 55 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑞 ∈ ( L ‘𝐵)) → 𝑀 ⊆ No
) |
57 | 56 | sselda 3943 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑞 ∈ ( L ‘𝐵)) ∧ 𝑚 ∈ 𝑀) → 𝑚 ∈ No
) |
58 | 4 | ad2antrr 724 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑞 ∈ ( L ‘𝐵)) ∧ 𝑚 ∈ 𝑀) → 𝐴 ∈ No
) |
59 | 53, 57, 58 | sleadd2d 34331 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑞 ∈ ( L ‘𝐵)) ∧ 𝑚 ∈ 𝑀) → (𝑞 ≤s 𝑚 ↔ (𝐴 +s 𝑞) ≤s (𝐴 +s 𝑚))) |
60 | 59 | rexbidva 3172 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑞 ∈ ( L ‘𝐵)) → (∃𝑚 ∈ 𝑀 𝑞 ≤s 𝑚 ↔ ∃𝑚 ∈ 𝑀 (𝐴 +s 𝑞) ≤s (𝐴 +s 𝑚))) |
61 | 60 | ralbidva 3171 |
. . . . . 6
⊢ (𝜑 → (∀𝑞 ∈ ( L ‘𝐵)∃𝑚 ∈ 𝑀 𝑞 ≤s 𝑚 ↔ ∀𝑞 ∈ ( L ‘𝐵)∃𝑚 ∈ 𝑀 (𝐴 +s 𝑞) ≤s (𝐴 +s 𝑚))) |
62 | 50, 61 | mpbid 231 |
. . . . 5
⊢ (𝜑 → ∀𝑞 ∈ ( L ‘𝐵)∃𝑚 ∈ 𝑀 (𝐴 +s 𝑞) ≤s (𝐴 +s 𝑚)) |
63 | | eqeq1 2740 |
. . . . . . . . . 10
⊢ (𝑧 = 𝑠 → (𝑧 = (𝐴 +s 𝑚) ↔ 𝑠 = (𝐴 +s 𝑚))) |
64 | 63 | rexbidv 3174 |
. . . . . . . . 9
⊢ (𝑧 = 𝑠 → (∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚) ↔ ∃𝑚 ∈ 𝑀 𝑠 = (𝐴 +s 𝑚))) |
65 | 64 | rexab 3651 |
. . . . . . . 8
⊢
(∃𝑠 ∈
{𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)} (𝐴 +s 𝑞) ≤s 𝑠 ↔ ∃𝑠(∃𝑚 ∈ 𝑀 𝑠 = (𝐴 +s 𝑚) ∧ (𝐴 +s 𝑞) ≤s 𝑠)) |
66 | | rexcom4 3270 |
. . . . . . . . 9
⊢
(∃𝑚 ∈
𝑀 ∃𝑠(𝑠 = (𝐴 +s 𝑚) ∧ (𝐴 +s 𝑞) ≤s 𝑠) ↔ ∃𝑠∃𝑚 ∈ 𝑀 (𝑠 = (𝐴 +s 𝑚) ∧ (𝐴 +s 𝑞) ≤s 𝑠)) |
67 | | ovex 7387 |
. . . . . . . . . . 11
⊢ (𝐴 +s 𝑚) ∈ V |
68 | | breq2 5108 |
. . . . . . . . . . 11
⊢ (𝑠 = (𝐴 +s 𝑚) → ((𝐴 +s 𝑞) ≤s 𝑠 ↔ (𝐴 +s 𝑞) ≤s (𝐴 +s 𝑚))) |
69 | 67, 68 | ceqsexv 3493 |
. . . . . . . . . 10
⊢
(∃𝑠(𝑠 = (𝐴 +s 𝑚) ∧ (𝐴 +s 𝑞) ≤s 𝑠) ↔ (𝐴 +s 𝑞) ≤s (𝐴 +s 𝑚)) |
70 | 69 | rexbii 3096 |
. . . . . . . . 9
⊢
(∃𝑚 ∈
𝑀 ∃𝑠(𝑠 = (𝐴 +s 𝑚) ∧ (𝐴 +s 𝑞) ≤s 𝑠) ↔ ∃𝑚 ∈ 𝑀 (𝐴 +s 𝑞) ≤s (𝐴 +s 𝑚)) |
71 | | r19.41v 3184 |
. . . . . . . . . 10
⊢
(∃𝑚 ∈
𝑀 (𝑠 = (𝐴 +s 𝑚) ∧ (𝐴 +s 𝑞) ≤s 𝑠) ↔ (∃𝑚 ∈ 𝑀 𝑠 = (𝐴 +s 𝑚) ∧ (𝐴 +s 𝑞) ≤s 𝑠)) |
72 | 71 | exbii 1850 |
. . . . . . . . 9
⊢
(∃𝑠∃𝑚 ∈ 𝑀 (𝑠 = (𝐴 +s 𝑚) ∧ (𝐴 +s 𝑞) ≤s 𝑠) ↔ ∃𝑠(∃𝑚 ∈ 𝑀 𝑠 = (𝐴 +s 𝑚) ∧ (𝐴 +s 𝑞) ≤s 𝑠)) |
73 | 66, 70, 72 | 3bitr3ri 301 |
. . . . . . . 8
⊢
(∃𝑠(∃𝑚 ∈ 𝑀 𝑠 = (𝐴 +s 𝑚) ∧ (𝐴 +s 𝑞) ≤s 𝑠) ↔ ∃𝑚 ∈ 𝑀 (𝐴 +s 𝑞) ≤s (𝐴 +s 𝑚)) |
74 | 65, 73 | bitri 274 |
. . . . . . 7
⊢
(∃𝑠 ∈
{𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)} (𝐴 +s 𝑞) ≤s 𝑠 ↔ ∃𝑚 ∈ 𝑀 (𝐴 +s 𝑞) ≤s (𝐴 +s 𝑚)) |
75 | | ssun2 4132 |
. . . . . . . 8
⊢ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)} ⊆ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)}) |
76 | | ssrexv 4010 |
. . . . . . . 8
⊢ ({𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)} ⊆ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)}) → (∃𝑠 ∈ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)} (𝐴 +s 𝑞) ≤s 𝑠 → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝐴 +s 𝑞) ≤s 𝑠)) |
77 | 75, 76 | ax-mp 5 |
. . . . . . 7
⊢
(∃𝑠 ∈
{𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)} (𝐴 +s 𝑞) ≤s 𝑠 → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝐴 +s 𝑞) ≤s 𝑠) |
78 | 74, 77 | sylbir 234 |
. . . . . 6
⊢
(∃𝑚 ∈
𝑀 (𝐴 +s 𝑞) ≤s (𝐴 +s 𝑚) → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝐴 +s 𝑞) ≤s 𝑠) |
79 | 78 | ralimi 3085 |
. . . . 5
⊢
(∀𝑞 ∈ (
L ‘𝐵)∃𝑚 ∈ 𝑀 (𝐴 +s 𝑞) ≤s (𝐴 +s 𝑚) → ∀𝑞 ∈ ( L ‘𝐵)∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝐴 +s 𝑞) ≤s 𝑠) |
80 | 62, 79 | syl 17 |
. . . 4
⊢ (𝜑 → ∀𝑞 ∈ ( L ‘𝐵)∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝐴 +s 𝑞) ≤s 𝑠) |
81 | | ralunb 4150 |
. . . . 5
⊢
(∀𝑟 ∈
({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)𝑎 = (𝑝 +s 𝐵)} ∪ {𝑏 ∣ ∃𝑞 ∈ ( L ‘𝐵)𝑏 = (𝐴 +s 𝑞)})∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠 ↔ (∀𝑟 ∈ {𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)𝑎 = (𝑝 +s 𝐵)}∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠 ∧ ∀𝑟 ∈ {𝑏 ∣ ∃𝑞 ∈ ( L ‘𝐵)𝑏 = (𝐴 +s 𝑞)}∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠)) |
82 | | eqeq1 2740 |
. . . . . . . . 9
⊢ (𝑎 = 𝑟 → (𝑎 = (𝑝 +s 𝐵) ↔ 𝑟 = (𝑝 +s 𝐵))) |
83 | 82 | rexbidv 3174 |
. . . . . . . 8
⊢ (𝑎 = 𝑟 → (∃𝑝 ∈ ( L ‘𝐴)𝑎 = (𝑝 +s 𝐵) ↔ ∃𝑝 ∈ ( L ‘𝐴)𝑟 = (𝑝 +s 𝐵))) |
84 | 83 | ralab 3648 |
. . . . . . 7
⊢
(∀𝑟 ∈
{𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)𝑎 = (𝑝 +s 𝐵)}∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠 ↔ ∀𝑟(∃𝑝 ∈ ( L ‘𝐴)𝑟 = (𝑝 +s 𝐵) → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠)) |
85 | | ralcom4 3268 |
. . . . . . . 8
⊢
(∀𝑝 ∈ (
L ‘𝐴)∀𝑟(𝑟 = (𝑝 +s 𝐵) → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠) ↔ ∀𝑟∀𝑝 ∈ ( L ‘𝐴)(𝑟 = (𝑝 +s 𝐵) → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠)) |
86 | | ovex 7387 |
. . . . . . . . . 10
⊢ (𝑝 +s 𝐵) ∈ V |
87 | | breq1 5107 |
. . . . . . . . . . 11
⊢ (𝑟 = (𝑝 +s 𝐵) → (𝑟 ≤s 𝑠 ↔ (𝑝 +s 𝐵) ≤s 𝑠)) |
88 | 87 | rexbidv 3174 |
. . . . . . . . . 10
⊢ (𝑟 = (𝑝 +s 𝐵) → (∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠 ↔ ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝑝 +s 𝐵) ≤s 𝑠)) |
89 | 86, 88 | ceqsalv 3480 |
. . . . . . . . 9
⊢
(∀𝑟(𝑟 = (𝑝 +s 𝐵) → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠) ↔ ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝑝 +s 𝐵) ≤s 𝑠) |
90 | 89 | ralbii 3095 |
. . . . . . . 8
⊢
(∀𝑝 ∈ (
L ‘𝐴)∀𝑟(𝑟 = (𝑝 +s 𝐵) → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠) ↔ ∀𝑝 ∈ ( L ‘𝐴)∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝑝 +s 𝐵) ≤s 𝑠) |
91 | | r19.23v 3178 |
. . . . . . . . 9
⊢
(∀𝑝 ∈ (
L ‘𝐴)(𝑟 = (𝑝 +s 𝐵) → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠) ↔ (∃𝑝 ∈ ( L ‘𝐴)𝑟 = (𝑝 +s 𝐵) → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠)) |
92 | 91 | albii 1821 |
. . . . . . . 8
⊢
(∀𝑟∀𝑝 ∈ ( L ‘𝐴)(𝑟 = (𝑝 +s 𝐵) → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠) ↔ ∀𝑟(∃𝑝 ∈ ( L ‘𝐴)𝑟 = (𝑝 +s 𝐵) → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠)) |
93 | 85, 90, 92 | 3bitr3ri 301 |
. . . . . . 7
⊢
(∀𝑟(∃𝑝 ∈ ( L ‘𝐴)𝑟 = (𝑝 +s 𝐵) → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠) ↔ ∀𝑝 ∈ ( L ‘𝐴)∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝑝 +s 𝐵) ≤s 𝑠) |
94 | 84, 93 | bitri 274 |
. . . . . 6
⊢
(∀𝑟 ∈
{𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)𝑎 = (𝑝 +s 𝐵)}∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠 ↔ ∀𝑝 ∈ ( L ‘𝐴)∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝑝 +s 𝐵) ≤s 𝑠) |
95 | | eqeq1 2740 |
. . . . . . . . 9
⊢ (𝑏 = 𝑟 → (𝑏 = (𝐴 +s 𝑞) ↔ 𝑟 = (𝐴 +s 𝑞))) |
96 | 95 | rexbidv 3174 |
. . . . . . . 8
⊢ (𝑏 = 𝑟 → (∃𝑞 ∈ ( L ‘𝐵)𝑏 = (𝐴 +s 𝑞) ↔ ∃𝑞 ∈ ( L ‘𝐵)𝑟 = (𝐴 +s 𝑞))) |
97 | 96 | ralab 3648 |
. . . . . . 7
⊢
(∀𝑟 ∈
{𝑏 ∣ ∃𝑞 ∈ ( L ‘𝐵)𝑏 = (𝐴 +s 𝑞)}∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠 ↔ ∀𝑟(∃𝑞 ∈ ( L ‘𝐵)𝑟 = (𝐴 +s 𝑞) → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠)) |
98 | | ralcom4 3268 |
. . . . . . . 8
⊢
(∀𝑞 ∈ (
L ‘𝐵)∀𝑟(𝑟 = (𝐴 +s 𝑞) → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠) ↔ ∀𝑟∀𝑞 ∈ ( L ‘𝐵)(𝑟 = (𝐴 +s 𝑞) → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠)) |
99 | | ovex 7387 |
. . . . . . . . . 10
⊢ (𝐴 +s 𝑞) ∈ V |
100 | | breq1 5107 |
. . . . . . . . . . 11
⊢ (𝑟 = (𝐴 +s 𝑞) → (𝑟 ≤s 𝑠 ↔ (𝐴 +s 𝑞) ≤s 𝑠)) |
101 | 100 | rexbidv 3174 |
. . . . . . . . . 10
⊢ (𝑟 = (𝐴 +s 𝑞) → (∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠 ↔ ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝐴 +s 𝑞) ≤s 𝑠)) |
102 | 99, 101 | ceqsalv 3480 |
. . . . . . . . 9
⊢
(∀𝑟(𝑟 = (𝐴 +s 𝑞) → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠) ↔ ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝐴 +s 𝑞) ≤s 𝑠) |
103 | 102 | ralbii 3095 |
. . . . . . . 8
⊢
(∀𝑞 ∈ (
L ‘𝐵)∀𝑟(𝑟 = (𝐴 +s 𝑞) → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠) ↔ ∀𝑞 ∈ ( L ‘𝐵)∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝐴 +s 𝑞) ≤s 𝑠) |
104 | | r19.23v 3178 |
. . . . . . . . 9
⊢
(∀𝑞 ∈ (
L ‘𝐵)(𝑟 = (𝐴 +s 𝑞) → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠) ↔ (∃𝑞 ∈ ( L ‘𝐵)𝑟 = (𝐴 +s 𝑞) → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠)) |
105 | 104 | albii 1821 |
. . . . . . . 8
⊢
(∀𝑟∀𝑞 ∈ ( L ‘𝐵)(𝑟 = (𝐴 +s 𝑞) → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠) ↔ ∀𝑟(∃𝑞 ∈ ( L ‘𝐵)𝑟 = (𝐴 +s 𝑞) → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠)) |
106 | 98, 103, 105 | 3bitr3ri 301 |
. . . . . . 7
⊢
(∀𝑟(∃𝑞 ∈ ( L ‘𝐵)𝑟 = (𝐴 +s 𝑞) → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠) ↔ ∀𝑞 ∈ ( L ‘𝐵)∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝐴 +s 𝑞) ≤s 𝑠) |
107 | 97, 106 | bitri 274 |
. . . . . 6
⊢
(∀𝑟 ∈
{𝑏 ∣ ∃𝑞 ∈ ( L ‘𝐵)𝑏 = (𝐴 +s 𝑞)}∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠 ↔ ∀𝑞 ∈ ( L ‘𝐵)∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝐴 +s 𝑞) ≤s 𝑠) |
108 | 94, 107 | anbi12i 627 |
. . . . 5
⊢
((∀𝑟 ∈
{𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)𝑎 = (𝑝 +s 𝐵)}∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠 ∧ ∀𝑟 ∈ {𝑏 ∣ ∃𝑞 ∈ ( L ‘𝐵)𝑏 = (𝐴 +s 𝑞)}∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠) ↔ (∀𝑝 ∈ ( L ‘𝐴)∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝑝 +s 𝐵) ≤s 𝑠 ∧ ∀𝑞 ∈ ( L ‘𝐵)∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝐴 +s 𝑞) ≤s 𝑠)) |
109 | 81, 108 | bitri 274 |
. . . 4
⊢
(∀𝑟 ∈
({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)𝑎 = (𝑝 +s 𝐵)} ∪ {𝑏 ∣ ∃𝑞 ∈ ( L ‘𝐵)𝑏 = (𝐴 +s 𝑞)})∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠 ↔ (∀𝑝 ∈ ( L ‘𝐴)∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝑝 +s 𝐵) ≤s 𝑠 ∧ ∀𝑞 ∈ ( L ‘𝐵)∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝐴 +s 𝑞) ≤s 𝑠)) |
110 | 49, 80, 109 | sylanbrc 583 |
. . 3
⊢ (𝜑 → ∀𝑟 ∈ ({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)𝑎 = (𝑝 +s 𝐵)} ∪ {𝑏 ∣ ∃𝑞 ∈ ( L ‘𝐵)𝑏 = (𝐴 +s 𝑞)})∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠) |
111 | 2, 1 | cofcutr2d 27241 |
. . . . . 6
⊢ (𝜑 → ∀𝑒 ∈ ( R ‘𝐴)∃𝑟 ∈ 𝑅 𝑟 ≤s 𝑒) |
112 | | ssltss2 27125 |
. . . . . . . . . . . 12
⊢ (𝐿 <<s 𝑅 → 𝑅 ⊆ No
) |
113 | 2, 112 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑅 ⊆ No
) |
114 | 113 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑒 ∈ ( R ‘𝐴)) → 𝑅 ⊆ No
) |
115 | 114 | sselda 3943 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑒 ∈ ( R ‘𝐴)) ∧ 𝑟 ∈ 𝑅) → 𝑟 ∈ No
) |
116 | | rightssno 27207 |
. . . . . . . . . . 11
⊢ ( R
‘𝐴) ⊆ No |
117 | 116 | sseli 3939 |
. . . . . . . . . 10
⊢ (𝑒 ∈ ( R ‘𝐴) → 𝑒 ∈ No
) |
118 | 117 | ad2antlr 725 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑒 ∈ ( R ‘𝐴)) ∧ 𝑟 ∈ 𝑅) → 𝑒 ∈ No
) |
119 | 8 | ad2antrr 724 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑒 ∈ ( R ‘𝐴)) ∧ 𝑟 ∈ 𝑅) → 𝐵 ∈ No
) |
120 | 115, 118,
119 | sleadd1d 34330 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑒 ∈ ( R ‘𝐴)) ∧ 𝑟 ∈ 𝑅) → (𝑟 ≤s 𝑒 ↔ (𝑟 +s 𝐵) ≤s (𝑒 +s 𝐵))) |
121 | 120 | rexbidva 3172 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑒 ∈ ( R ‘𝐴)) → (∃𝑟 ∈ 𝑅 𝑟 ≤s 𝑒 ↔ ∃𝑟 ∈ 𝑅 (𝑟 +s 𝐵) ≤s (𝑒 +s 𝐵))) |
122 | 121 | ralbidva 3171 |
. . . . . 6
⊢ (𝜑 → (∀𝑒 ∈ ( R ‘𝐴)∃𝑟 ∈ 𝑅 𝑟 ≤s 𝑒 ↔ ∀𝑒 ∈ ( R ‘𝐴)∃𝑟 ∈ 𝑅 (𝑟 +s 𝐵) ≤s (𝑒 +s 𝐵))) |
123 | 111, 122 | mpbid 231 |
. . . . 5
⊢ (𝜑 → ∀𝑒 ∈ ( R ‘𝐴)∃𝑟 ∈ 𝑅 (𝑟 +s 𝐵) ≤s (𝑒 +s 𝐵)) |
124 | | eqeq1 2740 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑏 → (𝑤 = (𝑟 +s 𝐵) ↔ 𝑏 = (𝑟 +s 𝐵))) |
125 | 124 | rexbidv 3174 |
. . . . . . . . 9
⊢ (𝑤 = 𝑏 → (∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵) ↔ ∃𝑟 ∈ 𝑅 𝑏 = (𝑟 +s 𝐵))) |
126 | 125 | rexab 3651 |
. . . . . . . 8
⊢
(∃𝑏 ∈
{𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)}𝑏 ≤s (𝑒 +s 𝐵) ↔ ∃𝑏(∃𝑟 ∈ 𝑅 𝑏 = (𝑟 +s 𝐵) ∧ 𝑏 ≤s (𝑒 +s 𝐵))) |
127 | | rexcom4 3270 |
. . . . . . . . 9
⊢
(∃𝑟 ∈
𝑅 ∃𝑏(𝑏 = (𝑟 +s 𝐵) ∧ 𝑏 ≤s (𝑒 +s 𝐵)) ↔ ∃𝑏∃𝑟 ∈ 𝑅 (𝑏 = (𝑟 +s 𝐵) ∧ 𝑏 ≤s (𝑒 +s 𝐵))) |
128 | | ovex 7387 |
. . . . . . . . . . 11
⊢ (𝑟 +s 𝐵) ∈ V |
129 | | breq1 5107 |
. . . . . . . . . . 11
⊢ (𝑏 = (𝑟 +s 𝐵) → (𝑏 ≤s (𝑒 +s 𝐵) ↔ (𝑟 +s 𝐵) ≤s (𝑒 +s 𝐵))) |
130 | 128, 129 | ceqsexv 3493 |
. . . . . . . . . 10
⊢
(∃𝑏(𝑏 = (𝑟 +s 𝐵) ∧ 𝑏 ≤s (𝑒 +s 𝐵)) ↔ (𝑟 +s 𝐵) ≤s (𝑒 +s 𝐵)) |
131 | 130 | rexbii 3096 |
. . . . . . . . 9
⊢
(∃𝑟 ∈
𝑅 ∃𝑏(𝑏 = (𝑟 +s 𝐵) ∧ 𝑏 ≤s (𝑒 +s 𝐵)) ↔ ∃𝑟 ∈ 𝑅 (𝑟 +s 𝐵) ≤s (𝑒 +s 𝐵)) |
132 | | r19.41v 3184 |
. . . . . . . . . 10
⊢
(∃𝑟 ∈
𝑅 (𝑏 = (𝑟 +s 𝐵) ∧ 𝑏 ≤s (𝑒 +s 𝐵)) ↔ (∃𝑟 ∈ 𝑅 𝑏 = (𝑟 +s 𝐵) ∧ 𝑏 ≤s (𝑒 +s 𝐵))) |
133 | 132 | exbii 1850 |
. . . . . . . . 9
⊢
(∃𝑏∃𝑟 ∈ 𝑅 (𝑏 = (𝑟 +s 𝐵) ∧ 𝑏 ≤s (𝑒 +s 𝐵)) ↔ ∃𝑏(∃𝑟 ∈ 𝑅 𝑏 = (𝑟 +s 𝐵) ∧ 𝑏 ≤s (𝑒 +s 𝐵))) |
134 | 127, 131,
133 | 3bitr3ri 301 |
. . . . . . . 8
⊢
(∃𝑏(∃𝑟 ∈ 𝑅 𝑏 = (𝑟 +s 𝐵) ∧ 𝑏 ≤s (𝑒 +s 𝐵)) ↔ ∃𝑟 ∈ 𝑅 (𝑟 +s 𝐵) ≤s (𝑒 +s 𝐵)) |
135 | 126, 134 | bitri 274 |
. . . . . . 7
⊢
(∃𝑏 ∈
{𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)}𝑏 ≤s (𝑒 +s 𝐵) ↔ ∃𝑟 ∈ 𝑅 (𝑟 +s 𝐵) ≤s (𝑒 +s 𝐵)) |
136 | | ssun1 4131 |
. . . . . . . 8
⊢ {𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ⊆ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)}) |
137 | | ssrexv 4010 |
. . . . . . . 8
⊢ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ⊆ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)}) → (∃𝑏 ∈ {𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)}𝑏 ≤s (𝑒 +s 𝐵) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝑒 +s 𝐵))) |
138 | 136, 137 | ax-mp 5 |
. . . . . . 7
⊢
(∃𝑏 ∈
{𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)}𝑏 ≤s (𝑒 +s 𝐵) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝑒 +s 𝐵)) |
139 | 135, 138 | sylbir 234 |
. . . . . 6
⊢
(∃𝑟 ∈
𝑅 (𝑟 +s 𝐵) ≤s (𝑒 +s 𝐵) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝑒 +s 𝐵)) |
140 | 139 | ralimi 3085 |
. . . . 5
⊢
(∀𝑒 ∈ (
R ‘𝐴)∃𝑟 ∈ 𝑅 (𝑟 +s 𝐵) ≤s (𝑒 +s 𝐵) → ∀𝑒 ∈ ( R ‘𝐴)∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝑒 +s 𝐵)) |
141 | 123, 140 | syl 17 |
. . . 4
⊢ (𝜑 → ∀𝑒 ∈ ( R ‘𝐴)∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝑒 +s 𝐵)) |
142 | 6, 5 | cofcutr2d 27241 |
. . . . . 6
⊢ (𝜑 → ∀𝑓 ∈ ( R ‘𝐵)∃𝑠 ∈ 𝑆 𝑠 ≤s 𝑓) |
143 | | ssltss2 27125 |
. . . . . . . . . . . 12
⊢ (𝑀 <<s 𝑆 → 𝑆 ⊆ No
) |
144 | 6, 143 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑆 ⊆ No
) |
145 | 144 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑓 ∈ ( R ‘𝐵)) → 𝑆 ⊆ No
) |
146 | 145 | sselda 3943 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓 ∈ ( R ‘𝐵)) ∧ 𝑠 ∈ 𝑆) → 𝑠 ∈ No
) |
147 | | rightssno 27207 |
. . . . . . . . . . 11
⊢ ( R
‘𝐵) ⊆ No |
148 | 147 | sseli 3939 |
. . . . . . . . . 10
⊢ (𝑓 ∈ ( R ‘𝐵) → 𝑓 ∈ No
) |
149 | 148 | ad2antlr 725 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓 ∈ ( R ‘𝐵)) ∧ 𝑠 ∈ 𝑆) → 𝑓 ∈ No
) |
150 | 4 | ad2antrr 724 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓 ∈ ( R ‘𝐵)) ∧ 𝑠 ∈ 𝑆) → 𝐴 ∈ No
) |
151 | 146, 149,
150 | sleadd2d 34331 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑓 ∈ ( R ‘𝐵)) ∧ 𝑠 ∈ 𝑆) → (𝑠 ≤s 𝑓 ↔ (𝐴 +s 𝑠) ≤s (𝐴 +s 𝑓))) |
152 | 151 | rexbidva 3172 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑓 ∈ ( R ‘𝐵)) → (∃𝑠 ∈ 𝑆 𝑠 ≤s 𝑓 ↔ ∃𝑠 ∈ 𝑆 (𝐴 +s 𝑠) ≤s (𝐴 +s 𝑓))) |
153 | 152 | ralbidva 3171 |
. . . . . 6
⊢ (𝜑 → (∀𝑓 ∈ ( R ‘𝐵)∃𝑠 ∈ 𝑆 𝑠 ≤s 𝑓 ↔ ∀𝑓 ∈ ( R ‘𝐵)∃𝑠 ∈ 𝑆 (𝐴 +s 𝑠) ≤s (𝐴 +s 𝑓))) |
154 | 142, 153 | mpbid 231 |
. . . . 5
⊢ (𝜑 → ∀𝑓 ∈ ( R ‘𝐵)∃𝑠 ∈ 𝑆 (𝐴 +s 𝑠) ≤s (𝐴 +s 𝑓)) |
155 | | eqeq1 2740 |
. . . . . . . . . 10
⊢ (𝑡 = 𝑏 → (𝑡 = (𝐴 +s 𝑠) ↔ 𝑏 = (𝐴 +s 𝑠))) |
156 | 155 | rexbidv 3174 |
. . . . . . . . 9
⊢ (𝑡 = 𝑏 → (∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠) ↔ ∃𝑠 ∈ 𝑆 𝑏 = (𝐴 +s 𝑠))) |
157 | 156 | rexab 3651 |
. . . . . . . 8
⊢
(∃𝑏 ∈
{𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)}𝑏 ≤s (𝐴 +s 𝑓) ↔ ∃𝑏(∃𝑠 ∈ 𝑆 𝑏 = (𝐴 +s 𝑠) ∧ 𝑏 ≤s (𝐴 +s 𝑓))) |
158 | | rexcom4 3270 |
. . . . . . . . 9
⊢
(∃𝑠 ∈
𝑆 ∃𝑏(𝑏 = (𝐴 +s 𝑠) ∧ 𝑏 ≤s (𝐴 +s 𝑓)) ↔ ∃𝑏∃𝑠 ∈ 𝑆 (𝑏 = (𝐴 +s 𝑠) ∧ 𝑏 ≤s (𝐴 +s 𝑓))) |
159 | | ovex 7387 |
. . . . . . . . . . 11
⊢ (𝐴 +s 𝑠) ∈ V |
160 | | breq1 5107 |
. . . . . . . . . . 11
⊢ (𝑏 = (𝐴 +s 𝑠) → (𝑏 ≤s (𝐴 +s 𝑓) ↔ (𝐴 +s 𝑠) ≤s (𝐴 +s 𝑓))) |
161 | 159, 160 | ceqsexv 3493 |
. . . . . . . . . 10
⊢
(∃𝑏(𝑏 = (𝐴 +s 𝑠) ∧ 𝑏 ≤s (𝐴 +s 𝑓)) ↔ (𝐴 +s 𝑠) ≤s (𝐴 +s 𝑓)) |
162 | 161 | rexbii 3096 |
. . . . . . . . 9
⊢
(∃𝑠 ∈
𝑆 ∃𝑏(𝑏 = (𝐴 +s 𝑠) ∧ 𝑏 ≤s (𝐴 +s 𝑓)) ↔ ∃𝑠 ∈ 𝑆 (𝐴 +s 𝑠) ≤s (𝐴 +s 𝑓)) |
163 | | r19.41v 3184 |
. . . . . . . . . 10
⊢
(∃𝑠 ∈
𝑆 (𝑏 = (𝐴 +s 𝑠) ∧ 𝑏 ≤s (𝐴 +s 𝑓)) ↔ (∃𝑠 ∈ 𝑆 𝑏 = (𝐴 +s 𝑠) ∧ 𝑏 ≤s (𝐴 +s 𝑓))) |
164 | 163 | exbii 1850 |
. . . . . . . . 9
⊢
(∃𝑏∃𝑠 ∈ 𝑆 (𝑏 = (𝐴 +s 𝑠) ∧ 𝑏 ≤s (𝐴 +s 𝑓)) ↔ ∃𝑏(∃𝑠 ∈ 𝑆 𝑏 = (𝐴 +s 𝑠) ∧ 𝑏 ≤s (𝐴 +s 𝑓))) |
165 | 158, 162,
164 | 3bitr3ri 301 |
. . . . . . . 8
⊢
(∃𝑏(∃𝑠 ∈ 𝑆 𝑏 = (𝐴 +s 𝑠) ∧ 𝑏 ≤s (𝐴 +s 𝑓)) ↔ ∃𝑠 ∈ 𝑆 (𝐴 +s 𝑠) ≤s (𝐴 +s 𝑓)) |
166 | 157, 165 | bitri 274 |
. . . . . . 7
⊢
(∃𝑏 ∈
{𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)}𝑏 ≤s (𝐴 +s 𝑓) ↔ ∃𝑠 ∈ 𝑆 (𝐴 +s 𝑠) ≤s (𝐴 +s 𝑓)) |
167 | | ssun2 4132 |
. . . . . . . 8
⊢ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)} ⊆ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)}) |
168 | | ssrexv 4010 |
. . . . . . . 8
⊢ ({𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)} ⊆ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)}) → (∃𝑏 ∈ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)}𝑏 ≤s (𝐴 +s 𝑓) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝐴 +s 𝑓))) |
169 | 167, 168 | ax-mp 5 |
. . . . . . 7
⊢
(∃𝑏 ∈
{𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)}𝑏 ≤s (𝐴 +s 𝑓) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝐴 +s 𝑓)) |
170 | 166, 169 | sylbir 234 |
. . . . . 6
⊢
(∃𝑠 ∈
𝑆 (𝐴 +s 𝑠) ≤s (𝐴 +s 𝑓) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝐴 +s 𝑓)) |
171 | 170 | ralimi 3085 |
. . . . 5
⊢
(∀𝑓 ∈ (
R ‘𝐵)∃𝑠 ∈ 𝑆 (𝐴 +s 𝑠) ≤s (𝐴 +s 𝑓) → ∀𝑓 ∈ ( R ‘𝐵)∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝐴 +s 𝑓)) |
172 | 154, 171 | syl 17 |
. . . 4
⊢ (𝜑 → ∀𝑓 ∈ ( R ‘𝐵)∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝐴 +s 𝑓)) |
173 | | ralunb 4150 |
. . . . 5
⊢
(∀𝑎 ∈
({𝑐 ∣ ∃𝑒 ∈ ( R ‘𝐴)𝑐 = (𝑒 +s 𝐵)} ∪ {𝑑 ∣ ∃𝑓 ∈ ( R ‘𝐵)𝑑 = (𝐴 +s 𝑓)})∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎 ↔ (∀𝑎 ∈ {𝑐 ∣ ∃𝑒 ∈ ( R ‘𝐴)𝑐 = (𝑒 +s 𝐵)}∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎 ∧ ∀𝑎 ∈ {𝑑 ∣ ∃𝑓 ∈ ( R ‘𝐵)𝑑 = (𝐴 +s 𝑓)}∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎)) |
174 | | eqeq1 2740 |
. . . . . . . . 9
⊢ (𝑐 = 𝑎 → (𝑐 = (𝑒 +s 𝐵) ↔ 𝑎 = (𝑒 +s 𝐵))) |
175 | 174 | rexbidv 3174 |
. . . . . . . 8
⊢ (𝑐 = 𝑎 → (∃𝑒 ∈ ( R ‘𝐴)𝑐 = (𝑒 +s 𝐵) ↔ ∃𝑒 ∈ ( R ‘𝐴)𝑎 = (𝑒 +s 𝐵))) |
176 | 175 | ralab 3648 |
. . . . . . 7
⊢
(∀𝑎 ∈
{𝑐 ∣ ∃𝑒 ∈ ( R ‘𝐴)𝑐 = (𝑒 +s 𝐵)}∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎 ↔ ∀𝑎(∃𝑒 ∈ ( R ‘𝐴)𝑎 = (𝑒 +s 𝐵) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎)) |
177 | | ralcom4 3268 |
. . . . . . . 8
⊢
(∀𝑒 ∈ (
R ‘𝐴)∀𝑎(𝑎 = (𝑒 +s 𝐵) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎) ↔ ∀𝑎∀𝑒 ∈ ( R ‘𝐴)(𝑎 = (𝑒 +s 𝐵) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎)) |
178 | | ovex 7387 |
. . . . . . . . . 10
⊢ (𝑒 +s 𝐵) ∈ V |
179 | | breq2 5108 |
. . . . . . . . . . 11
⊢ (𝑎 = (𝑒 +s 𝐵) → (𝑏 ≤s 𝑎 ↔ 𝑏 ≤s (𝑒 +s 𝐵))) |
180 | 179 | rexbidv 3174 |
. . . . . . . . . 10
⊢ (𝑎 = (𝑒 +s 𝐵) → (∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎 ↔ ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝑒 +s 𝐵))) |
181 | 178, 180 | ceqsalv 3480 |
. . . . . . . . 9
⊢
(∀𝑎(𝑎 = (𝑒 +s 𝐵) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎) ↔ ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝑒 +s 𝐵)) |
182 | 181 | ralbii 3095 |
. . . . . . . 8
⊢
(∀𝑒 ∈ (
R ‘𝐴)∀𝑎(𝑎 = (𝑒 +s 𝐵) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎) ↔ ∀𝑒 ∈ ( R ‘𝐴)∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝑒 +s 𝐵)) |
183 | | r19.23v 3178 |
. . . . . . . . 9
⊢
(∀𝑒 ∈ (
R ‘𝐴)(𝑎 = (𝑒 +s 𝐵) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎) ↔ (∃𝑒 ∈ ( R ‘𝐴)𝑎 = (𝑒 +s 𝐵) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎)) |
184 | 183 | albii 1821 |
. . . . . . . 8
⊢
(∀𝑎∀𝑒 ∈ ( R ‘𝐴)(𝑎 = (𝑒 +s 𝐵) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎) ↔ ∀𝑎(∃𝑒 ∈ ( R ‘𝐴)𝑎 = (𝑒 +s 𝐵) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎)) |
185 | 177, 182,
184 | 3bitr3ri 301 |
. . . . . . 7
⊢
(∀𝑎(∃𝑒 ∈ ( R ‘𝐴)𝑎 = (𝑒 +s 𝐵) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎) ↔ ∀𝑒 ∈ ( R ‘𝐴)∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝑒 +s 𝐵)) |
186 | 176, 185 | bitri 274 |
. . . . . 6
⊢
(∀𝑎 ∈
{𝑐 ∣ ∃𝑒 ∈ ( R ‘𝐴)𝑐 = (𝑒 +s 𝐵)}∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎 ↔ ∀𝑒 ∈ ( R ‘𝐴)∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝑒 +s 𝐵)) |
187 | | eqeq1 2740 |
. . . . . . . . 9
⊢ (𝑑 = 𝑎 → (𝑑 = (𝐴 +s 𝑓) ↔ 𝑎 = (𝐴 +s 𝑓))) |
188 | 187 | rexbidv 3174 |
. . . . . . . 8
⊢ (𝑑 = 𝑎 → (∃𝑓 ∈ ( R ‘𝐵)𝑑 = (𝐴 +s 𝑓) ↔ ∃𝑓 ∈ ( R ‘𝐵)𝑎 = (𝐴 +s 𝑓))) |
189 | 188 | ralab 3648 |
. . . . . . 7
⊢
(∀𝑎 ∈
{𝑑 ∣ ∃𝑓 ∈ ( R ‘𝐵)𝑑 = (𝐴 +s 𝑓)}∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎 ↔ ∀𝑎(∃𝑓 ∈ ( R ‘𝐵)𝑎 = (𝐴 +s 𝑓) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎)) |
190 | | ralcom4 3268 |
. . . . . . . 8
⊢
(∀𝑓 ∈ (
R ‘𝐵)∀𝑎(𝑎 = (𝐴 +s 𝑓) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎) ↔ ∀𝑎∀𝑓 ∈ ( R ‘𝐵)(𝑎 = (𝐴 +s 𝑓) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎)) |
191 | | ovex 7387 |
. . . . . . . . . 10
⊢ (𝐴 +s 𝑓) ∈ V |
192 | | breq2 5108 |
. . . . . . . . . . 11
⊢ (𝑎 = (𝐴 +s 𝑓) → (𝑏 ≤s 𝑎 ↔ 𝑏 ≤s (𝐴 +s 𝑓))) |
193 | 192 | rexbidv 3174 |
. . . . . . . . . 10
⊢ (𝑎 = (𝐴 +s 𝑓) → (∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎 ↔ ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝐴 +s 𝑓))) |
194 | 191, 193 | ceqsalv 3480 |
. . . . . . . . 9
⊢
(∀𝑎(𝑎 = (𝐴 +s 𝑓) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎) ↔ ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝐴 +s 𝑓)) |
195 | 194 | ralbii 3095 |
. . . . . . . 8
⊢
(∀𝑓 ∈ (
R ‘𝐵)∀𝑎(𝑎 = (𝐴 +s 𝑓) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎) ↔ ∀𝑓 ∈ ( R ‘𝐵)∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝐴 +s 𝑓)) |
196 | | r19.23v 3178 |
. . . . . . . . 9
⊢
(∀𝑓 ∈ (
R ‘𝐵)(𝑎 = (𝐴 +s 𝑓) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎) ↔ (∃𝑓 ∈ ( R ‘𝐵)𝑎 = (𝐴 +s 𝑓) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎)) |
197 | 196 | albii 1821 |
. . . . . . . 8
⊢
(∀𝑎∀𝑓 ∈ ( R ‘𝐵)(𝑎 = (𝐴 +s 𝑓) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎) ↔ ∀𝑎(∃𝑓 ∈ ( R ‘𝐵)𝑎 = (𝐴 +s 𝑓) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎)) |
198 | 190, 195,
197 | 3bitr3ri 301 |
. . . . . . 7
⊢
(∀𝑎(∃𝑓 ∈ ( R ‘𝐵)𝑎 = (𝐴 +s 𝑓) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎) ↔ ∀𝑓 ∈ ( R ‘𝐵)∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝐴 +s 𝑓)) |
199 | 189, 198 | bitri 274 |
. . . . . 6
⊢
(∀𝑎 ∈
{𝑑 ∣ ∃𝑓 ∈ ( R ‘𝐵)𝑑 = (𝐴 +s 𝑓)}∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎 ↔ ∀𝑓 ∈ ( R ‘𝐵)∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝐴 +s 𝑓)) |
200 | 186, 199 | anbi12i 627 |
. . . . 5
⊢
((∀𝑎 ∈
{𝑐 ∣ ∃𝑒 ∈ ( R ‘𝐴)𝑐 = (𝑒 +s 𝐵)}∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎 ∧ ∀𝑎 ∈ {𝑑 ∣ ∃𝑓 ∈ ( R ‘𝐵)𝑑 = (𝐴 +s 𝑓)}∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎) ↔ (∀𝑒 ∈ ( R ‘𝐴)∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝑒 +s 𝐵) ∧ ∀𝑓 ∈ ( R ‘𝐵)∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝐴 +s 𝑓))) |
201 | 173, 200 | bitri 274 |
. . . 4
⊢
(∀𝑎 ∈
({𝑐 ∣ ∃𝑒 ∈ ( R ‘𝐴)𝑐 = (𝑒 +s 𝐵)} ∪ {𝑑 ∣ ∃𝑓 ∈ ( R ‘𝐵)𝑑 = (𝐴 +s 𝑓)})∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎 ↔ (∀𝑒 ∈ ( R ‘𝐴)∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝑒 +s 𝐵) ∧ ∀𝑓 ∈ ( R ‘𝐵)∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝐴 +s 𝑓))) |
202 | 141, 172,
201 | sylanbrc 583 |
. . 3
⊢ (𝜑 → ∀𝑎 ∈ ({𝑐 ∣ ∃𝑒 ∈ ( R ‘𝐴)𝑐 = (𝑒 +s 𝐵)} ∪ {𝑑 ∣ ∃𝑓 ∈ ( R ‘𝐵)𝑑 = (𝐴 +s 𝑓)})∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎) |
203 | | eqid 2736 |
. . . . . . . 8
⊢ (𝑙 ∈ 𝐿 ↦ (𝑙 +s 𝐵)) = (𝑙 ∈ 𝐿 ↦ (𝑙 +s 𝐵)) |
204 | 203 | rnmpt 5909 |
. . . . . . 7
⊢ ran
(𝑙 ∈ 𝐿 ↦ (𝑙 +s 𝐵)) = {𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} |
205 | | ssltex1 27122 |
. . . . . . . . . 10
⊢ (𝐿 <<s 𝑅 → 𝐿 ∈ V) |
206 | 2, 205 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐿 ∈ V) |
207 | 206 | mptexd 7171 |
. . . . . . . 8
⊢ (𝜑 → (𝑙 ∈ 𝐿 ↦ (𝑙 +s 𝐵)) ∈ V) |
208 | | rnexg 7838 |
. . . . . . . 8
⊢ ((𝑙 ∈ 𝐿 ↦ (𝑙 +s 𝐵)) ∈ V → ran (𝑙 ∈ 𝐿 ↦ (𝑙 +s 𝐵)) ∈ V) |
209 | 207, 208 | syl 17 |
. . . . . . 7
⊢ (𝜑 → ran (𝑙 ∈ 𝐿 ↦ (𝑙 +s 𝐵)) ∈ V) |
210 | 204, 209 | eqeltrrid 2843 |
. . . . . 6
⊢ (𝜑 → {𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∈ V) |
211 | | eqid 2736 |
. . . . . . . 8
⊢ (𝑚 ∈ 𝑀 ↦ (𝐴 +s 𝑚)) = (𝑚 ∈ 𝑀 ↦ (𝐴 +s 𝑚)) |
212 | 211 | rnmpt 5909 |
. . . . . . 7
⊢ ran
(𝑚 ∈ 𝑀 ↦ (𝐴 +s 𝑚)) = {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)} |
213 | | ssltex1 27122 |
. . . . . . . . . 10
⊢ (𝑀 <<s 𝑆 → 𝑀 ∈ V) |
214 | 6, 213 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑀 ∈ V) |
215 | 214 | mptexd 7171 |
. . . . . . . 8
⊢ (𝜑 → (𝑚 ∈ 𝑀 ↦ (𝐴 +s 𝑚)) ∈ V) |
216 | | rnexg 7838 |
. . . . . . . 8
⊢ ((𝑚 ∈ 𝑀 ↦ (𝐴 +s 𝑚)) ∈ V → ran (𝑚 ∈ 𝑀 ↦ (𝐴 +s 𝑚)) ∈ V) |
217 | 215, 216 | syl 17 |
. . . . . . 7
⊢ (𝜑 → ran (𝑚 ∈ 𝑀 ↦ (𝐴 +s 𝑚)) ∈ V) |
218 | 212, 217 | eqeltrrid 2843 |
. . . . . 6
⊢ (𝜑 → {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)} ∈ V) |
219 | 210, 218 | unexd 7685 |
. . . . 5
⊢ (𝜑 → ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)}) ∈ V) |
220 | | snex 5387 |
. . . . . 6
⊢ {(𝐴 +s 𝐵)} ∈ V |
221 | 220 | a1i 11 |
. . . . 5
⊢ (𝜑 → {(𝐴 +s 𝐵)} ∈ V) |
222 | 24 | sselda 3943 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑙 ∈ 𝐿) → 𝑙 ∈ No
) |
223 | 8 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑙 ∈ 𝐿) → 𝐵 ∈ No
) |
224 | 222, 223 | addscld 34316 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑙 ∈ 𝐿) → (𝑙 +s 𝐵) ∈ No
) |
225 | | eleq1 2825 |
. . . . . . . . 9
⊢ (𝑦 = (𝑙 +s 𝐵) → (𝑦 ∈ No
↔ (𝑙 +s 𝐵) ∈
No )) |
226 | 224, 225 | syl5ibrcom 246 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑙 ∈ 𝐿) → (𝑦 = (𝑙 +s 𝐵) → 𝑦 ∈ No
)) |
227 | 226 | rexlimdva 3151 |
. . . . . . 7
⊢ (𝜑 → (∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵) → 𝑦 ∈ No
)) |
228 | 227 | abssdv 4024 |
. . . . . 6
⊢ (𝜑 → {𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ⊆ No
) |
229 | 4 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑀) → 𝐴 ∈ No
) |
230 | 55 | sselda 3943 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑀) → 𝑚 ∈ No
) |
231 | 229, 230 | addscld 34316 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑀) → (𝐴 +s 𝑚) ∈ No
) |
232 | | eleq1 2825 |
. . . . . . . . 9
⊢ (𝑧 = (𝐴 +s 𝑚) → (𝑧 ∈ No
↔ (𝐴 +s 𝑚) ∈
No )) |
233 | 231, 232 | syl5ibrcom 246 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑀) → (𝑧 = (𝐴 +s 𝑚) → 𝑧 ∈ No
)) |
234 | 233 | rexlimdva 3151 |
. . . . . . 7
⊢ (𝜑 → (∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚) → 𝑧 ∈ No
)) |
235 | 234 | abssdv 4024 |
. . . . . 6
⊢ (𝜑 → {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)} ⊆ No
) |
236 | 228, 235 | unssd 4145 |
. . . . 5
⊢ (𝜑 → ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)}) ⊆ No
) |
237 | 4, 8 | addscld 34316 |
. . . . . 6
⊢ (𝜑 → (𝐴 +s 𝐵) ∈ No
) |
238 | 237 | snssd 4768 |
. . . . 5
⊢ (𝜑 → {(𝐴 +s 𝐵)} ⊆ No
) |
239 | | velsn 4601 |
. . . . . . 7
⊢ (𝑏 ∈ {(𝐴 +s 𝐵)} ↔ 𝑏 = (𝐴 +s 𝐵)) |
240 | | elun 4107 |
. . . . . . . . . . 11
⊢ (𝑎 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)}) ↔ (𝑎 ∈ {𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∨ 𝑎 ∈ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})) |
241 | | vex 3448 |
. . . . . . . . . . . . 13
⊢ 𝑎 ∈ V |
242 | | eqeq1 2740 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = 𝑎 → (𝑦 = (𝑙 +s 𝐵) ↔ 𝑎 = (𝑙 +s 𝐵))) |
243 | 242 | rexbidv 3174 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 𝑎 → (∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵) ↔ ∃𝑙 ∈ 𝐿 𝑎 = (𝑙 +s 𝐵))) |
244 | 241, 243 | elab 3629 |
. . . . . . . . . . . 12
⊢ (𝑎 ∈ {𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ↔ ∃𝑙 ∈ 𝐿 𝑎 = (𝑙 +s 𝐵)) |
245 | | eqeq1 2740 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = 𝑎 → (𝑧 = (𝐴 +s 𝑚) ↔ 𝑎 = (𝐴 +s 𝑚))) |
246 | 245 | rexbidv 3174 |
. . . . . . . . . . . . 13
⊢ (𝑧 = 𝑎 → (∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚) ↔ ∃𝑚 ∈ 𝑀 𝑎 = (𝐴 +s 𝑚))) |
247 | 241, 246 | elab 3629 |
. . . . . . . . . . . 12
⊢ (𝑎 ∈ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)} ↔ ∃𝑚 ∈ 𝑀 𝑎 = (𝐴 +s 𝑚)) |
248 | 244, 247 | orbi12i 913 |
. . . . . . . . . . 11
⊢ ((𝑎 ∈ {𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∨ 𝑎 ∈ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)}) ↔ (∃𝑙 ∈ 𝐿 𝑎 = (𝑙 +s 𝐵) ∨ ∃𝑚 ∈ 𝑀 𝑎 = (𝐴 +s 𝑚))) |
249 | 240, 248 | bitri 274 |
. . . . . . . . . 10
⊢ (𝑎 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)}) ↔ (∃𝑙 ∈ 𝐿 𝑎 = (𝑙 +s 𝐵) ∨ ∃𝑚 ∈ 𝑀 𝑎 = (𝐴 +s 𝑚))) |
250 | | scutcut 27136 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐿 <<s 𝑅 → ((𝐿 |s 𝑅) ∈ No
∧ 𝐿 <<s {(𝐿 |s 𝑅)} ∧ {(𝐿 |s 𝑅)} <<s 𝑅)) |
251 | 2, 250 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((𝐿 |s 𝑅) ∈ No
∧ 𝐿 <<s {(𝐿 |s 𝑅)} ∧ {(𝐿 |s 𝑅)} <<s 𝑅)) |
252 | 251 | simp2d 1143 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐿 <<s {(𝐿 |s 𝑅)}) |
253 | 252 | adantr 481 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑙 ∈ 𝐿) → 𝐿 <<s {(𝐿 |s 𝑅)}) |
254 | | simpr 485 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑙 ∈ 𝐿) → 𝑙 ∈ 𝐿) |
255 | | ovex 7387 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐿 |s 𝑅) ∈ V |
256 | 255 | snid 4621 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐿 |s 𝑅) ∈ {(𝐿 |s 𝑅)} |
257 | 256 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑙 ∈ 𝐿) → (𝐿 |s 𝑅) ∈ {(𝐿 |s 𝑅)}) |
258 | 253, 254,
257 | ssltsepcd 27129 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑙 ∈ 𝐿) → 𝑙 <s (𝐿 |s 𝑅)) |
259 | 1 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑙 ∈ 𝐿) → 𝐴 = (𝐿 |s 𝑅)) |
260 | 258, 259 | breqtrrd 5132 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑙 ∈ 𝐿) → 𝑙 <s 𝐴) |
261 | 4 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑙 ∈ 𝐿) → 𝐴 ∈ No
) |
262 | 222, 261,
223 | sltadd1d 34333 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑙 ∈ 𝐿) → (𝑙 <s 𝐴 ↔ (𝑙 +s 𝐵) <s (𝐴 +s 𝐵))) |
263 | 260, 262 | mpbid 231 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑙 ∈ 𝐿) → (𝑙 +s 𝐵) <s (𝐴 +s 𝐵)) |
264 | | breq1 5107 |
. . . . . . . . . . . . 13
⊢ (𝑎 = (𝑙 +s 𝐵) → (𝑎 <s (𝐴 +s 𝐵) ↔ (𝑙 +s 𝐵) <s (𝐴 +s 𝐵))) |
265 | 263, 264 | syl5ibrcom 246 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑙 ∈ 𝐿) → (𝑎 = (𝑙 +s 𝐵) → 𝑎 <s (𝐴 +s 𝐵))) |
266 | 265 | rexlimdva 3151 |
. . . . . . . . . . 11
⊢ (𝜑 → (∃𝑙 ∈ 𝐿 𝑎 = (𝑙 +s 𝐵) → 𝑎 <s (𝐴 +s 𝐵))) |
267 | | scutcut 27136 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑀 <<s 𝑆 → ((𝑀 |s 𝑆) ∈ No
∧ 𝑀 <<s {(𝑀 |s 𝑆)} ∧ {(𝑀 |s 𝑆)} <<s 𝑆)) |
268 | 6, 267 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((𝑀 |s 𝑆) ∈ No
∧ 𝑀 <<s {(𝑀 |s 𝑆)} ∧ {(𝑀 |s 𝑆)} <<s 𝑆)) |
269 | 268 | simp2d 1143 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑀 <<s {(𝑀 |s 𝑆)}) |
270 | 269 | adantr 481 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑀) → 𝑀 <<s {(𝑀 |s 𝑆)}) |
271 | | simpr 485 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑀) → 𝑚 ∈ 𝑀) |
272 | | ovex 7387 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑀 |s 𝑆) ∈ V |
273 | 272 | snid 4621 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑀 |s 𝑆) ∈ {(𝑀 |s 𝑆)} |
274 | 273 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑀) → (𝑀 |s 𝑆) ∈ {(𝑀 |s 𝑆)}) |
275 | 270, 271,
274 | ssltsepcd 27129 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑀) → 𝑚 <s (𝑀 |s 𝑆)) |
276 | 5 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑀) → 𝐵 = (𝑀 |s 𝑆)) |
277 | 275, 276 | breqtrrd 5132 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑀) → 𝑚 <s 𝐵) |
278 | 8 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑀) → 𝐵 ∈ No
) |
279 | 230, 278,
229 | sltadd2d 34332 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑀) → (𝑚 <s 𝐵 ↔ (𝐴 +s 𝑚) <s (𝐴 +s 𝐵))) |
280 | 277, 279 | mpbid 231 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑀) → (𝐴 +s 𝑚) <s (𝐴 +s 𝐵)) |
281 | | breq1 5107 |
. . . . . . . . . . . . 13
⊢ (𝑎 = (𝐴 +s 𝑚) → (𝑎 <s (𝐴 +s 𝐵) ↔ (𝐴 +s 𝑚) <s (𝐴 +s 𝐵))) |
282 | 280, 281 | syl5ibrcom 246 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑀) → (𝑎 = (𝐴 +s 𝑚) → 𝑎 <s (𝐴 +s 𝐵))) |
283 | 282 | rexlimdva 3151 |
. . . . . . . . . . 11
⊢ (𝜑 → (∃𝑚 ∈ 𝑀 𝑎 = (𝐴 +s 𝑚) → 𝑎 <s (𝐴 +s 𝐵))) |
284 | 266, 283 | jaod 857 |
. . . . . . . . . 10
⊢ (𝜑 → ((∃𝑙 ∈ 𝐿 𝑎 = (𝑙 +s 𝐵) ∨ ∃𝑚 ∈ 𝑀 𝑎 = (𝐴 +s 𝑚)) → 𝑎 <s (𝐴 +s 𝐵))) |
285 | 249, 284 | biimtrid 241 |
. . . . . . . . 9
⊢ (𝜑 → (𝑎 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)}) → 𝑎 <s (𝐴 +s 𝐵))) |
286 | 285 | imp 407 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑎 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})) → 𝑎 <s (𝐴 +s 𝐵)) |
287 | | breq2 5108 |
. . . . . . . 8
⊢ (𝑏 = (𝐴 +s 𝐵) → (𝑎 <s 𝑏 ↔ 𝑎 <s (𝐴 +s 𝐵))) |
288 | 286, 287 | syl5ibrcom 246 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑎 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})) → (𝑏 = (𝐴 +s 𝐵) → 𝑎 <s 𝑏)) |
289 | 239, 288 | biimtrid 241 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑎 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})) → (𝑏 ∈ {(𝐴 +s 𝐵)} → 𝑎 <s 𝑏)) |
290 | 289 | 3impia 1117 |
. . . . 5
⊢ ((𝜑 ∧ 𝑎 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)}) ∧ 𝑏 ∈ {(𝐴 +s 𝐵)}) → 𝑎 <s 𝑏) |
291 | 219, 221,
236, 238, 290 | ssltd 27127 |
. . . 4
⊢ (𝜑 → ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)}) <<s {(𝐴 +s 𝐵)}) |
292 | 10 | sneqd 4597 |
. . . 4
⊢ (𝜑 → {(𝐴 +s 𝐵)} = {(({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)𝑎 = (𝑝 +s 𝐵)} ∪ {𝑏 ∣ ∃𝑞 ∈ ( L ‘𝐵)𝑏 = (𝐴 +s 𝑞)}) |s ({𝑐 ∣ ∃𝑒 ∈ ( R ‘𝐴)𝑐 = (𝑒 +s 𝐵)} ∪ {𝑑 ∣ ∃𝑓 ∈ ( R ‘𝐵)𝑑 = (𝐴 +s 𝑓)}))}) |
293 | 291, 292 | breqtrd 5130 |
. . 3
⊢ (𝜑 → ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)}) <<s {(({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)𝑎 = (𝑝 +s 𝐵)} ∪ {𝑏 ∣ ∃𝑞 ∈ ( L ‘𝐵)𝑏 = (𝐴 +s 𝑞)}) |s ({𝑐 ∣ ∃𝑒 ∈ ( R ‘𝐴)𝑐 = (𝑒 +s 𝐵)} ∪ {𝑑 ∣ ∃𝑓 ∈ ( R ‘𝐵)𝑑 = (𝐴 +s 𝑓)}))}) |
294 | | eqid 2736 |
. . . . . . . 8
⊢ (𝑟 ∈ 𝑅 ↦ (𝑟 +s 𝐵)) = (𝑟 ∈ 𝑅 ↦ (𝑟 +s 𝐵)) |
295 | 294 | rnmpt 5909 |
. . . . . . 7
⊢ ran
(𝑟 ∈ 𝑅 ↦ (𝑟 +s 𝐵)) = {𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} |
296 | | ssltex2 27123 |
. . . . . . . . . 10
⊢ (𝐿 <<s 𝑅 → 𝑅 ∈ V) |
297 | 2, 296 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑅 ∈ V) |
298 | 297 | mptexd 7171 |
. . . . . . . 8
⊢ (𝜑 → (𝑟 ∈ 𝑅 ↦ (𝑟 +s 𝐵)) ∈ V) |
299 | | rnexg 7838 |
. . . . . . . 8
⊢ ((𝑟 ∈ 𝑅 ↦ (𝑟 +s 𝐵)) ∈ V → ran (𝑟 ∈ 𝑅 ↦ (𝑟 +s 𝐵)) ∈ V) |
300 | 298, 299 | syl 17 |
. . . . . . 7
⊢ (𝜑 → ran (𝑟 ∈ 𝑅 ↦ (𝑟 +s 𝐵)) ∈ V) |
301 | 295, 300 | eqeltrrid 2843 |
. . . . . 6
⊢ (𝜑 → {𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∈ V) |
302 | | eqid 2736 |
. . . . . . . 8
⊢ (𝑠 ∈ 𝑆 ↦ (𝐴 +s 𝑠)) = (𝑠 ∈ 𝑆 ↦ (𝐴 +s 𝑠)) |
303 | 302 | rnmpt 5909 |
. . . . . . 7
⊢ ran
(𝑠 ∈ 𝑆 ↦ (𝐴 +s 𝑠)) = {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)} |
304 | | ssltex2 27123 |
. . . . . . . . . 10
⊢ (𝑀 <<s 𝑆 → 𝑆 ∈ V) |
305 | 6, 304 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑆 ∈ V) |
306 | 305 | mptexd 7171 |
. . . . . . . 8
⊢ (𝜑 → (𝑠 ∈ 𝑆 ↦ (𝐴 +s 𝑠)) ∈ V) |
307 | | rnexg 7838 |
. . . . . . . 8
⊢ ((𝑠 ∈ 𝑆 ↦ (𝐴 +s 𝑠)) ∈ V → ran (𝑠 ∈ 𝑆 ↦ (𝐴 +s 𝑠)) ∈ V) |
308 | 306, 307 | syl 17 |
. . . . . . 7
⊢ (𝜑 → ran (𝑠 ∈ 𝑆 ↦ (𝐴 +s 𝑠)) ∈ V) |
309 | 303, 308 | eqeltrrid 2843 |
. . . . . 6
⊢ (𝜑 → {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)} ∈ V) |
310 | 301, 309 | unexd 7685 |
. . . . 5
⊢ (𝜑 → ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)}) ∈ V) |
311 | 113 | sselda 3943 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → 𝑟 ∈ No
) |
312 | 8 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → 𝐵 ∈ No
) |
313 | 311, 312 | addscld 34316 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → (𝑟 +s 𝐵) ∈ No
) |
314 | | eleq1 2825 |
. . . . . . . . 9
⊢ (𝑤 = (𝑟 +s 𝐵) → (𝑤 ∈ No
↔ (𝑟 +s 𝐵) ∈
No )) |
315 | 313, 314 | syl5ibrcom 246 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → (𝑤 = (𝑟 +s 𝐵) → 𝑤 ∈ No
)) |
316 | 315 | rexlimdva 3151 |
. . . . . . 7
⊢ (𝜑 → (∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵) → 𝑤 ∈ No
)) |
317 | 316 | abssdv 4024 |
. . . . . 6
⊢ (𝜑 → {𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ⊆ No
) |
318 | 4 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝐴 ∈ No
) |
319 | 144 | sselda 3943 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝑠 ∈ No
) |
320 | 318, 319 | addscld 34316 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝐴 +s 𝑠) ∈ No
) |
321 | | eleq1 2825 |
. . . . . . . . 9
⊢ (𝑡 = (𝐴 +s 𝑠) → (𝑡 ∈ No
↔ (𝐴 +s 𝑠) ∈
No )) |
322 | 320, 321 | syl5ibrcom 246 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑡 = (𝐴 +s 𝑠) → 𝑡 ∈ No
)) |
323 | 322 | rexlimdva 3151 |
. . . . . . 7
⊢ (𝜑 → (∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠) → 𝑡 ∈ No
)) |
324 | 323 | abssdv 4024 |
. . . . . 6
⊢ (𝜑 → {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)} ⊆ No
) |
325 | 317, 324 | unssd 4145 |
. . . . 5
⊢ (𝜑 → ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)}) ⊆ No
) |
326 | | velsn 4601 |
. . . . . . 7
⊢ (𝑎 ∈ {(𝐴 +s 𝐵)} ↔ 𝑎 = (𝐴 +s 𝐵)) |
327 | | elun 4107 |
. . . . . . . . . . . . 13
⊢ (𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)}) ↔ (𝑏 ∈ {𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∨ 𝑏 ∈ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})) |
328 | | vex 3448 |
. . . . . . . . . . . . . . 15
⊢ 𝑏 ∈ V |
329 | 328, 125 | elab 3629 |
. . . . . . . . . . . . . 14
⊢ (𝑏 ∈ {𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ↔ ∃𝑟 ∈ 𝑅 𝑏 = (𝑟 +s 𝐵)) |
330 | 328, 156 | elab 3629 |
. . . . . . . . . . . . . 14
⊢ (𝑏 ∈ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)} ↔ ∃𝑠 ∈ 𝑆 𝑏 = (𝐴 +s 𝑠)) |
331 | 329, 330 | orbi12i 913 |
. . . . . . . . . . . . 13
⊢ ((𝑏 ∈ {𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∨ 𝑏 ∈ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)}) ↔ (∃𝑟 ∈ 𝑅 𝑏 = (𝑟 +s 𝐵) ∨ ∃𝑠 ∈ 𝑆 𝑏 = (𝐴 +s 𝑠))) |
332 | 327, 331 | bitri 274 |
. . . . . . . . . . . 12
⊢ (𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)}) ↔ (∃𝑟 ∈ 𝑅 𝑏 = (𝑟 +s 𝐵) ∨ ∃𝑠 ∈ 𝑆 𝑏 = (𝐴 +s 𝑠))) |
333 | 1 | adantr 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → 𝐴 = (𝐿 |s 𝑅)) |
334 | 251 | simp3d 1144 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → {(𝐿 |s 𝑅)} <<s 𝑅) |
335 | 334 | adantr 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → {(𝐿 |s 𝑅)} <<s 𝑅) |
336 | 256 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → (𝐿 |s 𝑅) ∈ {(𝐿 |s 𝑅)}) |
337 | | simpr 485 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → 𝑟 ∈ 𝑅) |
338 | 335, 336,
337 | ssltsepcd 27129 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → (𝐿 |s 𝑅) <s 𝑟) |
339 | 333, 338 | eqbrtrd 5126 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → 𝐴 <s 𝑟) |
340 | 4 | adantr 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → 𝐴 ∈ No
) |
341 | 340, 311,
312 | sltadd1d 34333 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → (𝐴 <s 𝑟 ↔ (𝐴 +s 𝐵) <s (𝑟 +s 𝐵))) |
342 | 339, 341 | mpbid 231 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → (𝐴 +s 𝐵) <s (𝑟 +s 𝐵)) |
343 | | breq2 5108 |
. . . . . . . . . . . . . . 15
⊢ (𝑏 = (𝑟 +s 𝐵) → ((𝐴 +s 𝐵) <s 𝑏 ↔ (𝐴 +s 𝐵) <s (𝑟 +s 𝐵))) |
344 | 342, 343 | syl5ibrcom 246 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → (𝑏 = (𝑟 +s 𝐵) → (𝐴 +s 𝐵) <s 𝑏)) |
345 | 344 | rexlimdva 3151 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (∃𝑟 ∈ 𝑅 𝑏 = (𝑟 +s 𝐵) → (𝐴 +s 𝐵) <s 𝑏)) |
346 | 5 | adantr 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝐵 = (𝑀 |s 𝑆)) |
347 | 268 | simp3d 1144 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → {(𝑀 |s 𝑆)} <<s 𝑆) |
348 | 347 | adantr 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → {(𝑀 |s 𝑆)} <<s 𝑆) |
349 | 273 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑀 |s 𝑆) ∈ {(𝑀 |s 𝑆)}) |
350 | | simpr 485 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝑠 ∈ 𝑆) |
351 | 348, 349,
350 | ssltsepcd 27129 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑀 |s 𝑆) <s 𝑠) |
352 | 346, 351 | eqbrtrd 5126 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝐵 <s 𝑠) |
353 | 8 | adantr 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝐵 ∈ No
) |
354 | 353, 319,
318 | sltadd2d 34332 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝐵 <s 𝑠 ↔ (𝐴 +s 𝐵) <s (𝐴 +s 𝑠))) |
355 | 352, 354 | mpbid 231 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝐴 +s 𝐵) <s (𝐴 +s 𝑠)) |
356 | | breq2 5108 |
. . . . . . . . . . . . . . 15
⊢ (𝑏 = (𝐴 +s 𝑠) → ((𝐴 +s 𝐵) <s 𝑏 ↔ (𝐴 +s 𝐵) <s (𝐴 +s 𝑠))) |
357 | 355, 356 | syl5ibrcom 246 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑏 = (𝐴 +s 𝑠) → (𝐴 +s 𝐵) <s 𝑏)) |
358 | 357 | rexlimdva 3151 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (∃𝑠 ∈ 𝑆 𝑏 = (𝐴 +s 𝑠) → (𝐴 +s 𝐵) <s 𝑏)) |
359 | 345, 358 | jaod 857 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((∃𝑟 ∈ 𝑅 𝑏 = (𝑟 +s 𝐵) ∨ ∃𝑠 ∈ 𝑆 𝑏 = (𝐴 +s 𝑠)) → (𝐴 +s 𝐵) <s 𝑏)) |
360 | 332, 359 | biimtrid 241 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)}) → (𝐴 +s 𝐵) <s 𝑏)) |
361 | 360 | imp 407 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})) → (𝐴 +s 𝐵) <s 𝑏) |
362 | | breq1 5107 |
. . . . . . . . . 10
⊢ (𝑎 = (𝐴 +s 𝐵) → (𝑎 <s 𝑏 ↔ (𝐴 +s 𝐵) <s 𝑏)) |
363 | 361, 362 | syl5ibrcom 246 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})) → (𝑎 = (𝐴 +s 𝐵) → 𝑎 <s 𝑏)) |
364 | 363 | ex 413 |
. . . . . . . 8
⊢ (𝜑 → (𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)}) → (𝑎 = (𝐴 +s 𝐵) → 𝑎 <s 𝑏))) |
365 | 364 | com23 86 |
. . . . . . 7
⊢ (𝜑 → (𝑎 = (𝐴 +s 𝐵) → (𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)}) → 𝑎 <s 𝑏))) |
366 | 326, 365 | biimtrid 241 |
. . . . . 6
⊢ (𝜑 → (𝑎 ∈ {(𝐴 +s 𝐵)} → (𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)}) → 𝑎 <s 𝑏))) |
367 | 366 | 3imp 1111 |
. . . . 5
⊢ ((𝜑 ∧ 𝑎 ∈ {(𝐴 +s 𝐵)} ∧ 𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})) → 𝑎 <s 𝑏) |
368 | 221, 310,
238, 325, 367 | ssltd 27127 |
. . . 4
⊢ (𝜑 → {(𝐴 +s 𝐵)} <<s ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})) |
369 | 292, 368 | eqbrtrrd 5128 |
. . 3
⊢ (𝜑 → {(({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)𝑎 = (𝑝 +s 𝐵)} ∪ {𝑏 ∣ ∃𝑞 ∈ ( L ‘𝐵)𝑏 = (𝐴 +s 𝑞)}) |s ({𝑐 ∣ ∃𝑒 ∈ ( R ‘𝐴)𝑐 = (𝑒 +s 𝐵)} ∪ {𝑑 ∣ ∃𝑓 ∈ ( R ‘𝐵)𝑑 = (𝐴 +s 𝑓)}))} <<s ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})) |
370 | 18, 110, 202, 293, 369 | cofcut1d 27236 |
. 2
⊢ (𝜑 → (({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)𝑎 = (𝑝 +s 𝐵)} ∪ {𝑏 ∣ ∃𝑞 ∈ ( L ‘𝐵)𝑏 = (𝐴 +s 𝑞)}) |s ({𝑐 ∣ ∃𝑒 ∈ ( R ‘𝐴)𝑐 = (𝑒 +s 𝐵)} ∪ {𝑑 ∣ ∃𝑓 ∈ ( R ‘𝐵)𝑑 = (𝐴 +s 𝑓)})) = (({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)}) |s ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)}))) |
371 | 10, 370 | eqtrd 2776 |
1
⊢ (𝜑 → (𝐴 +s 𝐵) = (({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)}) |s ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)}))) |