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Theorem mulscom 28099
Description: Surreal multiplication commutes. Part of theorem 7 of [Conway] p. 19. (Contributed by Scott Fenton, 6-Mar-2025.)
Assertion
Ref Expression
mulscom ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ·s 𝐵) = (𝐵 ·s 𝐴))
Proof of Theorem mulscom
Dummy variables 𝑥 𝑦 𝑥𝑂 𝑦𝑂 𝑎 𝑏 𝑐 𝑑 𝑝 𝑞 𝑟 𝑠 𝑡 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 7417 . . 3 (𝑥 = 𝑥𝑂 → (𝑥 ·s 𝑦) = (𝑥𝑂 ·s 𝑦))
2 oveq2 7418 . . 3 (𝑥 = 𝑥𝑂 → (𝑦 ·s 𝑥) = (𝑦 ·s 𝑥𝑂))
31, 2eqeq12d 2752 . 2 (𝑥 = 𝑥𝑂 → ((𝑥 ·s 𝑦) = (𝑦 ·s 𝑥) ↔ (𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂)))
4 oveq2 7418 . . 3 (𝑦 = 𝑦𝑂 → (𝑥𝑂 ·s 𝑦) = (𝑥𝑂 ·s 𝑦𝑂))
5 oveq1 7417 . . 3 (𝑦 = 𝑦𝑂 → (𝑦 ·s 𝑥𝑂) = (𝑦𝑂 ·s 𝑥𝑂))
64, 5eqeq12d 2752 . 2 (𝑦 = 𝑦𝑂 → ((𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ↔ (𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂)))
7 oveq1 7417 . . 3 (𝑥 = 𝑥𝑂 → (𝑥 ·s 𝑦𝑂) = (𝑥𝑂 ·s 𝑦𝑂))
8 oveq2 7418 . . 3 (𝑥 = 𝑥𝑂 → (𝑦𝑂 ·s 𝑥) = (𝑦𝑂 ·s 𝑥𝑂))
97, 8eqeq12d 2752 . 2 (𝑥 = 𝑥𝑂 → ((𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥) ↔ (𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂)))
10 oveq1 7417 . . 3 (𝑥 = 𝐴 → (𝑥 ·s 𝑦) = (𝐴 ·s 𝑦))
11 oveq2 7418 . . 3 (𝑥 = 𝐴 → (𝑦 ·s 𝑥) = (𝑦 ·s 𝐴))
1210, 11eqeq12d 2752 . 2 (𝑥 = 𝐴 → ((𝑥 ·s 𝑦) = (𝑦 ·s 𝑥) ↔ (𝐴 ·s 𝑦) = (𝑦 ·s 𝐴)))
13 oveq2 7418 . . 3 (𝑦 = 𝐵 → (𝐴 ·s 𝑦) = (𝐴 ·s 𝐵))
14 oveq1 7417 . . 3 (𝑦 = 𝐵 → (𝑦 ·s 𝐴) = (𝐵 ·s 𝐴))
1513, 14eqeq12d 2752 . 2 (𝑦 = 𝐵 → ((𝐴 ·s 𝑦) = (𝑦 ·s 𝐴) ↔ (𝐴 ·s 𝐵) = (𝐵 ·s 𝐴)))
16 oveq1 7417 . . . . . . . . . . . . . . 15 (𝑥𝑂 = 𝑝 → (𝑥𝑂 ·s 𝑦) = (𝑝 ·s 𝑦))
17 oveq2 7418 . . . . . . . . . . . . . . 15 (𝑥𝑂 = 𝑝 → (𝑦 ·s 𝑥𝑂) = (𝑦 ·s 𝑝))
1816, 17eqeq12d 2752 . . . . . . . . . . . . . 14 (𝑥𝑂 = 𝑝 → ((𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ↔ (𝑝 ·s 𝑦) = (𝑦 ·s 𝑝)))
19 simplr2 1217 . . . . . . . . . . . . . 14 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑝 ∈ ( L ‘𝑥) ∧ 𝑞 ∈ ( L ‘𝑦))) → ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂))
20 simprl 770 . . . . . . . . . . . . . . 15 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑝 ∈ ( L ‘𝑥) ∧ 𝑞 ∈ ( L ‘𝑦))) → 𝑝 ∈ ( L ‘𝑥))
21 elun1 4162 . . . . . . . . . . . . . . 15 (𝑝 ∈ ( L ‘𝑥) → 𝑝 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
2220, 21syl 17 . . . . . . . . . . . . . 14 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑝 ∈ ( L ‘𝑥) ∧ 𝑞 ∈ ( L ‘𝑦))) → 𝑝 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
2318, 19, 22rspcdva 3607 . . . . . . . . . . . . 13 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑝 ∈ ( L ‘𝑥) ∧ 𝑞 ∈ ( L ‘𝑦))) → (𝑝 ·s 𝑦) = (𝑦 ·s 𝑝))
24 oveq2 7418 . . . . . . . . . . . . . . 15 (𝑦𝑂 = 𝑞 → (𝑥 ·s 𝑦𝑂) = (𝑥 ·s 𝑞))
25 oveq1 7417 . . . . . . . . . . . . . . 15 (𝑦𝑂 = 𝑞 → (𝑦𝑂 ·s 𝑥) = (𝑞 ·s 𝑥))
2624, 25eqeq12d 2752 . . . . . . . . . . . . . 14 (𝑦𝑂 = 𝑞 → ((𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥) ↔ (𝑥 ·s 𝑞) = (𝑞 ·s 𝑥)))
27 simplr3 1218 . . . . . . . . . . . . . 14 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑝 ∈ ( L ‘𝑥) ∧ 𝑞 ∈ ( L ‘𝑦))) → ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))
28 simprr 772 . . . . . . . . . . . . . . 15 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑝 ∈ ( L ‘𝑥) ∧ 𝑞 ∈ ( L ‘𝑦))) → 𝑞 ∈ ( L ‘𝑦))
29 elun1 4162 . . . . . . . . . . . . . . 15 (𝑞 ∈ ( L ‘𝑦) → 𝑞 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
3028, 29syl 17 . . . . . . . . . . . . . 14 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑝 ∈ ( L ‘𝑥) ∧ 𝑞 ∈ ( L ‘𝑦))) → 𝑞 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
3126, 27, 30rspcdva 3607 . . . . . . . . . . . . 13 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑝 ∈ ( L ‘𝑥) ∧ 𝑞 ∈ ( L ‘𝑦))) → (𝑥 ·s 𝑞) = (𝑞 ·s 𝑥))
3223, 31oveq12d 7428 . . . . . . . . . . . 12 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑝 ∈ ( L ‘𝑥) ∧ 𝑞 ∈ ( L ‘𝑦))) → ((𝑝 ·s 𝑦) +s (𝑥 ·s 𝑞)) = ((𝑦 ·s 𝑝) +s (𝑞 ·s 𝑥)))
33 simpllr 775 . . . . . . . . . . . . . 14 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑝 ∈ ( L ‘𝑥) ∧ 𝑞 ∈ ( L ‘𝑦))) → 𝑦 ∈ No )
34 leftssno 27849 . . . . . . . . . . . . . . 15 ( L ‘𝑥) ⊆ No
3534, 20sselid 3961 . . . . . . . . . . . . . 14 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑝 ∈ ( L ‘𝑥) ∧ 𝑞 ∈ ( L ‘𝑦))) → 𝑝 ∈ No )
3633, 35mulscld 28095 . . . . . . . . . . . . 13 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑝 ∈ ( L ‘𝑥) ∧ 𝑞 ∈ ( L ‘𝑦))) → (𝑦 ·s 𝑝) ∈ No )
37 leftssno 27849 . . . . . . . . . . . . . . 15 ( L ‘𝑦) ⊆ No
3837, 28sselid 3961 . . . . . . . . . . . . . 14 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑝 ∈ ( L ‘𝑥) ∧ 𝑞 ∈ ( L ‘𝑦))) → 𝑞 ∈ No )
39 simplll 774 . . . . . . . . . . . . . 14 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑝 ∈ ( L ‘𝑥) ∧ 𝑞 ∈ ( L ‘𝑦))) → 𝑥 ∈ No )
4038, 39mulscld 28095 . . . . . . . . . . . . 13 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑝 ∈ ( L ‘𝑥) ∧ 𝑞 ∈ ( L ‘𝑦))) → (𝑞 ·s 𝑥) ∈ No )
4136, 40addscomd 27931 . . . . . . . . . . . 12 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑝 ∈ ( L ‘𝑥) ∧ 𝑞 ∈ ( L ‘𝑦))) → ((𝑦 ·s 𝑝) +s (𝑞 ·s 𝑥)) = ((𝑞 ·s 𝑥) +s (𝑦 ·s 𝑝)))
4232, 41eqtrd 2771 . . . . . . . . . . 11 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑝 ∈ ( L ‘𝑥) ∧ 𝑞 ∈ ( L ‘𝑦))) → ((𝑝 ·s 𝑦) +s (𝑥 ·s 𝑞)) = ((𝑞 ·s 𝑥) +s (𝑦 ·s 𝑝)))
43 oveq1 7417 . . . . . . . . . . . . 13 (𝑥𝑂 = 𝑝 → (𝑥𝑂 ·s 𝑦𝑂) = (𝑝 ·s 𝑦𝑂))
44 oveq2 7418 . . . . . . . . . . . . 13 (𝑥𝑂 = 𝑝 → (𝑦𝑂 ·s 𝑥𝑂) = (𝑦𝑂 ·s 𝑝))
4543, 44eqeq12d 2752 . . . . . . . . . . . 12 (𝑥𝑂 = 𝑝 → ((𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ↔ (𝑝 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑝)))
46 oveq2 7418 . . . . . . . . . . . . 13 (𝑦𝑂 = 𝑞 → (𝑝 ·s 𝑦𝑂) = (𝑝 ·s 𝑞))
47 oveq1 7417 . . . . . . . . . . . . 13 (𝑦𝑂 = 𝑞 → (𝑦𝑂 ·s 𝑝) = (𝑞 ·s 𝑝))
4846, 47eqeq12d 2752 . . . . . . . . . . . 12 (𝑦𝑂 = 𝑞 → ((𝑝 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑝) ↔ (𝑝 ·s 𝑞) = (𝑞 ·s 𝑝)))
49 simplr1 1216 . . . . . . . . . . . 12 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑝 ∈ ( L ‘𝑥) ∧ 𝑞 ∈ ( L ‘𝑦))) → ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂))
5045, 48, 49, 22, 30rspc2dv 3621 . . . . . . . . . . 11 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑝 ∈ ( L ‘𝑥) ∧ 𝑞 ∈ ( L ‘𝑦))) → (𝑝 ·s 𝑞) = (𝑞 ·s 𝑝))
5142, 50oveq12d 7428 . . . . . . . . . 10 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑝 ∈ ( L ‘𝑥) ∧ 𝑞 ∈ ( L ‘𝑦))) → (((𝑝 ·s 𝑦) +s (𝑥 ·s 𝑞)) -s (𝑝 ·s 𝑞)) = (((𝑞 ·s 𝑥) +s (𝑦 ·s 𝑝)) -s (𝑞 ·s 𝑝)))
5251eqeq2d 2747 . . . . . . . . 9 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑝 ∈ ( L ‘𝑥) ∧ 𝑞 ∈ ( L ‘𝑦))) → (𝑎 = (((𝑝 ·s 𝑦) +s (𝑥 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ↔ 𝑎 = (((𝑞 ·s 𝑥) +s (𝑦 ·s 𝑝)) -s (𝑞 ·s 𝑝))))
53522rexbidva 3208 . . . . . . . 8 (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) → (∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝 ·s 𝑦) +s (𝑥 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ↔ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑞 ·s 𝑥) +s (𝑦 ·s 𝑝)) -s (𝑞 ·s 𝑝))))
54 rexcom 3275 . . . . . . . 8 (∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑞 ·s 𝑥) +s (𝑦 ·s 𝑝)) -s (𝑞 ·s 𝑝)) ↔ ∃𝑞 ∈ ( L ‘𝑦)∃𝑝 ∈ ( L ‘𝑥)𝑎 = (((𝑞 ·s 𝑥) +s (𝑦 ·s 𝑝)) -s (𝑞 ·s 𝑝)))
5553, 54bitrdi 287 . . . . . . 7 (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) → (∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝 ·s 𝑦) +s (𝑥 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ↔ ∃𝑞 ∈ ( L ‘𝑦)∃𝑝 ∈ ( L ‘𝑥)𝑎 = (((𝑞 ·s 𝑥) +s (𝑦 ·s 𝑝)) -s (𝑞 ·s 𝑝))))
5655abbidv 2802 . . . . . 6 (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) → {𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝 ·s 𝑦) +s (𝑥 ·s 𝑞)) -s (𝑝 ·s 𝑞))} = {𝑎 ∣ ∃𝑞 ∈ ( L ‘𝑦)∃𝑝 ∈ ( L ‘𝑥)𝑎 = (((𝑞 ·s 𝑥) +s (𝑦 ·s 𝑝)) -s (𝑞 ·s 𝑝))})
57 oveq1 7417 . . . . . . . . . . . . . . 15 (𝑥𝑂 = 𝑟 → (𝑥𝑂 ·s 𝑦) = (𝑟 ·s 𝑦))
58 oveq2 7418 . . . . . . . . . . . . . . 15 (𝑥𝑂 = 𝑟 → (𝑦 ·s 𝑥𝑂) = (𝑦 ·s 𝑟))
5957, 58eqeq12d 2752 . . . . . . . . . . . . . 14 (𝑥𝑂 = 𝑟 → ((𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ↔ (𝑟 ·s 𝑦) = (𝑦 ·s 𝑟)))
60 simplr2 1217 . . . . . . . . . . . . . 14 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑟 ∈ ( R ‘𝑥) ∧ 𝑠 ∈ ( R ‘𝑦))) → ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂))
61 simprl 770 . . . . . . . . . . . . . . 15 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑟 ∈ ( R ‘𝑥) ∧ 𝑠 ∈ ( R ‘𝑦))) → 𝑟 ∈ ( R ‘𝑥))
62 elun2 4163 . . . . . . . . . . . . . . 15 (𝑟 ∈ ( R ‘𝑥) → 𝑟 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
6361, 62syl 17 . . . . . . . . . . . . . 14 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑟 ∈ ( R ‘𝑥) ∧ 𝑠 ∈ ( R ‘𝑦))) → 𝑟 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
6459, 60, 63rspcdva 3607 . . . . . . . . . . . . 13 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑟 ∈ ( R ‘𝑥) ∧ 𝑠 ∈ ( R ‘𝑦))) → (𝑟 ·s 𝑦) = (𝑦 ·s 𝑟))
65 oveq2 7418 . . . . . . . . . . . . . . 15 (𝑦𝑂 = 𝑠 → (𝑥 ·s 𝑦𝑂) = (𝑥 ·s 𝑠))
66 oveq1 7417 . . . . . . . . . . . . . . 15 (𝑦𝑂 = 𝑠 → (𝑦𝑂 ·s 𝑥) = (𝑠 ·s 𝑥))
6765, 66eqeq12d 2752 . . . . . . . . . . . . . 14 (𝑦𝑂 = 𝑠 → ((𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥) ↔ (𝑥 ·s 𝑠) = (𝑠 ·s 𝑥)))
68 simplr3 1218 . . . . . . . . . . . . . 14 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑟 ∈ ( R ‘𝑥) ∧ 𝑠 ∈ ( R ‘𝑦))) → ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))
69 simprr 772 . . . . . . . . . . . . . . 15 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑟 ∈ ( R ‘𝑥) ∧ 𝑠 ∈ ( R ‘𝑦))) → 𝑠 ∈ ( R ‘𝑦))
70 elun2 4163 . . . . . . . . . . . . . . 15 (𝑠 ∈ ( R ‘𝑦) → 𝑠 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
7169, 70syl 17 . . . . . . . . . . . . . 14 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑟 ∈ ( R ‘𝑥) ∧ 𝑠 ∈ ( R ‘𝑦))) → 𝑠 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
7267, 68, 71rspcdva 3607 . . . . . . . . . . . . 13 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑟 ∈ ( R ‘𝑥) ∧ 𝑠 ∈ ( R ‘𝑦))) → (𝑥 ·s 𝑠) = (𝑠 ·s 𝑥))
7364, 72oveq12d 7428 . . . . . . . . . . . 12 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑟 ∈ ( R ‘𝑥) ∧ 𝑠 ∈ ( R ‘𝑦))) → ((𝑟 ·s 𝑦) +s (𝑥 ·s 𝑠)) = ((𝑦 ·s 𝑟) +s (𝑠 ·s 𝑥)))
74 simpllr 775 . . . . . . . . . . . . . 14 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑟 ∈ ( R ‘𝑥) ∧ 𝑠 ∈ ( R ‘𝑦))) → 𝑦 ∈ No )
75 rightssno 27850 . . . . . . . . . . . . . . 15 ( R ‘𝑥) ⊆ No
7675, 61sselid 3961 . . . . . . . . . . . . . 14 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑟 ∈ ( R ‘𝑥) ∧ 𝑠 ∈ ( R ‘𝑦))) → 𝑟 ∈ No )
7774, 76mulscld 28095 . . . . . . . . . . . . 13 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑟 ∈ ( R ‘𝑥) ∧ 𝑠 ∈ ( R ‘𝑦))) → (𝑦 ·s 𝑟) ∈ No )
78 rightssno 27850 . . . . . . . . . . . . . . 15 ( R ‘𝑦) ⊆ No
7978, 69sselid 3961 . . . . . . . . . . . . . 14 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑟 ∈ ( R ‘𝑥) ∧ 𝑠 ∈ ( R ‘𝑦))) → 𝑠 ∈ No )
80 simplll 774 . . . . . . . . . . . . . 14 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑟 ∈ ( R ‘𝑥) ∧ 𝑠 ∈ ( R ‘𝑦))) → 𝑥 ∈ No )
8179, 80mulscld 28095 . . . . . . . . . . . . 13 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑟 ∈ ( R ‘𝑥) ∧ 𝑠 ∈ ( R ‘𝑦))) → (𝑠 ·s 𝑥) ∈ No )
8277, 81addscomd 27931 . . . . . . . . . . . 12 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑟 ∈ ( R ‘𝑥) ∧ 𝑠 ∈ ( R ‘𝑦))) → ((𝑦 ·s 𝑟) +s (𝑠 ·s 𝑥)) = ((𝑠 ·s 𝑥) +s (𝑦 ·s 𝑟)))
8373, 82eqtrd 2771 . . . . . . . . . . 11 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑟 ∈ ( R ‘𝑥) ∧ 𝑠 ∈ ( R ‘𝑦))) → ((𝑟 ·s 𝑦) +s (𝑥 ·s 𝑠)) = ((𝑠 ·s 𝑥) +s (𝑦 ·s 𝑟)))
84 oveq1 7417 . . . . . . . . . . . . 13 (𝑥𝑂 = 𝑟 → (𝑥𝑂 ·s 𝑦𝑂) = (𝑟 ·s 𝑦𝑂))
85 oveq2 7418 . . . . . . . . . . . . 13 (𝑥𝑂 = 𝑟 → (𝑦𝑂 ·s 𝑥𝑂) = (𝑦𝑂 ·s 𝑟))
8684, 85eqeq12d 2752 . . . . . . . . . . . 12 (𝑥𝑂 = 𝑟 → ((𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ↔ (𝑟 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑟)))
87 oveq2 7418 . . . . . . . . . . . . 13 (𝑦𝑂 = 𝑠 → (𝑟 ·s 𝑦𝑂) = (𝑟 ·s 𝑠))
88 oveq1 7417 . . . . . . . . . . . . 13 (𝑦𝑂 = 𝑠 → (𝑦𝑂 ·s 𝑟) = (𝑠 ·s 𝑟))
8987, 88eqeq12d 2752 . . . . . . . . . . . 12 (𝑦𝑂 = 𝑠 → ((𝑟 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑟) ↔ (𝑟 ·s 𝑠) = (𝑠 ·s 𝑟)))
90 simplr1 1216 . . . . . . . . . . . 12 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑟 ∈ ( R ‘𝑥) ∧ 𝑠 ∈ ( R ‘𝑦))) → ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂))
9186, 89, 90, 63, 71rspc2dv 3621 . . . . . . . . . . 11 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑟 ∈ ( R ‘𝑥) ∧ 𝑠 ∈ ( R ‘𝑦))) → (𝑟 ·s 𝑠) = (𝑠 ·s 𝑟))
9283, 91oveq12d 7428 . . . . . . . . . 10 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑟 ∈ ( R ‘𝑥) ∧ 𝑠 ∈ ( R ‘𝑦))) → (((𝑟 ·s 𝑦) +s (𝑥 ·s 𝑠)) -s (𝑟 ·s 𝑠)) = (((𝑠 ·s 𝑥) +s (𝑦 ·s 𝑟)) -s (𝑠 ·s 𝑟)))
9392eqeq2d 2747 . . . . . . . . 9 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑟 ∈ ( R ‘𝑥) ∧ 𝑠 ∈ ( R ‘𝑦))) → (𝑏 = (((𝑟 ·s 𝑦) +s (𝑥 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ↔ 𝑏 = (((𝑠 ·s 𝑥) +s (𝑦 ·s 𝑟)) -s (𝑠 ·s 𝑟))))
94932rexbidva 3208 . . . . . . . 8 (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) → (∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟 ·s 𝑦) +s (𝑥 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ↔ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑠 ·s 𝑥) +s (𝑦 ·s 𝑟)) -s (𝑠 ·s 𝑟))))
95 rexcom 3275 . . . . . . . 8 (∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑠 ·s 𝑥) +s (𝑦 ·s 𝑟)) -s (𝑠 ·s 𝑟)) ↔ ∃𝑠 ∈ ( R ‘𝑦)∃𝑟 ∈ ( R ‘𝑥)𝑏 = (((𝑠 ·s 𝑥) +s (𝑦 ·s 𝑟)) -s (𝑠 ·s 𝑟)))
9694, 95bitrdi 287 . . . . . . 7 (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) → (∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟 ·s 𝑦) +s (𝑥 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ↔ ∃𝑠 ∈ ( R ‘𝑦)∃𝑟 ∈ ( R ‘𝑥)𝑏 = (((𝑠 ·s 𝑥) +s (𝑦 ·s 𝑟)) -s (𝑠 ·s 𝑟))))
9796abbidv 2802 . . . . . 6 (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) → {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟 ·s 𝑦) +s (𝑥 ·s 𝑠)) -s (𝑟 ·s 𝑠))} = {𝑏 ∣ ∃𝑠 ∈ ( R ‘𝑦)∃𝑟 ∈ ( R ‘𝑥)𝑏 = (((𝑠 ·s 𝑥) +s (𝑦 ·s 𝑟)) -s (𝑠 ·s 𝑟))})
9856, 97uneq12d 4149 . . . . 5 (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) → ({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝 ·s 𝑦) +s (𝑥 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟 ·s 𝑦) +s (𝑥 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) = ({𝑎 ∣ ∃𝑞 ∈ ( L ‘𝑦)∃𝑝 ∈ ( L ‘𝑥)𝑎 = (((𝑞 ·s 𝑥) +s (𝑦 ·s 𝑝)) -s (𝑞 ·s 𝑝))} ∪ {𝑏 ∣ ∃𝑠 ∈ ( R ‘𝑦)∃𝑟 ∈ ( R ‘𝑥)𝑏 = (((𝑠 ·s 𝑥) +s (𝑦 ·s 𝑟)) -s (𝑠 ·s 𝑟))}))
99 oveq1 7417 . . . . . . . . . . . . . . . 16 (𝑥𝑂 = 𝑡 → (𝑥𝑂 ·s 𝑦) = (𝑡 ·s 𝑦))
100 oveq2 7418 . . . . . . . . . . . . . . . 16 (𝑥𝑂 = 𝑡 → (𝑦 ·s 𝑥𝑂) = (𝑦 ·s 𝑡))
10199, 100eqeq12d 2752 . . . . . . . . . . . . . . 15 (𝑥𝑂 = 𝑡 → ((𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ↔ (𝑡 ·s 𝑦) = (𝑦 ·s 𝑡)))
102 simplr2 1217 . . . . . . . . . . . . . . 15 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑡 ∈ ( L ‘𝑥) ∧ 𝑢 ∈ ( R ‘𝑦))) → ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂))
103 simprl 770 . . . . . . . . . . . . . . . 16 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑡 ∈ ( L ‘𝑥) ∧ 𝑢 ∈ ( R ‘𝑦))) → 𝑡 ∈ ( L ‘𝑥))
104 elun1 4162 . . . . . . . . . . . . . . . 16 (𝑡 ∈ ( L ‘𝑥) → 𝑡 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
105103, 104syl 17 . . . . . . . . . . . . . . 15 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑡 ∈ ( L ‘𝑥) ∧ 𝑢 ∈ ( R ‘𝑦))) → 𝑡 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
106101, 102, 105rspcdva 3607 . . . . . . . . . . . . . 14 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑡 ∈ ( L ‘𝑥) ∧ 𝑢 ∈ ( R ‘𝑦))) → (𝑡 ·s 𝑦) = (𝑦 ·s 𝑡))
107 oveq2 7418 . . . . . . . . . . . . . . . 16 (𝑦𝑂 = 𝑢 → (𝑥 ·s 𝑦𝑂) = (𝑥 ·s 𝑢))
108 oveq1 7417 . . . . . . . . . . . . . . . 16 (𝑦𝑂 = 𝑢 → (𝑦𝑂 ·s 𝑥) = (𝑢 ·s 𝑥))
109107, 108eqeq12d 2752 . . . . . . . . . . . . . . 15 (𝑦𝑂 = 𝑢 → ((𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥) ↔ (𝑥 ·s 𝑢) = (𝑢 ·s 𝑥)))
110 simplr3 1218 . . . . . . . . . . . . . . 15 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑡 ∈ ( L ‘𝑥) ∧ 𝑢 ∈ ( R ‘𝑦))) → ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))
111 simprr 772 . . . . . . . . . . . . . . . 16 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑡 ∈ ( L ‘𝑥) ∧ 𝑢 ∈ ( R ‘𝑦))) → 𝑢 ∈ ( R ‘𝑦))
112 elun2 4163 . . . . . . . . . . . . . . . 16 (𝑢 ∈ ( R ‘𝑦) → 𝑢 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
113111, 112syl 17 . . . . . . . . . . . . . . 15 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑡 ∈ ( L ‘𝑥) ∧ 𝑢 ∈ ( R ‘𝑦))) → 𝑢 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
114109, 110, 113rspcdva 3607 . . . . . . . . . . . . . 14 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑡 ∈ ( L ‘𝑥) ∧ 𝑢 ∈ ( R ‘𝑦))) → (𝑥 ·s 𝑢) = (𝑢 ·s 𝑥))
115106, 114oveq12d 7428 . . . . . . . . . . . . 13 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑡 ∈ ( L ‘𝑥) ∧ 𝑢 ∈ ( R ‘𝑦))) → ((𝑡 ·s 𝑦) +s (𝑥 ·s 𝑢)) = ((𝑦 ·s 𝑡) +s (𝑢 ·s 𝑥)))
116 simpllr 775 . . . . . . . . . . . . . . 15 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑡 ∈ ( L ‘𝑥) ∧ 𝑢 ∈ ( R ‘𝑦))) → 𝑦 ∈ No )
11734, 103sselid 3961 . . . . . . . . . . . . . . 15 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑡 ∈ ( L ‘𝑥) ∧ 𝑢 ∈ ( R ‘𝑦))) → 𝑡 ∈ No )
118116, 117mulscld 28095 . . . . . . . . . . . . . 14 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑡 ∈ ( L ‘𝑥) ∧ 𝑢 ∈ ( R ‘𝑦))) → (𝑦 ·s 𝑡) ∈ No )
11978, 111sselid 3961 . . . . . . . . . . . . . . 15 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑡 ∈ ( L ‘𝑥) ∧ 𝑢 ∈ ( R ‘𝑦))) → 𝑢 ∈ No )
120 simplll 774 . . . . . . . . . . . . . . 15 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑡 ∈ ( L ‘𝑥) ∧ 𝑢 ∈ ( R ‘𝑦))) → 𝑥 ∈ No )
121119, 120mulscld 28095 . . . . . . . . . . . . . 14 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑡 ∈ ( L ‘𝑥) ∧ 𝑢 ∈ ( R ‘𝑦))) → (𝑢 ·s 𝑥) ∈ No )
122118, 121addscomd 27931 . . . . . . . . . . . . 13 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑡 ∈ ( L ‘𝑥) ∧ 𝑢 ∈ ( R ‘𝑦))) → ((𝑦 ·s 𝑡) +s (𝑢 ·s 𝑥)) = ((𝑢 ·s 𝑥) +s (𝑦 ·s 𝑡)))
123115, 122eqtrd 2771 . . . . . . . . . . . 12 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑡 ∈ ( L ‘𝑥) ∧ 𝑢 ∈ ( R ‘𝑦))) → ((𝑡 ·s 𝑦) +s (𝑥 ·s 𝑢)) = ((𝑢 ·s 𝑥) +s (𝑦 ·s 𝑡)))
124 oveq1 7417 . . . . . . . . . . . . . 14 (𝑥𝑂 = 𝑡 → (𝑥𝑂 ·s 𝑦𝑂) = (𝑡 ·s 𝑦𝑂))
125 oveq2 7418 . . . . . . . . . . . . . 14 (𝑥𝑂 = 𝑡 → (𝑦𝑂 ·s 𝑥𝑂) = (𝑦𝑂 ·s 𝑡))
126124, 125eqeq12d 2752 . . . . . . . . . . . . 13 (𝑥𝑂 = 𝑡 → ((𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ↔ (𝑡 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑡)))
127 oveq2 7418 . . . . . . . . . . . . . 14 (𝑦𝑂 = 𝑢 → (𝑡 ·s 𝑦𝑂) = (𝑡 ·s 𝑢))
128 oveq1 7417 . . . . . . . . . . . . . 14 (𝑦𝑂 = 𝑢 → (𝑦𝑂 ·s 𝑡) = (𝑢 ·s 𝑡))
129127, 128eqeq12d 2752 . . . . . . . . . . . . 13 (𝑦𝑂 = 𝑢 → ((𝑡 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑡) ↔ (𝑡 ·s 𝑢) = (𝑢 ·s 𝑡)))
130 simplr1 1216 . . . . . . . . . . . . 13 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑡 ∈ ( L ‘𝑥) ∧ 𝑢 ∈ ( R ‘𝑦))) → ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂))
131126, 129, 130, 105, 113rspc2dv 3621 . . . . . . . . . . . 12 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑡 ∈ ( L ‘𝑥) ∧ 𝑢 ∈ ( R ‘𝑦))) → (𝑡 ·s 𝑢) = (𝑢 ·s 𝑡))
132123, 131oveq12d 7428 . . . . . . . . . . 11 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑡 ∈ ( L ‘𝑥) ∧ 𝑢 ∈ ( R ‘𝑦))) → (((𝑡 ·s 𝑦) +s (𝑥 ·s 𝑢)) -s (𝑡 ·s 𝑢)) = (((𝑢 ·s 𝑥) +s (𝑦 ·s 𝑡)) -s (𝑢 ·s 𝑡)))
133132eqeq2d 2747 . . . . . . . . . 10 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑡 ∈ ( L ‘𝑥) ∧ 𝑢 ∈ ( R ‘𝑦))) → (𝑐 = (((𝑡 ·s 𝑦) +s (𝑥 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ↔ 𝑐 = (((𝑢 ·s 𝑥) +s (𝑦 ·s 𝑡)) -s (𝑢 ·s 𝑡))))
1341332rexbidva 3208 . . . . . . . . 9 (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) → (∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡 ·s 𝑦) +s (𝑥 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ↔ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑢 ·s 𝑥) +s (𝑦 ·s 𝑡)) -s (𝑢 ·s 𝑡))))
135 rexcom 3275 . . . . . . . . 9 (∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑢 ·s 𝑥) +s (𝑦 ·s 𝑡)) -s (𝑢 ·s 𝑡)) ↔ ∃𝑢 ∈ ( R ‘𝑦)∃𝑡 ∈ ( L ‘𝑥)𝑐 = (((𝑢 ·s 𝑥) +s (𝑦 ·s 𝑡)) -s (𝑢 ·s 𝑡)))
136134, 135bitrdi 287 . . . . . . . 8 (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) → (∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡 ·s 𝑦) +s (𝑥 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ↔ ∃𝑢 ∈ ( R ‘𝑦)∃𝑡 ∈ ( L ‘𝑥)𝑐 = (((𝑢 ·s 𝑥) +s (𝑦 ·s 𝑡)) -s (𝑢 ·s 𝑡))))
137136abbidv 2802 . . . . . . 7 (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) → {𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡 ·s 𝑦) +s (𝑥 ·s 𝑢)) -s (𝑡 ·s 𝑢))} = {𝑐 ∣ ∃𝑢 ∈ ( R ‘𝑦)∃𝑡 ∈ ( L ‘𝑥)𝑐 = (((𝑢 ·s 𝑥) +s (𝑦 ·s 𝑡)) -s (𝑢 ·s 𝑡))})
138 oveq1 7417 . . . . . . . . . . . . . . . 16 (𝑥𝑂 = 𝑣 → (𝑥𝑂 ·s 𝑦) = (𝑣 ·s 𝑦))
139 oveq2 7418 . . . . . . . . . . . . . . . 16 (𝑥𝑂 = 𝑣 → (𝑦 ·s 𝑥𝑂) = (𝑦 ·s 𝑣))
140138, 139eqeq12d 2752 . . . . . . . . . . . . . . 15 (𝑥𝑂 = 𝑣 → ((𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ↔ (𝑣 ·s 𝑦) = (𝑦 ·s 𝑣)))
141 simplr2 1217 . . . . . . . . . . . . . . 15 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑣 ∈ ( R ‘𝑥) ∧ 𝑤 ∈ ( L ‘𝑦))) → ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂))
142 simprl 770 . . . . . . . . . . . . . . . 16 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑣 ∈ ( R ‘𝑥) ∧ 𝑤 ∈ ( L ‘𝑦))) → 𝑣 ∈ ( R ‘𝑥))
143 elun2 4163 . . . . . . . . . . . . . . . 16 (𝑣 ∈ ( R ‘𝑥) → 𝑣 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
144142, 143syl 17 . . . . . . . . . . . . . . 15 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑣 ∈ ( R ‘𝑥) ∧ 𝑤 ∈ ( L ‘𝑦))) → 𝑣 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
145140, 141, 144rspcdva 3607 . . . . . . . . . . . . . 14 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑣 ∈ ( R ‘𝑥) ∧ 𝑤 ∈ ( L ‘𝑦))) → (𝑣 ·s 𝑦) = (𝑦 ·s 𝑣))
146 oveq2 7418 . . . . . . . . . . . . . . . 16 (𝑦𝑂 = 𝑤 → (𝑥 ·s 𝑦𝑂) = (𝑥 ·s 𝑤))
147 oveq1 7417 . . . . . . . . . . . . . . . 16 (𝑦𝑂 = 𝑤 → (𝑦𝑂 ·s 𝑥) = (𝑤 ·s 𝑥))
148146, 147eqeq12d 2752 . . . . . . . . . . . . . . 15 (𝑦𝑂 = 𝑤 → ((𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥) ↔ (𝑥 ·s 𝑤) = (𝑤 ·s 𝑥)))
149 simplr3 1218 . . . . . . . . . . . . . . 15 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑣 ∈ ( R ‘𝑥) ∧ 𝑤 ∈ ( L ‘𝑦))) → ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))
150 simprr 772 . . . . . . . . . . . . . . . 16 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑣 ∈ ( R ‘𝑥) ∧ 𝑤 ∈ ( L ‘𝑦))) → 𝑤 ∈ ( L ‘𝑦))
151 elun1 4162 . . . . . . . . . . . . . . . 16 (𝑤 ∈ ( L ‘𝑦) → 𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
152150, 151syl 17 . . . . . . . . . . . . . . 15 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑣 ∈ ( R ‘𝑥) ∧ 𝑤 ∈ ( L ‘𝑦))) → 𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
153148, 149, 152rspcdva 3607 . . . . . . . . . . . . . 14 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑣 ∈ ( R ‘𝑥) ∧ 𝑤 ∈ ( L ‘𝑦))) → (𝑥 ·s 𝑤) = (𝑤 ·s 𝑥))
154145, 153oveq12d 7428 . . . . . . . . . . . . 13 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑣 ∈ ( R ‘𝑥) ∧ 𝑤 ∈ ( L ‘𝑦))) → ((𝑣 ·s 𝑦) +s (𝑥 ·s 𝑤)) = ((𝑦 ·s 𝑣) +s (𝑤 ·s 𝑥)))
155 simpllr 775 . . . . . . . . . . . . . . 15 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑣 ∈ ( R ‘𝑥) ∧ 𝑤 ∈ ( L ‘𝑦))) → 𝑦 ∈ No )
15675, 142sselid 3961 . . . . . . . . . . . . . . 15 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑣 ∈ ( R ‘𝑥) ∧ 𝑤 ∈ ( L ‘𝑦))) → 𝑣 ∈ No )
157155, 156mulscld 28095 . . . . . . . . . . . . . 14 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑣 ∈ ( R ‘𝑥) ∧ 𝑤 ∈ ( L ‘𝑦))) → (𝑦 ·s 𝑣) ∈ No )
15837, 150sselid 3961 . . . . . . . . . . . . . . 15 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑣 ∈ ( R ‘𝑥) ∧ 𝑤 ∈ ( L ‘𝑦))) → 𝑤 ∈ No )
159 simplll 774 . . . . . . . . . . . . . . 15 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑣 ∈ ( R ‘𝑥) ∧ 𝑤 ∈ ( L ‘𝑦))) → 𝑥 ∈ No )
160158, 159mulscld 28095 . . . . . . . . . . . . . 14 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑣 ∈ ( R ‘𝑥) ∧ 𝑤 ∈ ( L ‘𝑦))) → (𝑤 ·s 𝑥) ∈ No )
161157, 160addscomd 27931 . . . . . . . . . . . . 13 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑣 ∈ ( R ‘𝑥) ∧ 𝑤 ∈ ( L ‘𝑦))) → ((𝑦 ·s 𝑣) +s (𝑤 ·s 𝑥)) = ((𝑤 ·s 𝑥) +s (𝑦 ·s 𝑣)))
162154, 161eqtrd 2771 . . . . . . . . . . . 12 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑣 ∈ ( R ‘𝑥) ∧ 𝑤 ∈ ( L ‘𝑦))) → ((𝑣 ·s 𝑦) +s (𝑥 ·s 𝑤)) = ((𝑤 ·s 𝑥) +s (𝑦 ·s 𝑣)))
163 oveq1 7417 . . . . . . . . . . . . . 14 (𝑥𝑂 = 𝑣 → (𝑥𝑂 ·s 𝑦𝑂) = (𝑣 ·s 𝑦𝑂))
164 oveq2 7418 . . . . . . . . . . . . . 14 (𝑥𝑂 = 𝑣 → (𝑦𝑂 ·s 𝑥𝑂) = (𝑦𝑂 ·s 𝑣))
165163, 164eqeq12d 2752 . . . . . . . . . . . . 13 (𝑥𝑂 = 𝑣 → ((𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ↔ (𝑣 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑣)))
166 oveq2 7418 . . . . . . . . . . . . . 14 (𝑦𝑂 = 𝑤 → (𝑣 ·s 𝑦𝑂) = (𝑣 ·s 𝑤))
167 oveq1 7417 . . . . . . . . . . . . . 14 (𝑦𝑂 = 𝑤 → (𝑦𝑂 ·s 𝑣) = (𝑤 ·s 𝑣))
168166, 167eqeq12d 2752 . . . . . . . . . . . . 13 (𝑦𝑂 = 𝑤 → ((𝑣 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑣) ↔ (𝑣 ·s 𝑤) = (𝑤 ·s 𝑣)))
169 simplr1 1216 . . . . . . . . . . . . 13 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑣 ∈ ( R ‘𝑥) ∧ 𝑤 ∈ ( L ‘𝑦))) → ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂))
170165, 168, 169, 144, 152rspc2dv 3621 . . . . . . . . . . . 12 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑣 ∈ ( R ‘𝑥) ∧ 𝑤 ∈ ( L ‘𝑦))) → (𝑣 ·s 𝑤) = (𝑤 ·s 𝑣))
171162, 170oveq12d 7428 . . . . . . . . . . 11 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑣 ∈ ( R ‘𝑥) ∧ 𝑤 ∈ ( L ‘𝑦))) → (((𝑣 ·s 𝑦) +s (𝑥 ·s 𝑤)) -s (𝑣 ·s 𝑤)) = (((𝑤 ·s 𝑥) +s (𝑦 ·s 𝑣)) -s (𝑤 ·s 𝑣)))
172171eqeq2d 2747 . . . . . . . . . 10 ((((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) ∧ (𝑣 ∈ ( R ‘𝑥) ∧ 𝑤 ∈ ( L ‘𝑦))) → (𝑑 = (((𝑣 ·s 𝑦) +s (𝑥 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ↔ 𝑑 = (((𝑤 ·s 𝑥) +s (𝑦 ·s 𝑣)) -s (𝑤 ·s 𝑣))))
1731722rexbidva 3208 . . . . . . . . 9 (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) → (∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣 ·s 𝑦) +s (𝑥 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ↔ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑤 ·s 𝑥) +s (𝑦 ·s 𝑣)) -s (𝑤 ·s 𝑣))))
174 rexcom 3275 . . . . . . . . 9 (∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑤 ·s 𝑥) +s (𝑦 ·s 𝑣)) -s (𝑤 ·s 𝑣)) ↔ ∃𝑤 ∈ ( L ‘𝑦)∃𝑣 ∈ ( R ‘𝑥)𝑑 = (((𝑤 ·s 𝑥) +s (𝑦 ·s 𝑣)) -s (𝑤 ·s 𝑣)))
175173, 174bitrdi 287 . . . . . . . 8 (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) → (∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣 ·s 𝑦) +s (𝑥 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ↔ ∃𝑤 ∈ ( L ‘𝑦)∃𝑣 ∈ ( R ‘𝑥)𝑑 = (((𝑤 ·s 𝑥) +s (𝑦 ·s 𝑣)) -s (𝑤 ·s 𝑣))))
176175abbidv 2802 . . . . . . 7 (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) → {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣 ·s 𝑦) +s (𝑥 ·s 𝑤)) -s (𝑣 ·s 𝑤))} = {𝑑 ∣ ∃𝑤 ∈ ( L ‘𝑦)∃𝑣 ∈ ( R ‘𝑥)𝑑 = (((𝑤 ·s 𝑥) +s (𝑦 ·s 𝑣)) -s (𝑤 ·s 𝑣))})
177137, 176uneq12d 4149 . . . . . 6 (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) → ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡 ·s 𝑦) +s (𝑥 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣 ·s 𝑦) +s (𝑥 ·s 𝑤)) -s (𝑣 ·s 𝑤))}) = ({𝑐 ∣ ∃𝑢 ∈ ( R ‘𝑦)∃𝑡 ∈ ( L ‘𝑥)𝑐 = (((𝑢 ·s 𝑥) +s (𝑦 ·s 𝑡)) -s (𝑢 ·s 𝑡))} ∪ {𝑑 ∣ ∃𝑤 ∈ ( L ‘𝑦)∃𝑣 ∈ ( R ‘𝑥)𝑑 = (((𝑤 ·s 𝑥) +s (𝑦 ·s 𝑣)) -s (𝑤 ·s 𝑣))}))
178 uncom 4138 . . . . . 6 ({𝑐 ∣ ∃𝑢 ∈ ( R ‘𝑦)∃𝑡 ∈ ( L ‘𝑥)𝑐 = (((𝑢 ·s 𝑥) +s (𝑦 ·s 𝑡)) -s (𝑢 ·s 𝑡))} ∪ {𝑑 ∣ ∃𝑤 ∈ ( L ‘𝑦)∃𝑣 ∈ ( R ‘𝑥)𝑑 = (((𝑤 ·s 𝑥) +s (𝑦 ·s 𝑣)) -s (𝑤 ·s 𝑣))}) = ({𝑑 ∣ ∃𝑤 ∈ ( L ‘𝑦)∃𝑣 ∈ ( R ‘𝑥)𝑑 = (((𝑤 ·s 𝑥) +s (𝑦 ·s 𝑣)) -s (𝑤 ·s 𝑣))} ∪ {𝑐 ∣ ∃𝑢 ∈ ( R ‘𝑦)∃𝑡 ∈ ( L ‘𝑥)𝑐 = (((𝑢 ·s 𝑥) +s (𝑦 ·s 𝑡)) -s (𝑢 ·s 𝑡))})
179177, 178eqtrdi 2787 . . . . 5 (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) → ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡 ·s 𝑦) +s (𝑥 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣 ·s 𝑦) +s (𝑥 ·s 𝑤)) -s (𝑣 ·s 𝑤))}) = ({𝑑 ∣ ∃𝑤 ∈ ( L ‘𝑦)∃𝑣 ∈ ( R ‘𝑥)𝑑 = (((𝑤 ·s 𝑥) +s (𝑦 ·s 𝑣)) -s (𝑤 ·s 𝑣))} ∪ {𝑐 ∣ ∃𝑢 ∈ ( R ‘𝑦)∃𝑡 ∈ ( L ‘𝑥)𝑐 = (((𝑢 ·s 𝑥) +s (𝑦 ·s 𝑡)) -s (𝑢 ·s 𝑡))}))
18098, 179oveq12d 7428 . . . 4 (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) → (({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝 ·s 𝑦) +s (𝑥 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟 ·s 𝑦) +s (𝑥 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡 ·s 𝑦) +s (𝑥 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣 ·s 𝑦) +s (𝑥 ·s 𝑤)) -s (𝑣 ·s 𝑤))})) = (({𝑎 ∣ ∃𝑞 ∈ ( L ‘𝑦)∃𝑝 ∈ ( L ‘𝑥)𝑎 = (((𝑞 ·s 𝑥) +s (𝑦 ·s 𝑝)) -s (𝑞 ·s 𝑝))} ∪ {𝑏 ∣ ∃𝑠 ∈ ( R ‘𝑦)∃𝑟 ∈ ( R ‘𝑥)𝑏 = (((𝑠 ·s 𝑥) +s (𝑦 ·s 𝑟)) -s (𝑠 ·s 𝑟))}) |s ({𝑑 ∣ ∃𝑤 ∈ ( L ‘𝑦)∃𝑣 ∈ ( R ‘𝑥)𝑑 = (((𝑤 ·s 𝑥) +s (𝑦 ·s 𝑣)) -s (𝑤 ·s 𝑣))} ∪ {𝑐 ∣ ∃𝑢 ∈ ( R ‘𝑦)∃𝑡 ∈ ( L ‘𝑥)𝑐 = (((𝑢 ·s 𝑥) +s (𝑦 ·s 𝑡)) -s (𝑢 ·s 𝑡))})))
181 mulsval 28069 . . . . 5 ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → (𝑥 ·s 𝑦) = (({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝 ·s 𝑦) +s (𝑥 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟 ·s 𝑦) +s (𝑥 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡 ·s 𝑦) +s (𝑥 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣 ·s 𝑦) +s (𝑥 ·s 𝑤)) -s (𝑣 ·s 𝑤))})))
182181adantr 480 . . . 4 (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) → (𝑥 ·s 𝑦) = (({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝 ·s 𝑦) +s (𝑥 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟 ·s 𝑦) +s (𝑥 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡 ·s 𝑦) +s (𝑥 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣 ·s 𝑦) +s (𝑥 ·s 𝑤)) -s (𝑣 ·s 𝑤))})))
183 mulsval 28069 . . . . . 6 ((𝑦 ∈ No ∧ 𝑥 ∈ No ) → (𝑦 ·s 𝑥) = (({𝑎 ∣ ∃𝑞 ∈ ( L ‘𝑦)∃𝑝 ∈ ( L ‘𝑥)𝑎 = (((𝑞 ·s 𝑥) +s (𝑦 ·s 𝑝)) -s (𝑞 ·s 𝑝))} ∪ {𝑏 ∣ ∃𝑠 ∈ ( R ‘𝑦)∃𝑟 ∈ ( R ‘𝑥)𝑏 = (((𝑠 ·s 𝑥) +s (𝑦 ·s 𝑟)) -s (𝑠 ·s 𝑟))}) |s ({𝑑 ∣ ∃𝑤 ∈ ( L ‘𝑦)∃𝑣 ∈ ( R ‘𝑥)𝑑 = (((𝑤 ·s 𝑥) +s (𝑦 ·s 𝑣)) -s (𝑤 ·s 𝑣))} ∪ {𝑐 ∣ ∃𝑢 ∈ ( R ‘𝑦)∃𝑡 ∈ ( L ‘𝑥)𝑐 = (((𝑢 ·s 𝑥) +s (𝑦 ·s 𝑡)) -s (𝑢 ·s 𝑡))})))
184183ancoms 458 . . . . 5 ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → (𝑦 ·s 𝑥) = (({𝑎 ∣ ∃𝑞 ∈ ( L ‘𝑦)∃𝑝 ∈ ( L ‘𝑥)𝑎 = (((𝑞 ·s 𝑥) +s (𝑦 ·s 𝑝)) -s (𝑞 ·s 𝑝))} ∪ {𝑏 ∣ ∃𝑠 ∈ ( R ‘𝑦)∃𝑟 ∈ ( R ‘𝑥)𝑏 = (((𝑠 ·s 𝑥) +s (𝑦 ·s 𝑟)) -s (𝑠 ·s 𝑟))}) |s ({𝑑 ∣ ∃𝑤 ∈ ( L ‘𝑦)∃𝑣 ∈ ( R ‘𝑥)𝑑 = (((𝑤 ·s 𝑥) +s (𝑦 ·s 𝑣)) -s (𝑤 ·s 𝑣))} ∪ {𝑐 ∣ ∃𝑢 ∈ ( R ‘𝑦)∃𝑡 ∈ ( L ‘𝑥)𝑐 = (((𝑢 ·s 𝑥) +s (𝑦 ·s 𝑡)) -s (𝑢 ·s 𝑡))})))
185184adantr 480 . . . 4 (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) → (𝑦 ·s 𝑥) = (({𝑎 ∣ ∃𝑞 ∈ ( L ‘𝑦)∃𝑝 ∈ ( L ‘𝑥)𝑎 = (((𝑞 ·s 𝑥) +s (𝑦 ·s 𝑝)) -s (𝑞 ·s 𝑝))} ∪ {𝑏 ∣ ∃𝑠 ∈ ( R ‘𝑦)∃𝑟 ∈ ( R ‘𝑥)𝑏 = (((𝑠 ·s 𝑥) +s (𝑦 ·s 𝑟)) -s (𝑠 ·s 𝑟))}) |s ({𝑑 ∣ ∃𝑤 ∈ ( L ‘𝑦)∃𝑣 ∈ ( R ‘𝑥)𝑑 = (((𝑤 ·s 𝑥) +s (𝑦 ·s 𝑣)) -s (𝑤 ·s 𝑣))} ∪ {𝑐 ∣ ∃𝑢 ∈ ( R ‘𝑦)∃𝑡 ∈ ( L ‘𝑥)𝑐 = (((𝑢 ·s 𝑥) +s (𝑦 ·s 𝑡)) -s (𝑢 ·s 𝑡))})))
186180, 182, 1853eqtr4d 2781 . . 3 (((𝑥 ∈ No ∧ 𝑦 ∈ No ) ∧ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥))) → (𝑥 ·s 𝑦) = (𝑦 ·s 𝑥))
187186ex 412 . 2 ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → ((∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥𝑂 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥𝑂) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 𝑦) = (𝑦 ·s 𝑥𝑂) ∧ ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 ·s 𝑦𝑂) = (𝑦𝑂 ·s 𝑥)) → (𝑥 ·s 𝑦) = (𝑦 ·s 𝑥)))
1883, 6, 9, 12, 15, 187no2inds 27919 1 ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ·s 𝐵) = (𝐵 ·s 𝐴))
Colors of variables: wff setvar class
Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  {cab 2714  ∀wral 3052  ∃wrex 3061   ∪ cun 3929  ‘cfv 6536  (class class class)co 7410   No csur 27608   |s cscut 27751   L cleft 27810   R cright 27811   +s cadds 27923   -s csubs 27983   ·s cmuls 28066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-ot 4615  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-1o 8485  df-2o 8486  df-nadd 8683  df-no 27611  df-slt 27612  df-bday 27613  df-sle 27714  df-sslt 27750  df-scut 27752  df-0s 27793  df-made 27812  df-old 27813  df-left 27815  df-right 27816  df-norec 27902  df-norec2 27913  df-adds 27924  df-negs 27984  df-subs 27985  df-muls 28067
This theorem is referenced by:  mulscomd  28100  muls02  28101  mulslid  28102
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