MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mulsuniflem Structured version   Visualization version   GIF version

Theorem mulsuniflem 27533
Description: Lemma for mulsunif 27534. State the theorem with some extra distinct variable conditions. (Contributed by Scott Fenton, 8-Mar-2025.)
Hypotheses
Ref Expression
mulsuniflem.1 (𝜑𝐿 <<s 𝑅)
mulsuniflem.2 (𝜑𝑀 <<s 𝑆)
mulsuniflem.3 (𝜑𝐴 = (𝐿 |s 𝑅))
mulsuniflem.4 (𝜑𝐵 = (𝑀 |s 𝑆))
Assertion
Ref Expression
mulsuniflem (𝜑 → (𝐴 ·s 𝐵) = (({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) |s ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})))
Distinct variable groups:   𝐴,𝑎,𝑝,𝑞   𝐴,𝑏,𝑟,𝑠   𝐴,𝑐,𝑡,𝑢   𝐴,𝑑,𝑣,𝑤   𝐵,𝑎,𝑝,𝑞   𝐵,𝑏,𝑟,𝑠   𝐵,𝑐,𝑡,𝑢   𝐵,𝑑,𝑣,𝑤   𝐿,𝑎,𝑝,𝑞   𝐿,𝑐,𝑡,𝑢   𝑣,𝐿   𝐿,𝑟   𝑀,𝑎,𝑝,𝑞   𝑢,𝑀   𝑀,𝑑,𝑣,𝑤   𝑀,𝑠   𝑅,𝑝   𝑅,𝑏,𝑟,𝑠   𝑡,𝑅   𝑅,𝑑,𝑣,𝑤   𝑆,𝑏,𝑟,𝑠   𝑆,𝑐,𝑡,𝑢   𝑤,𝑆   𝑆,𝑞   𝜑,𝑎,𝑝,𝑞   𝜑,𝑏,𝑟,𝑠   𝜑,𝑐,𝑡,𝑢   𝜑,𝑑,𝑣,𝑤
Allowed substitution hints:   𝑅(𝑢,𝑞,𝑎,𝑐)   𝑆(𝑣,𝑝,𝑎,𝑑)   𝐿(𝑤,𝑠,𝑏,𝑑)   𝑀(𝑡,𝑟,𝑏,𝑐)

Proof of Theorem mulsuniflem
Dummy variables 𝑒 𝑓 𝑔 𝑖 𝑗 𝑘 𝑙 𝑚 𝑛 𝑥 𝑦 𝑧 𝑥𝑂 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mulsuniflem.3 . . . 4 (𝜑𝐴 = (𝐿 |s 𝑅))
2 mulsuniflem.1 . . . . 5 (𝜑𝐿 <<s 𝑅)
32scutcld 27233 . . . 4 (𝜑 → (𝐿 |s 𝑅) ∈ No )
41, 3eqeltrd 2833 . . 3 (𝜑𝐴 No )
5 mulsuniflem.4 . . . 4 (𝜑𝐵 = (𝑀 |s 𝑆))
6 mulsuniflem.2 . . . . 5 (𝜑𝑀 <<s 𝑆)
76scutcld 27233 . . . 4 (𝜑 → (𝑀 |s 𝑆) ∈ No )
85, 7eqeltrd 2833 . . 3 (𝜑𝐵 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))} ∪ { ∣ ∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵) = (((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))})))
104, 8, 9syl2anc 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))} ∪ { ∣ ∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵) = (((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))})))
114, 8mulscut2 27518 . . 3 (𝜑 → ({𝑒 ∣ ∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑒 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔))} ∪ { ∣ ∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵) = (((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))}))
122, 1cofcutr1d 27341 . . . . . 6 (𝜑 → ∀𝑓 ∈ ( L ‘𝐴)∃𝑝𝐿 𝑓 ≤s 𝑝)
136, 5cofcutr1d 27341 . . . . . . . . . 10 (𝜑 → ∀𝑔 ∈ ( L ‘𝐵)∃𝑞𝑀 𝑔 ≤s 𝑞)
1413adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ ∃𝑝𝐿 𝑓 ≤s 𝑝)) → ∀𝑔 ∈ ( L ‘𝐵)∃𝑞𝑀 𝑔 ≤s 𝑞)
15 reeanv 3226 . . . . . . . . . . . . . . 15 (∃𝑝𝐿𝑞𝑀 (𝑓 ≤s 𝑝𝑔 ≤s 𝑞) ↔ (∃𝑝𝐿 𝑓 ≤s 𝑝 ∧ ∃𝑞𝑀 𝑔 ≤s 𝑞))
16 leftssno 27304 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ( L ‘𝐴) ⊆ No
17 simprl 769 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵))) → 𝑓 ∈ ( L ‘𝐴))
1816, 17sselid 3977 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵))) → 𝑓 No )
1918adantrr 715 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝𝐿𝑞𝑀))) → 𝑓 No )
208adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝𝐿𝑞𝑀))) → 𝐵 No )
2119, 20mulscld 27520 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝𝐿𝑞𝑀))) → (𝑓 ·s 𝐵) ∈ No )
224adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝𝐿𝑞𝑀))) → 𝐴 No )
23 leftssno 27304 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ( L ‘𝐵) ⊆ No
24 simprr 771 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵))) → 𝑔 ∈ ( L ‘𝐵))
2523, 24sselid 3977 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵))) → 𝑔 No )
2625adantrr 715 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝𝐿𝑞𝑀))) → 𝑔 No )
2722, 26mulscld 27520 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝𝐿𝑞𝑀))) → (𝐴 ·s 𝑔) ∈ No )
2821, 27addscld 27393 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝𝐿𝑞𝑀))) → ((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) ∈ No )
2919, 26mulscld 27520 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝𝐿𝑞𝑀))) → (𝑓 ·s 𝑔) ∈ No )
3028, 29subscld 27464 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝𝐿𝑞𝑀))) → (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ∈ No )
3130adantrrr 723 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝𝐿𝑞𝑀) ∧ (𝑓 ≤s 𝑝𝑔 ≤s 𝑞)))) → (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ∈ No )
32 ssltss1 27219 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝐿 <<s 𝑅𝐿 No )
332, 32syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑𝐿 No )
3433adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑝𝐿𝑞𝑀)) → 𝐿 No )
35 simprl 769 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑝𝐿𝑞𝑀)) → 𝑝𝐿)
3634, 35sseldd 3980 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑝𝐿𝑞𝑀)) → 𝑝 No )
3736adantrl 714 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝𝐿𝑞𝑀))) → 𝑝 No )
3837, 20mulscld 27520 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝𝐿𝑞𝑀))) → (𝑝 ·s 𝐵) ∈ No )
3938, 27addscld 27393 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝𝐿𝑞𝑀))) → ((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑔)) ∈ No )
4037, 26mulscld 27520 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝𝐿𝑞𝑀))) → (𝑝 ·s 𝑔) ∈ No )
4139, 40subscld 27464 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝𝐿𝑞𝑀))) → (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑝 ·s 𝑔)) ∈ No )
4241adantrrr 723 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝𝐿𝑞𝑀) ∧ (𝑓 ≤s 𝑝𝑔 ≤s 𝑞)))) → (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑝 ·s 𝑔)) ∈ No )
43 ssltss1 27219 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑀 <<s 𝑆𝑀 No )
446, 43syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑𝑀 No )
4544adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑝𝐿𝑞𝑀)) → 𝑀 No )
46 simprr 771 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑝𝐿𝑞𝑀)) → 𝑞𝑀)
4745, 46sseldd 3980 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑝𝐿𝑞𝑀)) → 𝑞 No )
4847adantrl 714 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝𝐿𝑞𝑀))) → 𝑞 No )
4922, 48mulscld 27520 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝𝐿𝑞𝑀))) → (𝐴 ·s 𝑞) ∈ No )
5038, 49addscld 27393 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝𝐿𝑞𝑀))) → ((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) ∈ No )
5137, 48mulscld 27520 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝𝐿𝑞𝑀))) → (𝑝 ·s 𝑞) ∈ No )
5250, 51subscld 27464 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝𝐿𝑞𝑀))) → (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ∈ No )
5352adantrrr 723 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝𝐿𝑞𝑀) ∧ (𝑓 ≤s 𝑝𝑔 ≤s 𝑞)))) → (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ∈ No )
5418adantrr 715 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝𝐿𝑞𝑀) ∧ (𝑓 ≤s 𝑝𝑔 ≤s 𝑞)))) → 𝑓 No )
5537adantrrr 723 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝𝐿𝑞𝑀) ∧ (𝑓 ≤s 𝑝𝑔 ≤s 𝑞)))) → 𝑝 No )
5625adantrr 715 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝𝐿𝑞𝑀) ∧ (𝑓 ≤s 𝑝𝑔 ≤s 𝑞)))) → 𝑔 No )
578adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝𝐿𝑞𝑀) ∧ (𝑓 ≤s 𝑝𝑔 ≤s 𝑞)))) → 𝐵 No )
58 simprrl 779 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝𝐿𝑞𝑀) ∧ (𝑓 ≤s 𝑝𝑔 ≤s 𝑞))) → 𝑓 ≤s 𝑝)
5958adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝𝐿𝑞𝑀) ∧ (𝑓 ≤s 𝑝𝑔 ≤s 𝑞)))) → 𝑓 ≤s 𝑝)
608adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵))) → 𝐵 No )
61 ssltleft 27294 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝐵 No → ( L ‘𝐵) <<s {𝐵})
628, 61syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑 → ( L ‘𝐵) <<s {𝐵})
6362adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵))) → ( L ‘𝐵) <<s {𝐵})
64 snidg 4657 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝐵 No 𝐵 ∈ {𝐵})
658, 64syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑𝐵 ∈ {𝐵})
6665adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵))) → 𝐵 ∈ {𝐵})
6763, 24, 66ssltsepcd 27224 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵))) → 𝑔 <s 𝐵)
6825, 60, 67sltled 27201 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵))) → 𝑔 ≤s 𝐵)
6968adantrr 715 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝𝐿𝑞𝑀) ∧ (𝑓 ≤s 𝑝𝑔 ≤s 𝑞)))) → 𝑔 ≤s 𝐵)
7054, 55, 56, 57, 59, 69slemuld 27523 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝𝐿𝑞𝑀) ∧ (𝑓 ≤s 𝑝𝑔 ≤s 𝑞)))) → ((𝑓 ·s 𝐵) -s (𝑓 ·s 𝑔)) ≤s ((𝑝 ·s 𝐵) -s (𝑝 ·s 𝑔)))
7121, 29subscld 27464 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝𝐿𝑞𝑀))) → ((𝑓 ·s 𝐵) -s (𝑓 ·s 𝑔)) ∈ No )
7238, 40subscld 27464 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝𝐿𝑞𝑀))) → ((𝑝 ·s 𝐵) -s (𝑝 ·s 𝑔)) ∈ No )
7371, 72, 27sleadd1d 27407 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝𝐿𝑞𝑀))) → (((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))))
7473adantrrr 723 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝𝐿𝑞𝑀) ∧ (𝑓 ≤s 𝑝𝑔 ≤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))))
7570, 74mpbid 231 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝𝐿𝑞𝑀) ∧ (𝑓 ≤s 𝑝𝑔 ≤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)))
7621, 27, 29addsubsd 27478 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝𝐿𝑞𝑀))) → (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) = (((𝑓 ·s 𝐵) -s (𝑓 ·s 𝑔)) +s (𝐴 ·s 𝑔)))
7776adantrrr 723 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝𝐿𝑞𝑀) ∧ (𝑓 ≤s 𝑝𝑔 ≤s 𝑞)))) → (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) = (((𝑓 ·s 𝐵) -s (𝑓 ·s 𝑔)) +s (𝐴 ·s 𝑔)))
7838, 27, 40addsubsd 27478 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝𝐿𝑞𝑀))) → (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑝 ·s 𝑔)) = (((𝑝 ·s 𝐵) -s (𝑝 ·s 𝑔)) +s (𝐴 ·s 𝑔)))
7978adantrrr 723 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝𝐿𝑞𝑀) ∧ (𝑓 ≤s 𝑝𝑔 ≤s 𝑞)))) → (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑝 ·s 𝑔)) = (((𝑝 ·s 𝐵) -s (𝑝 ·s 𝑔)) +s (𝐴 ·s 𝑔)))
8075, 77, 793brtr4d 5174 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝𝐿𝑞𝑀) ∧ (𝑓 ≤s 𝑝𝑔 ≤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)))
814adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝𝐿𝑞𝑀) ∧ (𝑓 ≤s 𝑝𝑔 ≤s 𝑞)))) → 𝐴 No )
8248adantrrr 723 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝𝐿𝑞𝑀) ∧ (𝑓 ≤s 𝑝𝑔 ≤s 𝑞)))) → 𝑞 No )
834adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑝𝐿𝑞𝑀)) → 𝐴 No )
84 scutcut 27231 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝐿 <<s 𝑅 → ((𝐿 |s 𝑅) ∈ No 𝐿 <<s {(𝐿 |s 𝑅)} ∧ {(𝐿 |s 𝑅)} <<s 𝑅))
852, 84syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝜑 → ((𝐿 |s 𝑅) ∈ No 𝐿 <<s {(𝐿 |s 𝑅)} ∧ {(𝐿 |s 𝑅)} <<s 𝑅))
8685simp2d 1143 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝜑𝐿 <<s {(𝐿 |s 𝑅)})
8786adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ (𝑝𝐿𝑞𝑀)) → 𝐿 <<s {(𝐿 |s 𝑅)})
88 ovex 7427 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝐿 |s 𝑅) ∈ V
8988snid 4659 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝐿 |s 𝑅) ∈ {(𝐿 |s 𝑅)}
901, 89eqeltrdi 2841 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝜑𝐴 ∈ {(𝐿 |s 𝑅)})
9190adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ (𝑝𝐿𝑞𝑀)) → 𝐴 ∈ {(𝐿 |s 𝑅)})
9287, 35, 91ssltsepcd 27224 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑝𝐿𝑞𝑀)) → 𝑝 <s 𝐴)
9336, 83, 92sltled 27201 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑝𝐿𝑞𝑀)) → 𝑝 ≤s 𝐴)
9493adantrl 714 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝𝐿𝑞𝑀))) → 𝑝 ≤s 𝐴)
9594adantrrr 723 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝𝐿𝑞𝑀) ∧ (𝑓 ≤s 𝑝𝑔 ≤s 𝑞)))) → 𝑝 ≤s 𝐴)
96 simprrr 780 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝𝐿𝑞𝑀) ∧ (𝑓 ≤s 𝑝𝑔 ≤s 𝑞))) → 𝑔 ≤s 𝑞)
9796adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝𝐿𝑞𝑀) ∧ (𝑓 ≤s 𝑝𝑔 ≤s 𝑞)))) → 𝑔 ≤s 𝑞)
9855, 81, 56, 82, 95, 97slemuld 27523 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝𝐿𝑞𝑀) ∧ (𝑓 ≤s 𝑝𝑔 ≤s 𝑞)))) → ((𝑝 ·s 𝑞) -s (𝑝 ·s 𝑔)) ≤s ((𝐴 ·s 𝑞) -s (𝐴 ·s 𝑔)))
9951, 49, 40, 27slesubsub3bd 27484 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝𝐿𝑞𝑀))) → (((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))))
10027, 40subscld 27464 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝𝐿𝑞𝑀))) → ((𝐴 ·s 𝑔) -s (𝑝 ·s 𝑔)) ∈ No )
10149, 51subscld 27464 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝𝐿𝑞𝑀))) → ((𝐴 ·s 𝑞) -s (𝑝 ·s 𝑞)) ∈ No )
102100, 101, 38sleadd2d 27408 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝𝐿𝑞𝑀))) → (((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)))))
10399, 102bitrd 278 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝𝐿𝑞𝑀))) → (((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)))))
104103adantrrr 723 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝𝐿𝑞𝑀) ∧ (𝑓 ≤s 𝑝𝑔 ≤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)))))
10598, 104mpbid 231 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝𝐿𝑞𝑀) ∧ (𝑓 ≤s 𝑝𝑔 ≤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))))
10638, 27, 40addsubsassd 27477 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝𝐿𝑞𝑀))) → (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑝 ·s 𝑔)) = ((𝑝 ·s 𝐵) +s ((𝐴 ·s 𝑔) -s (𝑝 ·s 𝑔))))
107106adantrrr 723 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝𝐿𝑞𝑀) ∧ (𝑓 ≤s 𝑝𝑔 ≤s 𝑞)))) → (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑝 ·s 𝑔)) = ((𝑝 ·s 𝐵) +s ((𝐴 ·s 𝑔) -s (𝑝 ·s 𝑔))))
10838, 49, 51addsubsassd 27477 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝𝐿𝑞𝑀))) → (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) = ((𝑝 ·s 𝐵) +s ((𝐴 ·s 𝑞) -s (𝑝 ·s 𝑞))))
109108adantrrr 723 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝𝐿𝑞𝑀) ∧ (𝑓 ≤s 𝑝𝑔 ≤s 𝑞)))) → (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) = ((𝑝 ·s 𝐵) +s ((𝐴 ·s 𝑞) -s (𝑝 ·s 𝑞))))
110105, 107, 1093brtr4d 5174 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝𝐿𝑞𝑀) ∧ (𝑓 ≤s 𝑝𝑔 ≤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)))
11131, 42, 53, 80, 110sletrd 27194 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝𝐿𝑞𝑀) ∧ (𝑓 ≤s 𝑝𝑔 ≤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)))
112111anassrs 468 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵))) ∧ ((𝑝𝐿𝑞𝑀) ∧ (𝑓 ≤s 𝑝𝑔 ≤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)))
113112expr 457 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵))) ∧ (𝑝𝐿𝑞𝑀)) → ((𝑓 ≤s 𝑝𝑔 ≤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))))
114113reximdvva 3205 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵))) → (∃𝑝𝐿𝑞𝑀 (𝑓 ≤s 𝑝𝑔 ≤s 𝑞) → ∃𝑝𝐿𝑞𝑀 (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))))
115114expcom 414 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) → (𝜑 → (∃𝑝𝐿𝑞𝑀 (𝑓 ≤s 𝑝𝑔 ≤s 𝑞) → ∃𝑝𝐿𝑞𝑀 (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)))))
116115com23 86 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) → (∃𝑝𝐿𝑞𝑀 (𝑓 ≤s 𝑝𝑔 ≤s 𝑞) → (𝜑 → ∃𝑝𝐿𝑞𝑀 (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)))))
117116imp 407 . . . . . . . . . . . . . . 15 (((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ∃𝑝𝐿𝑞𝑀 (𝑓 ≤s 𝑝𝑔 ≤s 𝑞)) → (𝜑 → ∃𝑝𝐿𝑞𝑀 (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))))
11815, 117sylan2br 595 . . . . . . . . . . . . . 14 (((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (∃𝑝𝐿 𝑓 ≤s 𝑝 ∧ ∃𝑞𝑀 𝑔 ≤s 𝑞)) → (𝜑 → ∃𝑝𝐿𝑞𝑀 (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))))
119118an4s 658 . . . . . . . . . . . . 13 (((𝑓 ∈ ( L ‘𝐴) ∧ ∃𝑝𝐿 𝑓 ≤s 𝑝) ∧ (𝑔 ∈ ( L ‘𝐵) ∧ ∃𝑞𝑀 𝑔 ≤s 𝑞)) → (𝜑 → ∃𝑝𝐿𝑞𝑀 (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))))
120119impcom 408 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ ∃𝑝𝐿 𝑓 ≤s 𝑝) ∧ (𝑔 ∈ ( L ‘𝐵) ∧ ∃𝑞𝑀 𝑔 ≤s 𝑞))) → ∃𝑝𝐿𝑞𝑀 (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)))
121120anassrs 468 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ ∃𝑝𝐿 𝑓 ≤s 𝑝)) ∧ (𝑔 ∈ ( L ‘𝐵) ∧ ∃𝑞𝑀 𝑔 ≤s 𝑞)) → ∃𝑝𝐿𝑞𝑀 (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)))
122121expr 457 . . . . . . . . . 10 (((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ ∃𝑝𝐿 𝑓 ≤s 𝑝)) ∧ 𝑔 ∈ ( L ‘𝐵)) → (∃𝑞𝑀 𝑔 ≤s 𝑞 → ∃𝑝𝐿𝑞𝑀 (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))))
123122ralimdva 3167 . . . . . . . . 9 ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ ∃𝑝𝐿 𝑓 ≤s 𝑝)) → (∀𝑔 ∈ ( L ‘𝐵)∃𝑞𝑀 𝑔 ≤s 𝑞 → ∀𝑔 ∈ ( L ‘𝐵)∃𝑝𝐿𝑞𝑀 (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))))
12414, 123mpd 15 . . . . . . . 8 ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ ∃𝑝𝐿 𝑓 ≤s 𝑝)) → ∀𝑔 ∈ ( L ‘𝐵)∃𝑝𝐿𝑞𝑀 (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)))
125124expr 457 . . . . . . 7 ((𝜑𝑓 ∈ ( L ‘𝐴)) → (∃𝑝𝐿 𝑓 ≤s 𝑝 → ∀𝑔 ∈ ( L ‘𝐵)∃𝑝𝐿𝑞𝑀 (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))))
126125ralimdva 3167 . . . . . 6 (𝜑 → (∀𝑓 ∈ ( L ‘𝐴)∃𝑝𝐿 𝑓 ≤s 𝑝 → ∀𝑓 ∈ ( L ‘𝐴)∀𝑔 ∈ ( L ‘𝐵)∃𝑝𝐿𝑞𝑀 (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))))
12712, 126mpd 15 . . . . 5 (𝜑 → ∀𝑓 ∈ ( L ‘𝐴)∀𝑔 ∈ ( L ‘𝐵)∃𝑝𝐿𝑞𝑀 (((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))))
1291282rexbidv 3219 . . . . . . . . 9 (𝑎 = 𝑧 → (∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ↔ ∃𝑝𝐿𝑞𝑀 𝑧 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))))
130129rexab 3687 . . . . . . . 8 (∃𝑧 ∈ {𝑎 ∣ ∃𝑝𝐿𝑞𝑀 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(∃𝑝𝐿𝑞𝑀 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 (∃𝑝𝐿𝑞𝑀 (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) ↔ (∃𝑝𝐿𝑞𝑀 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))
132131exbii 1850 . . . . . . . 8 (∃𝑧𝑝𝐿𝑞𝑀 (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(∃𝑝𝐿𝑞𝑀 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 (∃𝑝𝐿𝑧𝑞𝑀 (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) ↔ ∃𝑧𝑝𝐿𝑞𝑀 (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 (∃𝑞𝑀𝑧(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) ↔ ∃𝑧𝑞𝑀 (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))))
137135, 136ceqsexv 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)))
138137rexbii 3094 . . . . . . . . . . 11 (∃𝑞𝑀𝑧(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) ↔ ∃𝑞𝑀 (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)))
139134, 138bitr3i 276 . . . . . . . . . 10 (∃𝑧𝑞𝑀 (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) ↔ ∃𝑞𝑀 (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)))
140139rexbii 3094 . . . . . . . . 9 (∃𝑝𝐿𝑧𝑞𝑀 (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) ↔ ∃𝑝𝐿𝑞𝑀 (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)))
141133, 140bitr3i 276 . . . . . . . 8 (∃𝑧𝑝𝐿𝑞𝑀 (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) ↔ ∃𝑝𝐿𝑞𝑀 (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) ≤s (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)))
142130, 132, 1413bitr2i 298 . . . . . . 7 (∃𝑧 ∈ {𝑎 ∣ ∃𝑝𝐿𝑞𝑀 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 ↔ ∃𝑝𝐿𝑞𝑀 (((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 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ⊆ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})
144 ssrexv 4048 . . . . . . . 8 ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ⊆ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) → (∃𝑧 ∈ {𝑎 ∣ ∃𝑝𝐿𝑞𝑀 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 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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))
145143, 144ax-mp 5 . . . . . . 7 (∃𝑧 ∈ {𝑎 ∣ ∃𝑝𝐿𝑞𝑀 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 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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)
146142, 145sylbir 234 . . . . . 6 (∃𝑝𝐿𝑞𝑀 (((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 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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)
1471462ralimi 3123 . . . . 5 (∀𝑓 ∈ ( L ‘𝐴)∀𝑔 ∈ ( L ‘𝐵)∃𝑝𝐿𝑞𝑀 (((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 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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)
148127, 147syl 17 . . . 4 (𝜑 → ∀𝑓 ∈ ( L ‘𝐴)∀𝑔 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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)
1492, 1cofcutr2d 27342 . . . . . 6 (𝜑 → ∀𝑖 ∈ ( R ‘𝐴)∃𝑟𝑅 𝑟 ≤s 𝑖)
1506, 5cofcutr2d 27342 . . . . . . . . . 10 (𝜑 → ∀𝑗 ∈ ( R ‘𝐵)∃𝑠𝑆 𝑠 ≤s 𝑗)
151150adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ ∃𝑟𝑅 𝑟 ≤s 𝑖)) → ∀𝑗 ∈ ( R ‘𝐵)∃𝑠𝑆 𝑠 ≤s 𝑗)
152 reeanv 3226 . . . . . . . . . . . . . . 15 (∃𝑟𝑅𝑠𝑆 (𝑟 ≤s 𝑖𝑠 ≤s 𝑗) ↔ (∃𝑟𝑅 𝑟 ≤s 𝑖 ∧ ∃𝑠𝑆 𝑠 ≤s 𝑗))
153 rightssno 27305 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ( R ‘𝐴) ⊆ No
154 simprl 769 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵))) → 𝑖 ∈ ( R ‘𝐴))
155153, 154sselid 3977 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵))) → 𝑖 No )
156155adantrr 715 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟𝑅𝑠𝑆))) → 𝑖 No )
1578adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟𝑅𝑠𝑆))) → 𝐵 No )
158156, 157mulscld 27520 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟𝑅𝑠𝑆))) → (𝑖 ·s 𝐵) ∈ No )
1594adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟𝑅𝑠𝑆))) → 𝐴 No )
160 rightssno 27305 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ( R ‘𝐵) ⊆ No
161 simprr 771 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵))) → 𝑗 ∈ ( R ‘𝐵))
162160, 161sselid 3977 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵))) → 𝑗 No )
163162adantrr 715 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟𝑅𝑠𝑆))) → 𝑗 No )
164159, 163mulscld 27520 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟𝑅𝑠𝑆))) → (𝐴 ·s 𝑗) ∈ No )
165158, 164addscld 27393 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟𝑅𝑠𝑆))) → ((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) ∈ No )
166156, 163mulscld 27520 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟𝑅𝑠𝑆))) → (𝑖 ·s 𝑗) ∈ No )
167165, 166subscld 27464 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟𝑅𝑠𝑆))) → (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ∈ No )
168167adantrrr 723 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟𝑅𝑠𝑆) ∧ (𝑟 ≤s 𝑖𝑠 ≤s 𝑗)))) → (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ∈ No )
169 ssltss2 27220 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝐿 <<s 𝑅𝑅 No )
1702, 169syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑𝑅 No )
171170adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → 𝑅 No )
172 simprl 769 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → 𝑟𝑅)
173171, 172sseldd 3980 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → 𝑟 No )
174173adantrl 714 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟𝑅𝑠𝑆))) → 𝑟 No )
175174, 157mulscld 27520 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟𝑅𝑠𝑆))) → (𝑟 ·s 𝐵) ∈ No )
176175, 164addscld 27393 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟𝑅𝑠𝑆))) → ((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑗)) ∈ No )
177174, 163mulscld 27520 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟𝑅𝑠𝑆))) → (𝑟 ·s 𝑗) ∈ No )
178176, 177subscld 27464 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟𝑅𝑠𝑆))) → (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑟 ·s 𝑗)) ∈ No )
179178adantrrr 723 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟𝑅𝑠𝑆) ∧ (𝑟 ≤s 𝑖𝑠 ≤s 𝑗)))) → (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑟 ·s 𝑗)) ∈ No )
180 ssltss2 27220 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑀 <<s 𝑆𝑆 No )
1816, 180syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑𝑆 No )
182181adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → 𝑆 No )
183 simprr 771 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → 𝑠𝑆)
184182, 183sseldd 3980 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → 𝑠 No )
185184adantrl 714 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟𝑅𝑠𝑆))) → 𝑠 No )
186159, 185mulscld 27520 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟𝑅𝑠𝑆))) → (𝐴 ·s 𝑠) ∈ No )
187175, 186addscld 27393 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟𝑅𝑠𝑆))) → ((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) ∈ No )
188173, 184mulscld 27520 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → (𝑟 ·s 𝑠) ∈ No )
189188adantrl 714 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟𝑅𝑠𝑆))) → (𝑟 ·s 𝑠) ∈ No )
190187, 189subscld 27464 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟𝑅𝑠𝑆))) → (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ∈ No )
191190adantrrr 723 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟𝑅𝑠𝑆) ∧ (𝑟 ≤s 𝑖𝑠 ≤s 𝑗)))) → (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ∈ No )
192174adantrrr 723 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟𝑅𝑠𝑆) ∧ (𝑟 ≤s 𝑖𝑠 ≤s 𝑗)))) → 𝑟 No )
193155adantrr 715 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟𝑅𝑠𝑆) ∧ (𝑟 ≤s 𝑖𝑠 ≤s 𝑗)))) → 𝑖 No )
1948adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟𝑅𝑠𝑆) ∧ (𝑟 ≤s 𝑖𝑠 ≤s 𝑗)))) → 𝐵 No )
195162adantrr 715 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟𝑅𝑠𝑆) ∧ (𝑟 ≤s 𝑖𝑠 ≤s 𝑗)))) → 𝑗 No )
196 simprrl 779 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟𝑅𝑠𝑆) ∧ (𝑟 ≤s 𝑖𝑠 ≤s 𝑗))) → 𝑟 ≤s 𝑖)
197196adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟𝑅𝑠𝑆) ∧ (𝑟 ≤s 𝑖𝑠 ≤s 𝑗)))) → 𝑟 ≤s 𝑖)
1988adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵))) → 𝐵 No )
199 ssltright 27295 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝐵 No → {𝐵} <<s ( R ‘𝐵))
2008, 199syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑 → {𝐵} <<s ( R ‘𝐵))
201200adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵))) → {𝐵} <<s ( R ‘𝐵))
20265adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵))) → 𝐵 ∈ {𝐵})
203201, 202, 161ssltsepcd 27224 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵))) → 𝐵 <s 𝑗)
204198, 162, 203sltled 27201 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵))) → 𝐵 ≤s 𝑗)
205204adantrr 715 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟𝑅𝑠𝑆) ∧ (𝑟 ≤s 𝑖𝑠 ≤s 𝑗)))) → 𝐵 ≤s 𝑗)
206192, 193, 194, 195, 197, 205slemuld 27523 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟𝑅𝑠𝑆) ∧ (𝑟 ≤s 𝑖𝑠 ≤s 𝑗)))) → ((𝑟 ·s 𝑗) -s (𝑟 ·s 𝐵)) ≤s ((𝑖 ·s 𝑗) -s (𝑖 ·s 𝐵)))
207177, 175, 166, 158slesubsub2bd 27483 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟𝑅𝑠𝑆))) → (((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))))
208158, 166subscld 27464 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟𝑅𝑠𝑆))) → ((𝑖 ·s 𝐵) -s (𝑖 ·s 𝑗)) ∈ No )
209175, 177subscld 27464 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟𝑅𝑠𝑆))) → ((𝑟 ·s 𝐵) -s (𝑟 ·s 𝑗)) ∈ No )
210208, 209, 164sleadd1d 27407 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟𝑅𝑠𝑆))) → (((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))))
211207, 210bitrd 278 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟𝑅𝑠𝑆))) → (((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))))
212211adantrrr 723 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟𝑅𝑠𝑆) ∧ (𝑟 ≤s 𝑖𝑠 ≤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))))
213206, 212mpbid 231 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟𝑅𝑠𝑆) ∧ (𝑟 ≤s 𝑖𝑠 ≤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)))
214158, 164, 166addsubsd 27478 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟𝑅𝑠𝑆))) → (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) = (((𝑖 ·s 𝐵) -s (𝑖 ·s 𝑗)) +s (𝐴 ·s 𝑗)))
215214adantrrr 723 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟𝑅𝑠𝑆) ∧ (𝑟 ≤s 𝑖𝑠 ≤s 𝑗)))) → (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) = (((𝑖 ·s 𝐵) -s (𝑖 ·s 𝑗)) +s (𝐴 ·s 𝑗)))
216175, 164, 177addsubsd 27478 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟𝑅𝑠𝑆))) → (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑟 ·s 𝑗)) = (((𝑟 ·s 𝐵) -s (𝑟 ·s 𝑗)) +s (𝐴 ·s 𝑗)))
217216adantrrr 723 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟𝑅𝑠𝑆) ∧ (𝑟 ≤s 𝑖𝑠 ≤s 𝑗)))) → (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑟 ·s 𝑗)) = (((𝑟 ·s 𝐵) -s (𝑟 ·s 𝑗)) +s (𝐴 ·s 𝑗)))
218213, 215, 2173brtr4d 5174 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟𝑅𝑠𝑆) ∧ (𝑟 ≤s 𝑖𝑠 ≤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)))
2194adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟𝑅𝑠𝑆) ∧ (𝑟 ≤s 𝑖𝑠 ≤s 𝑗)))) → 𝐴 No )
220185adantrrr 723 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟𝑅𝑠𝑆) ∧ (𝑟 ≤s 𝑖𝑠 ≤s 𝑗)))) → 𝑠 No )
2214adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → 𝐴 No )
22285simp3d 1144 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝜑 → {(𝐿 |s 𝑅)} <<s 𝑅)
223222adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → {(𝐿 |s 𝑅)} <<s 𝑅)
22490adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → 𝐴 ∈ {(𝐿 |s 𝑅)})
225223, 224, 172ssltsepcd 27224 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → 𝐴 <s 𝑟)
226221, 173, 225sltled 27201 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑟𝑅𝑠𝑆)) → 𝐴 ≤s 𝑟)
227226adantrl 714 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟𝑅𝑠𝑆))) → 𝐴 ≤s 𝑟)
228227adantrrr 723 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟𝑅𝑠𝑆) ∧ (𝑟 ≤s 𝑖𝑠 ≤s 𝑗)))) → 𝐴 ≤s 𝑟)
229 simprrr 780 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟𝑅𝑠𝑆) ∧ (𝑟 ≤s 𝑖𝑠 ≤s 𝑗))) → 𝑠 ≤s 𝑗)
230229adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟𝑅𝑠𝑆) ∧ (𝑟 ≤s 𝑖𝑠 ≤s 𝑗)))) → 𝑠 ≤s 𝑗)
231219, 192, 220, 195, 228, 230slemuld 27523 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟𝑅𝑠𝑆) ∧ (𝑟 ≤s 𝑖𝑠 ≤s 𝑗)))) → ((𝐴 ·s 𝑗) -s (𝐴 ·s 𝑠)) ≤s ((𝑟 ·s 𝑗) -s (𝑟 ·s 𝑠)))
232164, 177, 186, 189slesubsubbd 27482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟𝑅𝑠𝑆))) → (((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))))
233164, 177subscld 27464 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟𝑅𝑠𝑆))) → ((𝐴 ·s 𝑗) -s (𝑟 ·s 𝑗)) ∈ No )
234186, 189subscld 27464 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟𝑅𝑠𝑆))) → ((𝐴 ·s 𝑠) -s (𝑟 ·s 𝑠)) ∈ No )
235233, 234, 175sleadd2d 27408 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟𝑅𝑠𝑆))) → (((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)))))
236232, 235bitrd 278 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟𝑅𝑠𝑆))) → (((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)))))
237236adantrrr 723 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟𝑅𝑠𝑆) ∧ (𝑟 ≤s 𝑖𝑠 ≤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)))))
238231, 237mpbid 231 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟𝑅𝑠𝑆) ∧ (𝑟 ≤s 𝑖𝑠 ≤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))))
239175, 164, 177addsubsassd 27477 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟𝑅𝑠𝑆))) → (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑟 ·s 𝑗)) = ((𝑟 ·s 𝐵) +s ((𝐴 ·s 𝑗) -s (𝑟 ·s 𝑗))))
240239adantrrr 723 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟𝑅𝑠𝑆) ∧ (𝑟 ≤s 𝑖𝑠 ≤s 𝑗)))) → (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑟 ·s 𝑗)) = ((𝑟 ·s 𝐵) +s ((𝐴 ·s 𝑗) -s (𝑟 ·s 𝑗))))
241175, 186, 189addsubsassd 27477 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟𝑅𝑠𝑆))) → (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) = ((𝑟 ·s 𝐵) +s ((𝐴 ·s 𝑠) -s (𝑟 ·s 𝑠))))
242241adantrrr 723 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟𝑅𝑠𝑆) ∧ (𝑟 ≤s 𝑖𝑠 ≤s 𝑗)))) → (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) = ((𝑟 ·s 𝐵) +s ((𝐴 ·s 𝑠) -s (𝑟 ·s 𝑠))))
243238, 240, 2423brtr4d 5174 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟𝑅𝑠𝑆) ∧ (𝑟 ≤s 𝑖𝑠 ≤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)))
244168, 179, 191, 218, 243sletrd 27194 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟𝑅𝑠𝑆) ∧ (𝑟 ≤s 𝑖𝑠 ≤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)))
245244anassrs 468 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵))) ∧ ((𝑟𝑅𝑠𝑆) ∧ (𝑟 ≤s 𝑖𝑠 ≤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)))
246245expr 457 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵))) ∧ (𝑟𝑅𝑠𝑆)) → ((𝑟 ≤s 𝑖𝑠 ≤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))))
247246reximdvva 3205 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵))) → (∃𝑟𝑅𝑠𝑆 (𝑟 ≤s 𝑖𝑠 ≤s 𝑗) → ∃𝑟𝑅𝑠𝑆 (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))))
248247expcom 414 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) → (𝜑 → (∃𝑟𝑅𝑠𝑆 (𝑟 ≤s 𝑖𝑠 ≤s 𝑗) → ∃𝑟𝑅𝑠𝑆 (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)))))
249248com23 86 . . . . . . . . . . . . . . . 16 ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) → (∃𝑟𝑅𝑠𝑆 (𝑟 ≤s 𝑖𝑠 ≤s 𝑗) → (𝜑 → ∃𝑟𝑅𝑠𝑆 (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)))))
250249imp 407 . . . . . . . . . . . . . . 15 (((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ∃𝑟𝑅𝑠𝑆 (𝑟 ≤s 𝑖𝑠 ≤s 𝑗)) → (𝜑 → ∃𝑟𝑅𝑠𝑆 (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))))
251152, 250sylan2br 595 . . . . . . . . . . . . . 14 (((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (∃𝑟𝑅 𝑟 ≤s 𝑖 ∧ ∃𝑠𝑆 𝑠 ≤s 𝑗)) → (𝜑 → ∃𝑟𝑅𝑠𝑆 (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))))
252251an4s 658 . . . . . . . . . . . . 13 (((𝑖 ∈ ( R ‘𝐴) ∧ ∃𝑟𝑅 𝑟 ≤s 𝑖) ∧ (𝑗 ∈ ( R ‘𝐵) ∧ ∃𝑠𝑆 𝑠 ≤s 𝑗)) → (𝜑 → ∃𝑟𝑅𝑠𝑆 (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))))
253252impcom 408 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ ∃𝑟𝑅 𝑟 ≤s 𝑖) ∧ (𝑗 ∈ ( R ‘𝐵) ∧ ∃𝑠𝑆 𝑠 ≤s 𝑗))) → ∃𝑟𝑅𝑠𝑆 (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)))
254253anassrs 468 . . . . . . . . . . 11 (((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ ∃𝑟𝑅 𝑟 ≤s 𝑖)) ∧ (𝑗 ∈ ( R ‘𝐵) ∧ ∃𝑠𝑆 𝑠 ≤s 𝑗)) → ∃𝑟𝑅𝑠𝑆 (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)))
255254expr 457 . . . . . . . . . 10 (((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ ∃𝑟𝑅 𝑟 ≤s 𝑖)) ∧ 𝑗 ∈ ( R ‘𝐵)) → (∃𝑠𝑆 𝑠 ≤s 𝑗 → ∃𝑟𝑅𝑠𝑆 (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))))
256255ralimdva 3167 . . . . . . . . 9 ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ ∃𝑟𝑅 𝑟 ≤s 𝑖)) → (∀𝑗 ∈ ( R ‘𝐵)∃𝑠𝑆 𝑠 ≤s 𝑗 → ∀𝑗 ∈ ( R ‘𝐵)∃𝑟𝑅𝑠𝑆 (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))))
257151, 256mpd 15 . . . . . . . 8 ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ ∃𝑟𝑅 𝑟 ≤s 𝑖)) → ∀𝑗 ∈ ( R ‘𝐵)∃𝑟𝑅𝑠𝑆 (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)))
258257expr 457 . . . . . . 7 ((𝜑𝑖 ∈ ( R ‘𝐴)) → (∃𝑟𝑅 𝑟 ≤s 𝑖 → ∀𝑗 ∈ ( R ‘𝐵)∃𝑟𝑅𝑠𝑆 (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))))
259258ralimdva 3167 . . . . . 6 (𝜑 → (∀𝑖 ∈ ( R ‘𝐴)∃𝑟𝑅 𝑟 ≤s 𝑖 → ∀𝑖 ∈ ( R ‘𝐴)∀𝑗 ∈ ( R ‘𝐵)∃𝑟𝑅𝑠𝑆 (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))))
260149, 259mpd 15 . . . . 5 (𝜑 → ∀𝑖 ∈ ( R ‘𝐴)∀𝑗 ∈ ( R ‘𝐵)∃𝑟𝑅𝑠𝑆 (((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))))
2622612rexbidv 3219 . . . . . . . . 9 (𝑏 = 𝑧 → (∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ↔ ∃𝑟𝑅𝑠𝑆 𝑧 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))))
263262rexab 3687 . . . . . . . 8 (∃𝑧 ∈ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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(∃𝑟𝑅𝑠𝑆 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 (∃𝑟𝑅𝑠𝑆 (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) ↔ (∃𝑟𝑅𝑠𝑆 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))
265264exbii 1850 . . . . . . . 8 (∃𝑧𝑟𝑅𝑠𝑆 (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(∃𝑟𝑅𝑠𝑆 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 (∃𝑟𝑅𝑧𝑠𝑆 (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) ↔ ∃𝑧𝑟𝑅𝑠𝑆 (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 (∃𝑠𝑆𝑧(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) ↔ ∃𝑧𝑠𝑆 (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))))
270268, 269ceqsexv 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)))
271270rexbii 3094 . . . . . . . . . . 11 (∃𝑠𝑆𝑧(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) ↔ ∃𝑠𝑆 (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)))
272267, 271bitr3i 276 . . . . . . . . . 10 (∃𝑧𝑠𝑆 (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) ↔ ∃𝑠𝑆 (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)))
273272rexbii 3094 . . . . . . . . 9 (∃𝑟𝑅𝑧𝑠𝑆 (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) ↔ ∃𝑟𝑅𝑠𝑆 (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)))
274266, 273bitr3i 276 . . . . . . . 8 (∃𝑧𝑟𝑅𝑠𝑆 (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) ↔ ∃𝑟𝑅𝑠𝑆 (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ≤s (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)))
275263, 265, 2743bitr2i 298 . . . . . . 7 (∃𝑧 ∈ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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 ↔ ∃𝑟𝑅𝑠𝑆 (((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 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))} ⊆ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})
277 ssrexv 4048 . . . . . . . 8 ({𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))} ⊆ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) → (∃𝑧 ∈ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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))
278276, 277ax-mp 5 . . . . . . 7 (∃𝑧 ∈ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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)
279275, 278sylbir 234 . . . . . 6 (∃𝑟𝑅𝑠𝑆 (((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 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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)
2802792ralimi 3123 . . . . 5 (∀𝑖 ∈ ( R ‘𝐴)∀𝑗 ∈ ( R ‘𝐵)∃𝑟𝑅𝑠𝑆 (((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 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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)
281260, 280syl 17 . . . 4 (𝜑 → ∀𝑖 ∈ ( R ‘𝐴)∀𝑗 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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))} ∪ { ∣ ∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵) = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗))})∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧 ∧ ∀𝑥𝑂 ∈ { ∣ ∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵) = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗))}∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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))))
2842832rexbidv 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))))
285284ralab 3684 . . . . . . 7 (∀𝑥𝑂 ∈ {𝑒 ∣ ∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑒 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔))}∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧))
287286ralbii 3093 . . . . . . . . 9 (∀𝑓 ∈ ( L ‘𝐴)∀𝑔 ∈ ( L ‘𝐵)(𝑥𝑂 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧))
289287, 288bitri 274 . . . . . . . 8 (∀𝑓 ∈ ( L ‘𝐴)∀𝑔 ∈ ( L ‘𝐵)(𝑥𝑂 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧))
290289albii 1821 . . . . . . 7 (∀𝑥𝑂𝑓 ∈ ( L ‘𝐴)∀𝑔 ∈ ( L ‘𝐵)(𝑥𝑂 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧))
291 ralcom4 3283 . . . . . . . 8 (∀𝑓 ∈ ( L ‘𝐴)∀𝑥𝑂𝑔 ∈ ( L ‘𝐵)(𝑥𝑂 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧) ↔ ∀𝑥𝑂𝑓 ∈ ( L ‘𝐴)∀𝑔 ∈ ( L ‘𝐵)(𝑥𝑂 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧) ↔ ∀𝑥𝑂𝑔 ∈ ( L ‘𝐵)(𝑥𝑂 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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))
295294rexbidv 3178 . . . . . . . . . . . 12 (𝑥𝑂 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) → (∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧 ↔ ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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))
296293, 295ceqsalv 3510 . . . . . . . . . . 11 (∀𝑥𝑂(𝑥𝑂 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧) ↔ ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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)
297296ralbii 3093 . . . . . . . . . 10 (∀𝑔 ∈ ( L ‘𝐵)∀𝑥𝑂(𝑥𝑂 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧) ↔ ∀𝑔 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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)
298292, 297bitr3i 276 . . . . . . . . 9 (∀𝑥𝑂𝑔 ∈ ( L ‘𝐵)(𝑥𝑂 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧) ↔ ∀𝑔 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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)
299298ralbii 3093 . . . . . . . 8 (∀𝑓 ∈ ( L ‘𝐴)∀𝑥𝑂𝑔 ∈ ( L ‘𝐵)(𝑥𝑂 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧) ↔ ∀𝑓 ∈ ( L ‘𝐴)∀𝑔 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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)
300291, 299bitr3i 276 . . . . . . 7 (∀𝑥𝑂𝑓 ∈ ( L ‘𝐴)∀𝑔 ∈ ( L ‘𝐵)(𝑥𝑂 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧) ↔ ∀𝑓 ∈ ( L ‘𝐴)∀𝑔 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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)
301285, 290, 3003bitr2i 298 . . . . . 6 (∀𝑥𝑂 ∈ {𝑒 ∣ ∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑒 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔))}∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧 ↔ ∀𝑓 ∈ ( L ‘𝐴)∀𝑔 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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 ( = 𝑥𝑂 → ( = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) ↔ 𝑥𝑂 = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗))))
3033022rexbidv 3219 . . . . . . . 8 ( = 𝑥𝑂 → (∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵) = (((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))))
304303ralab 3684 . . . . . . 7 (∀𝑥𝑂 ∈ { ∣ ∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵) = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗))}∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧))
306305ralbii 3093 . . . . . . . . 9 (∀𝑖 ∈ ( R ‘𝐴)∀𝑗 ∈ ( R ‘𝐵)(𝑥𝑂 = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧))
308306, 307bitri 274 . . . . . . . 8 (∀𝑖 ∈ ( R ‘𝐴)∀𝑗 ∈ ( R ‘𝐵)(𝑥𝑂 = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧))
309308albii 1821 . . . . . . 7 (∀𝑥𝑂𝑖 ∈ ( R ‘𝐴)∀𝑗 ∈ ( R ‘𝐵)(𝑥𝑂 = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧))
310 ralcom4 3283 . . . . . . . 8 (∀𝑖 ∈ ( R ‘𝐴)∀𝑥𝑂𝑗 ∈ ( R ‘𝐵)(𝑥𝑂 = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧) ↔ ∀𝑥𝑂𝑖 ∈ ( R ‘𝐴)∀𝑗 ∈ ( R ‘𝐵)(𝑥𝑂 = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧) ↔ ∀𝑥𝑂𝑗 ∈ ( R ‘𝐵)(𝑥𝑂 = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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))
314313rexbidv 3178 . . . . . . . . . . . 12 (𝑥𝑂 = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) → (∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧 ↔ ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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))
315312, 314ceqsalv 3510 . . . . . . . . . . 11 (∀𝑥𝑂(𝑥𝑂 = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧) ↔ ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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)
316315ralbii 3093 . . . . . . . . . 10 (∀𝑗 ∈ ( R ‘𝐵)∀𝑥𝑂(𝑥𝑂 = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧) ↔ ∀𝑗 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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)
317311, 316bitr3i 276 . . . . . . . . 9 (∀𝑥𝑂𝑗 ∈ ( R ‘𝐵)(𝑥𝑂 = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧) ↔ ∀𝑗 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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)
318317ralbii 3093 . . . . . . . 8 (∀𝑖 ∈ ( R ‘𝐴)∀𝑥𝑂𝑗 ∈ ( R ‘𝐵)(𝑥𝑂 = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧) ↔ ∀𝑖 ∈ ( R ‘𝐴)∀𝑗 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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)
319310, 318bitr3i 276 . . . . . . 7 (∀𝑥𝑂𝑖 ∈ ( R ‘𝐴)∀𝑗 ∈ ( R ‘𝐵)(𝑥𝑂 = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧) ↔ ∀𝑖 ∈ ( R ‘𝐴)∀𝑗 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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)
320304, 309, 3193bitr2i 298 . . . . . 6 (∀𝑥𝑂 ∈ { ∣ ∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵) = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗))}∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧 ↔ ∀𝑖 ∈ ( R ‘𝐴)∀𝑗 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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)
321301, 320anbi12i 627 . . . . 5 ((∀𝑥𝑂 ∈ {𝑒 ∣ ∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑒 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔))}∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧 ∧ ∀𝑥𝑂 ∈ { ∣ ∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵) = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗))}∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧) ↔ (∀𝑓 ∈ ( L ‘𝐴)∀𝑔 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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))
322282, 321bitri 274 . . . 4 (∀𝑥𝑂 ∈ ({𝑒 ∣ ∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑒 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔))} ∪ { ∣ ∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵) = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗))})∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧 ↔ (∀𝑓 ∈ ( L ‘𝐴)∀𝑔 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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))
323148, 281, 322sylanbrc 583 . . 3 (𝜑 → ∀𝑥𝑂 ∈ ({𝑒 ∣ ∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑒 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔))} ∪ { ∣ ∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵) = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗))})∃𝑧 ∈ ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))})𝑥𝑂 ≤s 𝑧)
3242, 1cofcutr1d 27341 . . . . . 6 (𝜑 → ∀𝑙 ∈ ( L ‘𝐴)∃𝑡𝐿 𝑙 ≤s 𝑡)
3256, 5cofcutr2d 27342 . . . . . . . . . 10 (𝜑 → ∀𝑚 ∈ ( R ‘𝐵)∃𝑢𝑆 𝑢 ≤s 𝑚)
326325adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ ∃𝑡𝐿 𝑙 ≤s 𝑡)) → ∀𝑚 ∈ ( R ‘𝐵)∃𝑢𝑆 𝑢 ≤s 𝑚)
327 reeanv 3226 . . . . . . . . . . . . . . 15 (∃𝑡𝐿𝑢𝑆 (𝑙 ≤s 𝑡𝑢 ≤s 𝑚) ↔ (∃𝑡𝐿 𝑙 ≤s 𝑡 ∧ ∃𝑢𝑆 𝑢 ≤s 𝑚))
32833adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑡𝐿𝑢𝑆)) → 𝐿 No )
329 simprl 769 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑡𝐿𝑢𝑆)) → 𝑡𝐿)
330328, 329sseldd 3980 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑡𝐿𝑢𝑆)) → 𝑡 No )
331330adantrl 714 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡𝐿𝑢𝑆))) → 𝑡 No )
3328adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡𝐿𝑢𝑆))) → 𝐵 No )
333331, 332mulscld 27520 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡𝐿𝑢𝑆))) → (𝑡 ·s 𝐵) ∈ No )
3344adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡𝐿𝑢𝑆))) → 𝐴 No )
335181adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑡𝐿𝑢𝑆)) → 𝑆 No )
336 simprr 771 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑡𝐿𝑢𝑆)) → 𝑢𝑆)
337335, 336sseldd 3980 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑡𝐿𝑢𝑆)) → 𝑢 No )
338337adantrl 714 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡𝐿𝑢𝑆))) → 𝑢 No )
339334, 338mulscld 27520 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡𝐿𝑢𝑆))) → (𝐴 ·s 𝑢) ∈ No )
340333, 339addscld 27393 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡𝐿𝑢𝑆))) → ((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) ∈ No )
341331, 338mulscld 27520 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡𝐿𝑢𝑆))) → (𝑡 ·s 𝑢) ∈ No )
342340, 341subscld 27464 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡𝐿𝑢𝑆))) → (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ∈ No )
343342adantrrr 723 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡𝐿𝑢𝑆) ∧ (𝑙 ≤s 𝑡𝑢 ≤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))
34516, 344sselid 3977 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → 𝑙 No )
3468adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → 𝐵 No )
347345, 346mulscld 27520 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → (𝑙 ·s 𝐵) ∈ No )
348347adantrr 715 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡𝐿𝑢𝑆))) → (𝑙 ·s 𝐵) ∈ No )
349348, 339addscld 27393 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡𝐿𝑢𝑆))) → ((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑢)) ∈ No )
350345adantrr 715 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡𝐿𝑢𝑆))) → 𝑙 No )
351350, 338mulscld 27520 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡𝐿𝑢𝑆))) → (𝑙 ·s 𝑢) ∈ No )
352349, 351subscld 27464 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡𝐿𝑢𝑆))) → (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑙 ·s 𝑢)) ∈ No )
353352adantrrr 723 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡𝐿𝑢𝑆) ∧ (𝑙 ≤s 𝑡𝑢 ≤s 𝑚)))) → (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑙 ·s 𝑢)) ∈ No )
3544adantr 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))
356160, 355sselid 3977 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → 𝑚 No )
357354, 356mulscld 27520 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → (𝐴 ·s 𝑚) ∈ No )
358357adantrr 715 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡𝐿𝑢𝑆))) → (𝐴 ·s 𝑚) ∈ No )
359348, 358addscld 27393 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡𝐿𝑢𝑆))) → ((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) ∈ No )
360345, 356mulscld 27520 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → (𝑙 ·s 𝑚) ∈ No )
361360adantrr 715 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡𝐿𝑢𝑆))) → (𝑙 ·s 𝑚) ∈ No )
362359, 361subscld 27464 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡𝐿𝑢𝑆))) → (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) ∈ No )
363362adantrrr 723 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡𝐿𝑢𝑆) ∧ (𝑙 ≤s 𝑡𝑢 ≤s 𝑚)))) → (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) ∈ No )
364345adantrr 715 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡𝐿𝑢𝑆) ∧ (𝑙 ≤s 𝑡𝑢 ≤s 𝑚)))) → 𝑙 No )
365331adantrrr 723 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡𝐿𝑢𝑆) ∧ (𝑙 ≤s 𝑡𝑢 ≤s 𝑚)))) → 𝑡 No )
3668adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡𝐿𝑢𝑆) ∧ (𝑙 ≤s 𝑡𝑢 ≤s 𝑚)))) → 𝐵 No )
367338adantrrr 723 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡𝐿𝑢𝑆) ∧ (𝑙 ≤s 𝑡𝑢 ≤s 𝑚)))) → 𝑢 No )
368 simprrl 779 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡𝐿𝑢𝑆) ∧ (𝑙 ≤s 𝑡𝑢 ≤s 𝑚))) → 𝑙 ≤s 𝑡)
369368adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡𝐿𝑢𝑆) ∧ (𝑙 ≤s 𝑡𝑢 ≤s 𝑚)))) → 𝑙 ≤s 𝑡)
3708adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑡𝐿𝑢𝑆)) → 𝐵 No )
371 scutcut 27231 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑀 <<s 𝑆 → ((𝑀 |s 𝑆) ∈ No 𝑀 <<s {(𝑀 |s 𝑆)} ∧ {(𝑀 |s 𝑆)} <<s 𝑆))
3726, 371syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝜑 → ((𝑀 |s 𝑆) ∈ No 𝑀 <<s {(𝑀 |s 𝑆)} ∧ {(𝑀 |s 𝑆)} <<s 𝑆))
373372simp3d 1144 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝜑 → {(𝑀 |s 𝑆)} <<s 𝑆)
374373adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ (𝑡𝐿𝑢𝑆)) → {(𝑀 |s 𝑆)} <<s 𝑆)
375 ovex 7427 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑀 |s 𝑆) ∈ V
376375snid 4659 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑀 |s 𝑆) ∈ {(𝑀 |s 𝑆)}
3775, 376eqeltrdi 2841 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝜑𝐵 ∈ {(𝑀 |s 𝑆)})
378377adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ (𝑡𝐿𝑢𝑆)) → 𝐵 ∈ {(𝑀 |s 𝑆)})
379374, 378, 336ssltsepcd 27224 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑡𝐿𝑢𝑆)) → 𝐵 <s 𝑢)
380370, 337, 379sltled 27201 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑡𝐿𝑢𝑆)) → 𝐵 ≤s 𝑢)
381380adantrl 714 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡𝐿𝑢𝑆))) → 𝐵 ≤s 𝑢)
382381adantrrr 723 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡𝐿𝑢𝑆) ∧ (𝑙 ≤s 𝑡𝑢 ≤s 𝑚)))) → 𝐵 ≤s 𝑢)
383364, 365, 366, 367, 369, 382slemuld 27523 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡𝐿𝑢𝑆) ∧ (𝑙 ≤s 𝑡𝑢 ≤s 𝑚)))) → ((𝑙 ·s 𝑢) -s (𝑙 ·s 𝐵)) ≤s ((𝑡 ·s 𝑢) -s (𝑡 ·s 𝐵)))
384351, 348, 341, 333slesubsub2bd 27483 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡𝐿𝑢𝑆))) → (((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))))
385333, 341subscld 27464 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡𝐿𝑢𝑆))) → ((𝑡 ·s 𝐵) -s (𝑡 ·s 𝑢)) ∈ No )
386348, 351subscld 27464 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡𝐿𝑢𝑆))) → ((𝑙 ·s 𝐵) -s (𝑙 ·s 𝑢)) ∈ No )
387385, 386, 339sleadd1d 27407 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡𝐿𝑢𝑆))) → (((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))))
388384, 387bitrd 278 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡𝐿𝑢𝑆))) → (((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))))
389388adantrrr 723 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡𝐿𝑢𝑆) ∧ (𝑙 ≤s 𝑡𝑢 ≤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))))
390383, 389mpbid 231 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡𝐿𝑢𝑆) ∧ (𝑙 ≤s 𝑡𝑢 ≤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)))
391333, 339, 341addsubsd 27478 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡𝐿𝑢𝑆))) → (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) = (((𝑡 ·s 𝐵) -s (𝑡 ·s 𝑢)) +s (𝐴 ·s 𝑢)))
392391adantrrr 723 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡𝐿𝑢𝑆) ∧ (𝑙 ≤s 𝑡𝑢 ≤s 𝑚)))) → (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) = (((𝑡 ·s 𝐵) -s (𝑡 ·s 𝑢)) +s (𝐴 ·s 𝑢)))
393348, 339, 351addsubsd 27478 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡𝐿𝑢𝑆))) → (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑙 ·s 𝑢)) = (((𝑙 ·s 𝐵) -s (𝑙 ·s 𝑢)) +s (𝐴 ·s 𝑢)))
394393adantrrr 723 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡𝐿𝑢𝑆) ∧ (𝑙 ≤s 𝑡𝑢 ≤s 𝑚)))) → (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑙 ·s 𝑢)) = (((𝑙 ·s 𝐵) -s (𝑙 ·s 𝑢)) +s (𝐴 ·s 𝑢)))
395390, 392, 3943brtr4d 5174 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡𝐿𝑢𝑆) ∧ (𝑙 ≤s 𝑡𝑢 ≤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)))
3964adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡𝐿𝑢𝑆) ∧ (𝑙 ≤s 𝑡𝑢 ≤s 𝑚)))) → 𝐴 No )
397356adantrr 715 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡𝐿𝑢𝑆) ∧ (𝑙 ≤s 𝑡𝑢 ≤s 𝑚)))) → 𝑚 No )
398 ssltleft 27294 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝐴 No → ( L ‘𝐴) <<s {𝐴})
3994, 398syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑 → ( L ‘𝐴) <<s {𝐴})
400399adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → ( L ‘𝐴) <<s {𝐴})
401 snidg 4657 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝐴 No 𝐴 ∈ {𝐴})
4024, 401syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑𝐴 ∈ {𝐴})
403402adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → 𝐴 ∈ {𝐴})
404400, 344, 403ssltsepcd 27224 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → 𝑙 <s 𝐴)
405345, 354, 404sltled 27201 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → 𝑙 ≤s 𝐴)
406405adantrr 715 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡𝐿𝑢𝑆) ∧ (𝑙 ≤s 𝑡𝑢 ≤s 𝑚)))) → 𝑙 ≤s 𝐴)
407 simprrr 780 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡𝐿𝑢𝑆) ∧ (𝑙 ≤s 𝑡𝑢 ≤s 𝑚))) → 𝑢 ≤s 𝑚)
408407adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡𝐿𝑢𝑆) ∧ (𝑙 ≤s 𝑡𝑢 ≤s 𝑚)))) → 𝑢 ≤s 𝑚)
409364, 396, 367, 397, 406, 408slemuld 27523 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡𝐿𝑢𝑆) ∧ (𝑙 ≤s 𝑡𝑢 ≤s 𝑚)))) → ((𝑙 ·s 𝑚) -s (𝑙 ·s 𝑢)) ≤s ((𝐴 ·s 𝑚) -s (𝐴 ·s 𝑢)))
410361, 358, 351, 339slesubsub3bd 27484 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡𝐿𝑢𝑆))) → (((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))))
411339, 351subscld 27464 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡𝐿𝑢𝑆))) → ((𝐴 ·s 𝑢) -s (𝑙 ·s 𝑢)) ∈ No )
412358, 361subscld 27464 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡𝐿𝑢𝑆))) → ((𝐴 ·s 𝑚) -s (𝑙 ·s 𝑚)) ∈ No )
413411, 412, 348sleadd2d 27408 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡𝐿𝑢𝑆))) → (((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)))))
414410, 413bitrd 278 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡𝐿𝑢𝑆))) → (((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)))))
415414adantrrr 723 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡𝐿𝑢𝑆) ∧ (𝑙 ≤s 𝑡𝑢 ≤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)))))
416409, 415mpbid 231 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡𝐿𝑢𝑆) ∧ (𝑙 ≤s 𝑡𝑢 ≤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))))
417348, 339, 351addsubsassd 27477 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡𝐿𝑢𝑆))) → (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑙 ·s 𝑢)) = ((𝑙 ·s 𝐵) +s ((𝐴 ·s 𝑢) -s (𝑙 ·s 𝑢))))
418417adantrrr 723 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡𝐿𝑢𝑆) ∧ (𝑙 ≤s 𝑡𝑢 ≤s 𝑚)))) → (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑙 ·s 𝑢)) = ((𝑙 ·s 𝐵) +s ((𝐴 ·s 𝑢) -s (𝑙 ·s 𝑢))))
419348, 358, 361addsubsassd 27477 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡𝐿𝑢𝑆))) → (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) = ((𝑙 ·s 𝐵) +s ((𝐴 ·s 𝑚) -s (𝑙 ·s 𝑚))))
420419adantrrr 723 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡𝐿𝑢𝑆) ∧ (𝑙 ≤s 𝑡𝑢 ≤s 𝑚)))) → (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) = ((𝑙 ·s 𝐵) +s ((𝐴 ·s 𝑚) -s (𝑙 ·s 𝑚))))
421416, 418, 4203brtr4d 5174 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡𝐿𝑢𝑆) ∧ (𝑙 ≤s 𝑡𝑢 ≤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)))
422343, 353, 363, 395, 421sletrd 27194 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡𝐿𝑢𝑆) ∧ (𝑙 ≤s 𝑡𝑢 ≤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)))
423422anassrs 468 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) ∧ ((𝑡𝐿𝑢𝑆) ∧ (𝑙 ≤s 𝑡𝑢 ≤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)))
424423expr 457 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) ∧ (𝑡𝐿𝑢𝑆)) → ((𝑙 ≤s 𝑡𝑢 ≤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))))
425424reximdvva 3205 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → (∃𝑡𝐿𝑢𝑆 (𝑙 ≤s 𝑡𝑢 ≤s 𝑚) → ∃𝑡𝐿𝑢𝑆 (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))))
426425expcom 414 . . . . . . . . . . . . . . . . 17 ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) → (𝜑 → (∃𝑡𝐿𝑢𝑆 (𝑙 ≤s 𝑡𝑢 ≤s 𝑚) → ∃𝑡𝐿𝑢𝑆 (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)))))
427426com23 86 . . . . . . . . . . . . . . . 16 ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) → (∃𝑡𝐿𝑢𝑆 (𝑙 ≤s 𝑡𝑢 ≤s 𝑚) → (𝜑 → ∃𝑡𝐿𝑢𝑆 (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)))))
428427imp 407 . . . . . . . . . . . . . . 15 (((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ∃𝑡𝐿𝑢𝑆 (𝑙 ≤s 𝑡𝑢 ≤s 𝑚)) → (𝜑 → ∃𝑡𝐿𝑢𝑆 (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))))
429327, 428sylan2br 595 . . . . . . . . . . . . . 14 (((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (∃𝑡𝐿 𝑙 ≤s 𝑡 ∧ ∃𝑢𝑆 𝑢 ≤s 𝑚)) → (𝜑 → ∃𝑡𝐿𝑢𝑆 (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))))
430429an4s 658 . . . . . . . . . . . . 13 (((𝑙 ∈ ( L ‘𝐴) ∧ ∃𝑡𝐿 𝑙 ≤s 𝑡) ∧ (𝑚 ∈ ( R ‘𝐵) ∧ ∃𝑢𝑆 𝑢 ≤s 𝑚)) → (𝜑 → ∃𝑡𝐿𝑢𝑆 (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))))
431430impcom 408 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ ∃𝑡𝐿 𝑙 ≤s 𝑡) ∧ (𝑚 ∈ ( R ‘𝐵) ∧ ∃𝑢𝑆 𝑢 ≤s 𝑚))) → ∃𝑡𝐿𝑢𝑆 (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)))
432431anassrs 468 . . . . . . . . . . 11 (((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ ∃𝑡𝐿 𝑙 ≤s 𝑡)) ∧ (𝑚 ∈ ( R ‘𝐵) ∧ ∃𝑢𝑆 𝑢 ≤s 𝑚)) → ∃𝑡𝐿𝑢𝑆 (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)))
433432expr 457 . . . . . . . . . 10 (((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ ∃𝑡𝐿 𝑙 ≤s 𝑡)) ∧ 𝑚 ∈ ( R ‘𝐵)) → (∃𝑢𝑆 𝑢 ≤s 𝑚 → ∃𝑡𝐿𝑢𝑆 (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))))
434433ralimdva 3167 . . . . . . . . 9 ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ ∃𝑡𝐿 𝑙 ≤s 𝑡)) → (∀𝑚 ∈ ( R ‘𝐵)∃𝑢𝑆 𝑢 ≤s 𝑚 → ∀𝑚 ∈ ( R ‘𝐵)∃𝑡𝐿𝑢𝑆 (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))))
435326, 434mpd 15 . . . . . . . 8 ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ ∃𝑡𝐿 𝑙 ≤s 𝑡)) → ∀𝑚 ∈ ( R ‘𝐵)∃𝑡𝐿𝑢𝑆 (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)))
436435expr 457 . . . . . . 7 ((𝜑𝑙 ∈ ( L ‘𝐴)) → (∃𝑡𝐿 𝑙 ≤s 𝑡 → ∀𝑚 ∈ ( R ‘𝐵)∃𝑡𝐿𝑢𝑆 (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))))
437436ralimdva 3167 . . . . . 6 (𝜑 → (∀𝑙 ∈ ( L ‘𝐴)∃𝑡𝐿 𝑙 ≤s 𝑡 → ∀𝑙 ∈ ( L ‘𝐴)∀𝑚 ∈ ( R ‘𝐵)∃𝑡𝐿𝑢𝑆 (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))))
438324, 437mpd 15 . . . . 5 (𝜑 → ∀𝑙 ∈ ( L ‘𝐴)∀𝑚 ∈ ( R ‘𝐵)∃𝑡𝐿𝑢𝑆 (((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))))
4404392rexbidv 3219 . . . . . . . . 9 (𝑐 = 𝑧 → (∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ↔ ∃𝑡𝐿𝑢𝑆 𝑧 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))))
441440rexab 3687 . . . . . . . 8 (∃𝑧 ∈ {𝑐 ∣ ∃𝑡𝐿𝑢𝑆 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(∃𝑡𝐿𝑢𝑆 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 (∃𝑡𝐿𝑢𝑆 (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))) ↔ (∃𝑡𝐿𝑢𝑆 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))))
443442exbii 1850 . . . . . . . 8 (∃𝑧𝑡𝐿𝑢𝑆 (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(∃𝑡𝐿𝑢𝑆 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 (∃𝑡𝐿𝑧𝑢𝑆 (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))) ↔ ∃𝑧𝑡𝐿𝑢𝑆 (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 (∃𝑢𝑆𝑧(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))) ↔ ∃𝑧𝑢𝑆 (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))))
448446, 447ceqsexv 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)))
449448rexbii 3094 . . . . . . . . . . 11 (∃𝑢𝑆𝑧(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))) ↔ ∃𝑢𝑆 (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)))
450445, 449bitr3i 276 . . . . . . . . . 10 (∃𝑧𝑢𝑆 (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))) ↔ ∃𝑢𝑆 (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)))
451450rexbii 3094 . . . . . . . . 9 (∃𝑡𝐿𝑧𝑢𝑆 (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))) ↔ ∃𝑡𝐿𝑢𝑆 (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)))
452444, 451bitr3i 276 . . . . . . . 8 (∃𝑧𝑡𝐿𝑢𝑆 (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))) ↔ ∃𝑡𝐿𝑢𝑆 (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)))
453441, 443, 4523bitr2i 298 . . . . . . 7 (∃𝑧 ∈ {𝑐 ∣ ∃𝑡𝐿𝑢𝑆 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)) ↔ ∃𝑡𝐿𝑢𝑆 (((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 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ⊆ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})
455 ssrexv 4048 . . . . . . . 8 ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ⊆ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}) → (∃𝑧 ∈ {𝑐 ∣ ∃𝑡𝐿𝑢𝑆 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 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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))))
456454, 455ax-mp 5 . . . . . . 7 (∃𝑧 ∈ {𝑐 ∣ ∃𝑡𝐿𝑢𝑆 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 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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)))
457453, 456sylbir 234 . . . . . 6 (∃𝑡𝐿𝑢𝑆 (((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 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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)))
4584572ralimi 3123 . . . . 5 (∀𝑙 ∈ ( L ‘𝐴)∀𝑚 ∈ ( R ‘𝐵)∃𝑡𝐿𝑢𝑆 (((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 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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)))
459438, 458syl 17 . . . 4 (𝜑 → ∀𝑙 ∈ ( L ‘𝐴)∀𝑚 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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)))
4602, 1cofcutr2d 27342 . . . . . 6 (𝜑 → ∀𝑥 ∈ ( R ‘𝐴)∃𝑣𝑅 𝑣 ≤s 𝑥)
4616, 5cofcutr1d 27341 . . . . . . . . . 10 (𝜑 → ∀𝑦 ∈ ( L ‘𝐵)∃𝑤𝑀 𝑦 ≤s 𝑤)
462461adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ ∃𝑣𝑅 𝑣 ≤s 𝑥)) → ∀𝑦 ∈ ( L ‘𝐵)∃𝑤𝑀 𝑦 ≤s 𝑤)
463 reeanv 3226 . . . . . . . . . . . . . . 15 (∃𝑣𝑅𝑤𝑀 (𝑣 ≤s 𝑥𝑦 ≤s 𝑤) ↔ (∃𝑣𝑅 𝑣 ≤s 𝑥 ∧ ∃𝑤𝑀 𝑦 ≤s 𝑤))
464170adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑣𝑅𝑤𝑀)) → 𝑅 No )
465 simprl 769 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑣𝑅𝑤𝑀)) → 𝑣𝑅)
466464, 465sseldd 3980 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑣𝑅𝑤𝑀)) → 𝑣 No )
4678adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑣𝑅𝑤𝑀)) → 𝐵 No )
468466, 467mulscld 27520 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑣𝑅𝑤𝑀)) → (𝑣 ·s 𝐵) ∈ No )
469468adantrl 714 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣𝑅𝑤𝑀))) → (𝑣 ·s 𝐵) ∈ No )
4704adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑣𝑅𝑤𝑀)) → 𝐴 No )
47144adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑣𝑅𝑤𝑀)) → 𝑀 No )
472 simprr 771 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑣𝑅𝑤𝑀)) → 𝑤𝑀)
473471, 472sseldd 3980 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑣𝑅𝑤𝑀)) → 𝑤 No )
474470, 473mulscld 27520 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑣𝑅𝑤𝑀)) → (𝐴 ·s 𝑤) ∈ No )
475474adantrl 714 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣𝑅𝑤𝑀))) → (𝐴 ·s 𝑤) ∈ No )
476469, 475addscld 27393 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣𝑅𝑤𝑀))) → ((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) ∈ No )
477466, 473mulscld 27520 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑣𝑅𝑤𝑀)) → (𝑣 ·s 𝑤) ∈ No )
478477adantrl 714 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣𝑅𝑤𝑀))) → (𝑣 ·s 𝑤) ∈ No )
479476, 478subscld 27464 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣𝑅𝑤𝑀))) → (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ∈ No )
480479adantrrr 723 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣𝑅𝑤𝑀) ∧ (𝑣 ≤s 𝑥𝑦 ≤s 𝑤)))) → (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ∈ No )
4814adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣𝑅𝑤𝑀))) → 𝐴 No )
482 simprr 771 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → 𝑦 ∈ ( L ‘𝐵))
48323, 482sselid 3977 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → 𝑦 No )
484483adantrr 715 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣𝑅𝑤𝑀))) → 𝑦 No )
485481, 484mulscld 27520 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣𝑅𝑤𝑀))) → (𝐴 ·s 𝑦) ∈ No )
486469, 485addscld 27393 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣𝑅𝑤𝑀))) → ((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑦)) ∈ No )
487466adantrl 714 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣𝑅𝑤𝑀))) → 𝑣 No )
488487, 484mulscld 27520 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣𝑅𝑤𝑀))) → (𝑣 ·s 𝑦) ∈ No )
489486, 488subscld 27464 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣𝑅𝑤𝑀))) → (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑣 ·s 𝑦)) ∈ No )
490489adantrrr 723 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣𝑅𝑤𝑀) ∧ (𝑣 ≤s 𝑥𝑦 ≤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))
492153, 491sselid 3977 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → 𝑥 No )
4938adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → 𝐵 No )
494492, 493mulscld 27520 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → (𝑥 ·s 𝐵) ∈ No )
495494adantrr 715 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣𝑅𝑤𝑀))) → (𝑥 ·s 𝐵) ∈ No )
496495, 485addscld 27393 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣𝑅𝑤𝑀))) → ((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) ∈ No )
497492, 483mulscld 27520 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → (𝑥 ·s 𝑦) ∈ No )
498497adantrr 715 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣𝑅𝑤𝑀))) → (𝑥 ·s 𝑦) ∈ No )
499496, 498subscld 27464 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣𝑅𝑤𝑀))) → (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) ∈ No )
500499adantrrr 723 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣𝑅𝑤𝑀) ∧ (𝑣 ≤s 𝑥𝑦 ≤s 𝑤)))) → (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) ∈ No )
5014adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣𝑅𝑤𝑀) ∧ (𝑣 ≤s 𝑥𝑦 ≤s 𝑤)))) → 𝐴 No )
502487adantrrr 723 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣𝑅𝑤𝑀) ∧ (𝑣 ≤s 𝑥𝑦 ≤s 𝑤)))) → 𝑣 No )
503483adantrr 715 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣𝑅𝑤𝑀) ∧ (𝑣 ≤s 𝑥𝑦 ≤s 𝑤)))) → 𝑦 No )
504473adantrl 714 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣𝑅𝑤𝑀))) → 𝑤 No )
505504adantrrr 723 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣𝑅𝑤𝑀) ∧ (𝑣 ≤s 𝑥𝑦 ≤s 𝑤)))) → 𝑤 No )
5061sneqd 4635 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝜑 → {𝐴} = {(𝐿 |s 𝑅)})
507506, 222eqbrtrd 5164 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑 → {𝐴} <<s 𝑅)
508507adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣𝑅𝑤𝑀))) → {𝐴} <<s 𝑅)
509481, 401syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣𝑅𝑤𝑀))) → 𝐴 ∈ {𝐴})
510 simprrl 779 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣𝑅𝑤𝑀))) → 𝑣𝑅)
511508, 509, 510ssltsepcd 27224 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣𝑅𝑤𝑀))) → 𝐴 <s 𝑣)
512481, 487, 511sltled 27201 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣𝑅𝑤𝑀))) → 𝐴 ≤s 𝑣)
513512adantrrr 723 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣𝑅𝑤𝑀) ∧ (𝑣 ≤s 𝑥𝑦 ≤s 𝑤)))) → 𝐴 ≤s 𝑣)
514 simprrr 780 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣𝑅𝑤𝑀) ∧ (𝑣 ≤s 𝑥𝑦 ≤s 𝑤))) → 𝑦 ≤s 𝑤)
515514adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣𝑅𝑤𝑀) ∧ (𝑣 ≤s 𝑥𝑦 ≤s 𝑤)))) → 𝑦 ≤s 𝑤)
516501, 502, 503, 505, 513, 515slemuld 27523 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣𝑅𝑤𝑀) ∧ (𝑣 ≤s 𝑥𝑦 ≤s 𝑤)))) → ((𝐴 ·s 𝑤) -s (𝐴 ·s 𝑦)) ≤s ((𝑣 ·s 𝑤) -s (𝑣 ·s 𝑦)))
517475, 478, 485, 488slesubsubbd 27482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣𝑅𝑤𝑀))) → (((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))))
518475, 478subscld 27464 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣𝑅𝑤𝑀))) → ((𝐴 ·s 𝑤) -s (𝑣 ·s 𝑤)) ∈ No )
519485, 488subscld 27464 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣𝑅𝑤𝑀))) → ((𝐴 ·s 𝑦) -s (𝑣 ·s 𝑦)) ∈ No )
520518, 519, 469sleadd2d 27408 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣𝑅𝑤𝑀))) → (((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)))))
521517, 520bitrd 278 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣𝑅𝑤𝑀))) → (((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)))))
522521adantrrr 723 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣𝑅𝑤𝑀) ∧ (𝑣 ≤s 𝑥𝑦 ≤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)))))
523516, 522mpbid 231 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣𝑅𝑤𝑀) ∧ (𝑣 ≤s 𝑥𝑦 ≤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))))
524469, 475, 478addsubsassd 27477 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣𝑅𝑤𝑀))) → (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) = ((𝑣 ·s 𝐵) +s ((𝐴 ·s 𝑤) -s (𝑣 ·s 𝑤))))
525524adantrrr 723 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣𝑅𝑤𝑀) ∧ (𝑣 ≤s 𝑥𝑦 ≤s 𝑤)))) → (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) = ((𝑣 ·s 𝐵) +s ((𝐴 ·s 𝑤) -s (𝑣 ·s 𝑤))))
526469, 485, 488addsubsassd 27477 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣𝑅𝑤𝑀))) → (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑣 ·s 𝑦)) = ((𝑣 ·s 𝐵) +s ((𝐴 ·s 𝑦) -s (𝑣 ·s 𝑦))))
527526adantrrr 723 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣𝑅𝑤𝑀) ∧ (𝑣 ≤s 𝑥𝑦 ≤s 𝑤)))) → (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑣 ·s 𝑦)) = ((𝑣 ·s 𝐵) +s ((𝐴 ·s 𝑦) -s (𝑣 ·s 𝑦))))
528523, 525, 5273brtr4d 5174 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣𝑅𝑤𝑀) ∧ (𝑣 ≤s 𝑥𝑦 ≤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)))
529492adantrr 715 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣𝑅𝑤𝑀) ∧ (𝑣 ≤s 𝑥𝑦 ≤s 𝑤)))) → 𝑥 No )
5308adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣𝑅𝑤𝑀) ∧ (𝑣 ≤s 𝑥𝑦 ≤s 𝑤)))) → 𝐵 No )
531 simprrl 779 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣𝑅𝑤𝑀) ∧ (𝑣 ≤s 𝑥𝑦 ≤s 𝑤))) → 𝑣 ≤s 𝑥)
532531adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣𝑅𝑤𝑀) ∧ (𝑣 ≤s 𝑥𝑦 ≤s 𝑤)))) → 𝑣 ≤s 𝑥)
533493, 61syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → ( L ‘𝐵) <<s {𝐵})
53465adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → 𝐵 ∈ {𝐵})
535533, 482, 534ssltsepcd 27224 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → 𝑦 <s 𝐵)
536483, 493, 535sltled 27201 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → 𝑦 ≤s 𝐵)
537536adantrr 715 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣𝑅𝑤𝑀) ∧ (𝑣 ≤s 𝑥𝑦 ≤s 𝑤)))) → 𝑦 ≤s 𝐵)
538502, 529, 503, 530, 532, 537slemuld 27523 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣𝑅𝑤𝑀) ∧ (𝑣 ≤s 𝑥𝑦 ≤s 𝑤)))) → ((𝑣 ·s 𝐵) -s (𝑣 ·s 𝑦)) ≤s ((𝑥 ·s 𝐵) -s (𝑥 ·s 𝑦)))
539469, 488subscld 27464 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣𝑅𝑤𝑀))) → ((𝑣 ·s 𝐵) -s (𝑣 ·s 𝑦)) ∈ No )
540539adantrrr 723 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣𝑅𝑤𝑀) ∧ (𝑣 ≤s 𝑥𝑦 ≤s 𝑤)))) → ((𝑣 ·s 𝐵) -s (𝑣 ·s 𝑦)) ∈ No )
541495, 498subscld 27464 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣𝑅𝑤𝑀))) → ((𝑥 ·s 𝐵) -s (𝑥 ·s 𝑦)) ∈ No )
542541adantrrr 723 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣𝑅𝑤𝑀) ∧ (𝑣 ≤s 𝑥𝑦 ≤s 𝑤)))) → ((𝑥 ·s 𝐵) -s (𝑥 ·s 𝑦)) ∈ No )
543485adantrrr 723 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣𝑅𝑤𝑀) ∧ (𝑣 ≤s 𝑥𝑦 ≤s 𝑤)))) → (𝐴 ·s 𝑦) ∈ No )
544540, 542, 543sleadd1d 27407 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣𝑅𝑤𝑀) ∧ (𝑣 ≤s 𝑥𝑦 ≤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))))
545538, 544mpbid 231 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣𝑅𝑤𝑀) ∧ (𝑣 ≤s 𝑥𝑦 ≤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)))
546469, 485, 488addsubsd 27478 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣𝑅𝑤𝑀))) → (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑣 ·s 𝑦)) = (((𝑣 ·s 𝐵) -s (𝑣 ·s 𝑦)) +s (𝐴 ·s 𝑦)))
547546adantrrr 723 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣𝑅𝑤𝑀) ∧ (𝑣 ≤s 𝑥𝑦 ≤s 𝑤)))) → (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑣 ·s 𝑦)) = (((𝑣 ·s 𝐵) -s (𝑣 ·s 𝑦)) +s (𝐴 ·s 𝑦)))
5484adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → 𝐴 No )
549548, 483mulscld 27520 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → (𝐴 ·s 𝑦) ∈ No )
550494, 549, 497addsubsd 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)))
551550adantrr 715 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣𝑅𝑤𝑀) ∧ (𝑣 ≤s 𝑥𝑦 ≤s 𝑤)))) → (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) = (((𝑥 ·s 𝐵) -s (𝑥 ·s 𝑦)) +s (𝐴 ·s 𝑦)))
552545, 547, 5513brtr4d 5174 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣𝑅𝑤𝑀) ∧ (𝑣 ≤s 𝑥𝑦 ≤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)))
553480, 490, 500, 528, 552sletrd 27194 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣𝑅𝑤𝑀) ∧ (𝑣 ≤s 𝑥𝑦 ≤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)))
554553anassrs 468 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) ∧ ((𝑣𝑅𝑤𝑀) ∧ (𝑣 ≤s 𝑥𝑦 ≤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)))
555554expr 457 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) ∧ (𝑣𝑅𝑤𝑀)) → ((𝑣 ≤s 𝑥𝑦 ≤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))))
556555reximdvva 3205 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → (∃𝑣𝑅𝑤𝑀 (𝑣 ≤s 𝑥𝑦 ≤s 𝑤) → ∃𝑣𝑅𝑤𝑀 (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))))
557556expcom 414 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) → (𝜑 → (∃𝑣𝑅𝑤𝑀 (𝑣 ≤s 𝑥𝑦 ≤s 𝑤) → ∃𝑣𝑅𝑤𝑀 (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)))))
558557com23 86 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) → (∃𝑣𝑅𝑤𝑀 (𝑣 ≤s 𝑥𝑦 ≤s 𝑤) → (𝜑 → ∃𝑣𝑅𝑤𝑀 (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)))))
559558imp 407 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ∃𝑣𝑅𝑤𝑀 (𝑣 ≤s 𝑥𝑦 ≤s 𝑤)) → (𝜑 → ∃𝑣𝑅𝑤𝑀 (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))))
560463, 559sylan2br 595 . . . . . . . . . . . . . 14 (((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (∃𝑣𝑅 𝑣 ≤s 𝑥 ∧ ∃𝑤𝑀 𝑦 ≤s 𝑤)) → (𝜑 → ∃𝑣𝑅𝑤𝑀 (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))))
561560an4s 658 . . . . . . . . . . . . 13 (((𝑥 ∈ ( R ‘𝐴) ∧ ∃𝑣𝑅 𝑣 ≤s 𝑥) ∧ (𝑦 ∈ ( L ‘𝐵) ∧ ∃𝑤𝑀 𝑦 ≤s 𝑤)) → (𝜑 → ∃𝑣𝑅𝑤𝑀 (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))))
562561impcom 408 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ ∃𝑣𝑅 𝑣 ≤s 𝑥) ∧ (𝑦 ∈ ( L ‘𝐵) ∧ ∃𝑤𝑀 𝑦 ≤s 𝑤))) → ∃𝑣𝑅𝑤𝑀 (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)))
563562anassrs 468 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ ∃𝑣𝑅 𝑣 ≤s 𝑥)) ∧ (𝑦 ∈ ( L ‘𝐵) ∧ ∃𝑤𝑀 𝑦 ≤s 𝑤)) → ∃𝑣𝑅𝑤𝑀 (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)))
564563expr 457 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ ∃𝑣𝑅 𝑣 ≤s 𝑥)) ∧ 𝑦 ∈ ( L ‘𝐵)) → (∃𝑤𝑀 𝑦 ≤s 𝑤 → ∃𝑣𝑅𝑤𝑀 (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))))
565564ralimdva 3167 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ ∃𝑣𝑅 𝑣 ≤s 𝑥)) → (∀𝑦 ∈ ( L ‘𝐵)∃𝑤𝑀 𝑦 ≤s 𝑤 → ∀𝑦 ∈ ( L ‘𝐵)∃𝑣𝑅𝑤𝑀 (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))))
566462, 565mpd 15 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ ∃𝑣𝑅 𝑣 ≤s 𝑥)) → ∀𝑦 ∈ ( L ‘𝐵)∃𝑣𝑅𝑤𝑀 (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)))
567566expr 457 . . . . . . 7 ((𝜑𝑥 ∈ ( R ‘𝐴)) → (∃𝑣𝑅 𝑣 ≤s 𝑥 → ∀𝑦 ∈ ( L ‘𝐵)∃𝑣𝑅𝑤𝑀 (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))))
568567ralimdva 3167 . . . . . 6 (𝜑 → (∀𝑥 ∈ ( R ‘𝐴)∃𝑣𝑅 𝑣 ≤s 𝑥 → ∀𝑥 ∈ ( R ‘𝐴)∀𝑦 ∈ ( L ‘𝐵)∃𝑣𝑅𝑤𝑀 (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))))
569460, 568mpd 15 . . . . 5 (𝜑 → ∀𝑥 ∈ ( R ‘𝐴)∀𝑦 ∈ ( L ‘𝐵)∃𝑣𝑅𝑤𝑀 (((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))))
5715702rexbidv 3219 . . . . . . . . 9 (𝑑 = 𝑧 → (∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ↔ ∃𝑣𝑅𝑤𝑀 𝑧 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))))
572571rexab 3687 . . . . . . . 8 (∃𝑧 ∈ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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(∃𝑣𝑅𝑤𝑀 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 (∃𝑣𝑅𝑤𝑀 (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))) ↔ (∃𝑣𝑅𝑤𝑀 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))))
574573exbii 1850 . . . . . . . 8 (∃𝑧𝑣𝑅𝑤𝑀 (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(∃𝑣𝑅𝑤𝑀 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 (∃𝑣𝑅𝑧𝑤𝑀 (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))) ↔ ∃𝑧𝑣𝑅𝑤𝑀 (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 (∃𝑤𝑀𝑧(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))) ↔ ∃𝑧𝑤𝑀 (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))))
579577, 578ceqsexv 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)))
580579rexbii 3094 . . . . . . . . . . 11 (∃𝑤𝑀𝑧(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))) ↔ ∃𝑤𝑀 (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)))
581576, 580bitr3i 276 . . . . . . . . . 10 (∃𝑧𝑤𝑀 (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))) ↔ ∃𝑤𝑀 (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)))
582581rexbii 3094 . . . . . . . . 9 (∃𝑣𝑅𝑧𝑤𝑀 (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))) ↔ ∃𝑣𝑅𝑤𝑀 (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)))
583575, 582bitr3i 276 . . . . . . . 8 (∃𝑧𝑣𝑅𝑤𝑀 (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))) ↔ ∃𝑣𝑅𝑤𝑀 (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)))
584572, 574, 5833bitr2i 298 . . . . . . 7 (∃𝑧 ∈ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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)) ↔ ∃𝑣𝑅𝑤𝑀 (((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 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))} ⊆ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})
586 ssrexv 4048 . . . . . . . 8 ({𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))} ⊆ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}) → (∃𝑧 ∈ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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))))
587585, 586ax-mp 5 . . . . . . 7 (∃𝑧 ∈ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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)))
588584, 587sylbir 234 . . . . . 6 (∃𝑣𝑅𝑤𝑀 (((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 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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)))
5895882ralimi 3123 . . . . 5 (∀𝑥 ∈ ( R ‘𝐴)∀𝑦 ∈ ( L ‘𝐵)∃𝑣𝑅𝑤𝑀 (((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 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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)))
590569, 589syl 17 . . . 4 (𝜑 → ∀𝑥 ∈ ( R ‘𝐴)∀𝑦 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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))))
5935922rexbidv 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))))
594593ralab 3684 . . . . . . 7 (∀𝑥𝑂 ∈ {𝑘 ∣ ∃𝑙 ∈ ( L ‘𝐴)∃𝑚 ∈ ( R ‘𝐵)𝑘 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))}∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂))
596595ralbii 3093 . . . . . . . . 9 (∀𝑙 ∈ ( L ‘𝐴)∀𝑚 ∈ ( R ‘𝐵)(𝑥𝑂 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂))
598596, 597bitri 274 . . . . . . . 8 (∀𝑙 ∈ ( L ‘𝐴)∀𝑚 ∈ ( R ‘𝐵)(𝑥𝑂 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂))
599598albii 1821 . . . . . . 7 (∀𝑥𝑂𝑙 ∈ ( L ‘𝐴)∀𝑚 ∈ ( R ‘𝐵)(𝑥𝑂 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂))
600 ralcom4 3283 . . . . . . . 8 (∀𝑙 ∈ ( L ‘𝐴)∀𝑥𝑂𝑚 ∈ ( R ‘𝐵)(𝑥𝑂 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂) ↔ ∀𝑥𝑂𝑙 ∈ ( L ‘𝐴)∀𝑚 ∈ ( R ‘𝐵)(𝑥𝑂 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂) ↔ ∀𝑥𝑂𝑚 ∈ ( R ‘𝐵)(𝑥𝑂 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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 𝑥𝑂𝑧 ≤s (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))))
604603rexbidv 3178 . . . . . . . . . . . 12 (𝑥𝑂 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) → (∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂 ↔ ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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))))
605602, 604ceqsalv 3510 . . . . . . . . . . 11 (∀𝑥𝑂(𝑥𝑂 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂) ↔ ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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)))
606605ralbii 3093 . . . . . . . . . 10 (∀𝑚 ∈ ( R ‘𝐵)∀𝑥𝑂(𝑥𝑂 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂) ↔ ∀𝑚 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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)))
607601, 606bitr3i 276 . . . . . . . . 9 (∀𝑥𝑂𝑚 ∈ ( R ‘𝐵)(𝑥𝑂 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂) ↔ ∀𝑚 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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)))
608607ralbii 3093 . . . . . . . 8 (∀𝑙 ∈ ( L ‘𝐴)∀𝑥𝑂𝑚 ∈ ( R ‘𝐵)(𝑥𝑂 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂) ↔ ∀𝑙 ∈ ( L ‘𝐴)∀𝑚 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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)))
609600, 608bitr3i 276 . . . . . . 7 (∀𝑥𝑂𝑙 ∈ ( L ‘𝐴)∀𝑚 ∈ ( R ‘𝐵)(𝑥𝑂 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂) ↔ ∀𝑙 ∈ ( L ‘𝐴)∀𝑚 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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)))
610594, 599, 6093bitr2i 298 . . . . . 6 (∀𝑥𝑂 ∈ {𝑘 ∣ ∃𝑙 ∈ ( L ‘𝐴)∃𝑚 ∈ ( R ‘𝐵)𝑘 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))}∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂 ↔ ∀𝑙 ∈ ( L ‘𝐴)∀𝑚 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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))))
6126112rexbidv 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))))
613612ralab 3684 . . . . . . 7 (∀𝑥𝑂 ∈ {𝑛 ∣ ∃𝑥 ∈ ( R ‘𝐴)∃𝑦 ∈ ( L ‘𝐵)𝑛 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))}∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂))
615614ralbii 3093 . . . . . . . . 9 (∀𝑥 ∈ ( R ‘𝐴)∀𝑦 ∈ ( L ‘𝐵)(𝑥𝑂 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂))
617615, 616bitri 274 . . . . . . . 8 (∀𝑥 ∈ ( R ‘𝐴)∀𝑦 ∈ ( L ‘𝐵)(𝑥𝑂 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂))
618617albii 1821 . . . . . . 7 (∀𝑥𝑂𝑥 ∈ ( R ‘𝐴)∀𝑦 ∈ ( L ‘𝐵)(𝑥𝑂 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂))
619 ralcom4 3283 . . . . . . . 8 (∀𝑥 ∈ ( R ‘𝐴)∀𝑥𝑂𝑦 ∈ ( L ‘𝐵)(𝑥𝑂 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂) ↔ ∀𝑥𝑂𝑥 ∈ ( R ‘𝐴)∀𝑦 ∈ ( L ‘𝐵)(𝑥𝑂 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂) ↔ ∀𝑥𝑂𝑦 ∈ ( L ‘𝐵)(𝑥𝑂 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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 𝑥𝑂𝑧 ≤s (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))))
623622rexbidv 3178 . . . . . . . . . . . 12 (𝑥𝑂 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) → (∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂 ↔ ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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))))
624621, 623ceqsalv 3510 . . . . . . . . . . 11 (∀𝑥𝑂(𝑥𝑂 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂) ↔ ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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)))
625624ralbii 3093 . . . . . . . . . 10 (∀𝑦 ∈ ( L ‘𝐵)∀𝑥𝑂(𝑥𝑂 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂) ↔ ∀𝑦 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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)))
626620, 625bitr3i 276 . . . . . . . . 9 (∀𝑥𝑂𝑦 ∈ ( L ‘𝐵)(𝑥𝑂 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂) ↔ ∀𝑦 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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)))
627626ralbii 3093 . . . . . . . 8 (∀𝑥 ∈ ( R ‘𝐴)∀𝑥𝑂𝑦 ∈ ( L ‘𝐵)(𝑥𝑂 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂) ↔ ∀𝑥 ∈ ( R ‘𝐴)∀𝑦 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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)))
628619, 627bitr3i 276 . . . . . . 7 (∀𝑥𝑂𝑥 ∈ ( R ‘𝐴)∀𝑦 ∈ ( L ‘𝐵)(𝑥𝑂 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂) ↔ ∀𝑥 ∈ ( R ‘𝐴)∀𝑦 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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)))
629613, 618, 6283bitr2i 298 . . . . . 6 (∀𝑥𝑂 ∈ {𝑛 ∣ ∃𝑥 ∈ ( R ‘𝐴)∃𝑦 ∈ ( L ‘𝐵)𝑛 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦))}∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂 ↔ ∀𝑥 ∈ ( R ‘𝐴)∀𝑦 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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)))
630610, 629anbi12i 627 . . . . 5 ((∀𝑥𝑂 ∈ {𝑘 ∣ ∃𝑙 ∈ ( L ‘𝐴)∃𝑚 ∈ ( R ‘𝐵)𝑘 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))}∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂) ↔ (∀𝑙 ∈ ( L ‘𝐴)∀𝑚 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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))))
631591, 630bitri 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 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂 ↔ (∀𝑙 ∈ ( L ‘𝐴)∀𝑚 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 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))))
632459, 590, 631sylanbrc 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 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})𝑧 ≤s 𝑥𝑂)
6332, 6, 1, 5ssltmul1 27531 . . . 4 (𝜑 → ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) <<s {(𝐴 ·s 𝐵)})
63410sneqd 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))} ∪ { ∣ ∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵) = (((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))}))})
635633, 634breqtrd 5168 . . 3 (𝜑 → ({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 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))} ∪ { ∣ ∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵) = (((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))}))})
6362, 6, 1, 5ssltmul2 27532 . . . 4 (𝜑 → {(𝐴 ·s 𝐵)} <<s ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}))
637634, 636eqbrtrrd 5166 . . 3 (𝜑 → {(({𝑒 ∣ ∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑒 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔))} ∪ { ∣ ∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵) = (((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 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}))
63811, 323, 632, 635, 637cofcut1d 27337 . 2 (𝜑 → (({𝑒 ∣ ∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑒 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔))} ∪ { ∣ ∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵) = (((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 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) |s ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})))
63910, 638eqtrd 2772 1 (𝜑 → (𝐴 ·s 𝐵) = (({𝑎 ∣ ∃𝑝𝐿𝑞𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟𝑅𝑠𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) |s ({𝑐 ∣ ∃𝑡𝐿𝑢𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣𝑅𝑤𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087  wal 1539   = wceq 1541  wex 1781  wcel 2106  {cab 2709  wral 3061  wrex 3070  cun 3943  wss 3945  {csn 4623   class class class wbr 5142  cfv 6533  (class class class)co 7394   No csur 27072   ≤s csle 27176   <<s csslt 27211   |s cscut 27213   L cleft 27269   R cright 27270   +s cadds 27372   -s csubs 27424   ·s cmuls 27491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5357  ax-pr 5421  ax-un 7709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-ot 4632  df-uni 4903  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-se 5626  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-pred 6290  df-ord 6357  df-on 6358  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7350  df-ov 7397  df-oprab 7398  df-mpo 7399  df-1st 7959  df-2nd 7960  df-frecs 8250  df-wrecs 8281  df-recs 8355  df-1o 8450  df-2o 8451  df-nadd 8650  df-no 27075  df-slt 27076  df-bday 27077  df-sle 27177  df-sslt 27212  df-scut 27214  df-0s 27254  df-made 27271  df-old 27272  df-left 27274  df-right 27275  df-norec 27351  df-norec2 27362  df-adds 27373  df-negs 27425  df-subs 27426  df-muls 27492
This theorem is referenced by:  mulsunif  27534
  Copyright terms: Public domain W3C validator