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Theorem addscom 28014
Description: Surreal addition commutes. Part of Theorem 3 of [Conway] p. 17. (Contributed by Scott Fenton, 20-Aug-2024.)
Assertion
Ref Expression
addscom ((𝐴 No 𝐵 No ) → (𝐴 +s 𝐵) = (𝐵 +s 𝐴))

Proof of Theorem addscom
Dummy variables 𝑤 𝑥 𝑦 𝑙 𝑟 𝑥𝑂 𝑦𝑂 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 7438 . . 3 (𝑥 = 𝑥𝑂 → (𝑥 +s 𝑦) = (𝑥𝑂 +s 𝑦))
2 oveq2 7439 . . 3 (𝑥 = 𝑥𝑂 → (𝑦 +s 𝑥) = (𝑦 +s 𝑥𝑂))
31, 2eqeq12d 2751 . 2 (𝑥 = 𝑥𝑂 → ((𝑥 +s 𝑦) = (𝑦 +s 𝑥) ↔ (𝑥𝑂 +s 𝑦) = (𝑦 +s 𝑥𝑂)))
4 oveq2 7439 . . 3 (𝑦 = 𝑦𝑂 → (𝑥𝑂 +s 𝑦) = (𝑥𝑂 +s 𝑦𝑂))
5 oveq1 7438 . . 3 (𝑦 = 𝑦𝑂 → (𝑦 +s 𝑥𝑂) = (𝑦𝑂 +s 𝑥𝑂))
64, 5eqeq12d 2751 . 2 (𝑦 = 𝑦𝑂 → ((𝑥𝑂 +s 𝑦) = (𝑦 +s 𝑥𝑂) ↔ (𝑥𝑂 +s 𝑦𝑂) = (𝑦𝑂 +s 𝑥𝑂)))
7 oveq1 7438 . . 3 (𝑥 = 𝑥𝑂 → (𝑥 +s 𝑦𝑂) = (𝑥𝑂 +s 𝑦𝑂))
8 oveq2 7439 . . 3 (𝑥 = 𝑥𝑂 → (𝑦𝑂 +s 𝑥) = (𝑦𝑂 +s 𝑥𝑂))
97, 8eqeq12d 2751 . 2 (𝑥 = 𝑥𝑂 → ((𝑥 +s 𝑦𝑂) = (𝑦𝑂 +s 𝑥) ↔ (𝑥𝑂 +s 𝑦𝑂) = (𝑦𝑂 +s 𝑥𝑂)))
10 oveq1 7438 . . 3 (𝑥 = 𝐴 → (𝑥 +s 𝑦) = (𝐴 +s 𝑦))
11 oveq2 7439 . . 3 (𝑥 = 𝐴 → (𝑦 +s 𝑥) = (𝑦 +s 𝐴))
1210, 11eqeq12d 2751 . 2 (𝑥 = 𝐴 → ((𝑥 +s 𝑦) = (𝑦 +s 𝑥) ↔ (𝐴 +s 𝑦) = (𝑦 +s 𝐴)))
13 oveq2 7439 . . 3 (𝑦 = 𝐵 → (𝐴 +s 𝑦) = (𝐴 +s 𝐵))
14 oveq1 7438 . . 3 (𝑦 = 𝐵 → (𝑦 +s 𝐴) = (𝐵 +s 𝐴))
1513, 14eqeq12d 2751 . 2 (𝑦 = 𝐵 → ((𝐴 +s 𝑦) = (𝑦 +s 𝐴) ↔ (𝐴 +s 𝐵) = (𝐵 +s 𝐴)))
16 simpr2 1194 . . . . . . . . . . 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))) → ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s 𝑦) = (𝑦 +s 𝑥𝑂))
17 elun1 4192 . . . . . . . . . . 11 (𝑙 ∈ ( L ‘𝑥) → 𝑙 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
18 oveq1 7438 . . . . . . . . . . . . 13 (𝑥𝑂 = 𝑙 → (𝑥𝑂 +s 𝑦) = (𝑙 +s 𝑦))
19 oveq2 7439 . . . . . . . . . . . . 13 (𝑥𝑂 = 𝑙 → (𝑦 +s 𝑥𝑂) = (𝑦 +s 𝑙))
2018, 19eqeq12d 2751 . . . . . . . . . . . 12 (𝑥𝑂 = 𝑙 → ((𝑥𝑂 +s 𝑦) = (𝑦 +s 𝑥𝑂) ↔ (𝑙 +s 𝑦) = (𝑦 +s 𝑙)))
2120rspccva 3621 . . . . . . . . . . 11 ((∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s 𝑦) = (𝑦 +s 𝑥𝑂) ∧ 𝑙 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))) → (𝑙 +s 𝑦) = (𝑦 +s 𝑙))
2216, 17, 21syl2an 596 . . . . . . . . . 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))) ∧ 𝑙 ∈ ( L ‘𝑥)) → (𝑙 +s 𝑦) = (𝑦 +s 𝑙))
2322eqeq2d 2746 . . . . . . . . 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))) ∧ 𝑙 ∈ ( L ‘𝑥)) → (𝑤 = (𝑙 +s 𝑦) ↔ 𝑤 = (𝑦 +s 𝑙)))
2423rexbidva 3175 . . . . . . . 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))) → (∃𝑙 ∈ ( L ‘𝑥)𝑤 = (𝑙 +s 𝑦) ↔ ∃𝑙 ∈ ( L ‘𝑥)𝑤 = (𝑦 +s 𝑙)))
2524abbidv 2806 . . . . . . 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))) → {𝑤 ∣ ∃𝑙 ∈ ( L ‘𝑥)𝑤 = (𝑙 +s 𝑦)} = {𝑤 ∣ ∃𝑙 ∈ ( L ‘𝑥)𝑤 = (𝑦 +s 𝑙)})
26 simpr3 1195 . . . . . . . . . . 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))) → ∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 +s 𝑦𝑂) = (𝑦𝑂 +s 𝑥))
27 elun1 4192 . . . . . . . . . . 11 (𝑙 ∈ ( L ‘𝑦) → 𝑙 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
28 oveq2 7439 . . . . . . . . . . . . 13 (𝑦𝑂 = 𝑙 → (𝑥 +s 𝑦𝑂) = (𝑥 +s 𝑙))
29 oveq1 7438 . . . . . . . . . . . . 13 (𝑦𝑂 = 𝑙 → (𝑦𝑂 +s 𝑥) = (𝑙 +s 𝑥))
3028, 29eqeq12d 2751 . . . . . . . . . . . 12 (𝑦𝑂 = 𝑙 → ((𝑥 +s 𝑦𝑂) = (𝑦𝑂 +s 𝑥) ↔ (𝑥 +s 𝑙) = (𝑙 +s 𝑥)))
3130rspccva 3621 . . . . . . . . . . 11 ((∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 +s 𝑦𝑂) = (𝑦𝑂 +s 𝑥) ∧ 𝑙 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))) → (𝑥 +s 𝑙) = (𝑙 +s 𝑥))
3226, 27, 31syl2an 596 . . . . . . . . . 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))) ∧ 𝑙 ∈ ( L ‘𝑦)) → (𝑥 +s 𝑙) = (𝑙 +s 𝑥))
3332eqeq2d 2746 . . . . . . . . 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))) ∧ 𝑙 ∈ ( L ‘𝑦)) → (𝑧 = (𝑥 +s 𝑙) ↔ 𝑧 = (𝑙 +s 𝑥)))
3433rexbidva 3175 . . . . . . . 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))) → (∃𝑙 ∈ ( L ‘𝑦)𝑧 = (𝑥 +s 𝑙) ↔ ∃𝑙 ∈ ( L ‘𝑦)𝑧 = (𝑙 +s 𝑥)))
3534abbidv 2806 . . . . . . 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))) → {𝑧 ∣ ∃𝑙 ∈ ( L ‘𝑦)𝑧 = (𝑥 +s 𝑙)} = {𝑧 ∣ ∃𝑙 ∈ ( L ‘𝑦)𝑧 = (𝑙 +s 𝑥)})
3625, 35uneq12d 4179 . . . . . 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))) → ({𝑤 ∣ ∃𝑙 ∈ ( L ‘𝑥)𝑤 = (𝑙 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑙 ∈ ( L ‘𝑦)𝑧 = (𝑥 +s 𝑙)}) = ({𝑤 ∣ ∃𝑙 ∈ ( L ‘𝑥)𝑤 = (𝑦 +s 𝑙)} ∪ {𝑧 ∣ ∃𝑙 ∈ ( L ‘𝑦)𝑧 = (𝑙 +s 𝑥)}))
37 uncom 4168 . . . . . 6 ({𝑤 ∣ ∃𝑙 ∈ ( L ‘𝑥)𝑤 = (𝑦 +s 𝑙)} ∪ {𝑧 ∣ ∃𝑙 ∈ ( L ‘𝑦)𝑧 = (𝑙 +s 𝑥)}) = ({𝑧 ∣ ∃𝑙 ∈ ( L ‘𝑦)𝑧 = (𝑙 +s 𝑥)} ∪ {𝑤 ∣ ∃𝑙 ∈ ( L ‘𝑥)𝑤 = (𝑦 +s 𝑙)})
3836, 37eqtrdi 2791 . . . . 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))) → ({𝑤 ∣ ∃𝑙 ∈ ( L ‘𝑥)𝑤 = (𝑙 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑙 ∈ ( L ‘𝑦)𝑧 = (𝑥 +s 𝑙)}) = ({𝑧 ∣ ∃𝑙 ∈ ( L ‘𝑦)𝑧 = (𝑙 +s 𝑥)} ∪ {𝑤 ∣ ∃𝑙 ∈ ( L ‘𝑥)𝑤 = (𝑦 +s 𝑙)}))
39 elun2 4193 . . . . . . . . . . 11 (𝑟 ∈ ( R ‘𝑥) → 𝑟 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
40 oveq1 7438 . . . . . . . . . . . . 13 (𝑥𝑂 = 𝑟 → (𝑥𝑂 +s 𝑦) = (𝑟 +s 𝑦))
41 oveq2 7439 . . . . . . . . . . . . 13 (𝑥𝑂 = 𝑟 → (𝑦 +s 𝑥𝑂) = (𝑦 +s 𝑟))
4240, 41eqeq12d 2751 . . . . . . . . . . . 12 (𝑥𝑂 = 𝑟 → ((𝑥𝑂 +s 𝑦) = (𝑦 +s 𝑥𝑂) ↔ (𝑟 +s 𝑦) = (𝑦 +s 𝑟)))
4342rspccva 3621 . . . . . . . . . . 11 ((∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s 𝑦) = (𝑦 +s 𝑥𝑂) ∧ 𝑟 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))) → (𝑟 +s 𝑦) = (𝑦 +s 𝑟))
4416, 39, 43syl2an 596 . . . . . . . . . 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)) → (𝑟 +s 𝑦) = (𝑦 +s 𝑟))
4544eqeq2d 2746 . . . . . . . . 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)) → (𝑤 = (𝑟 +s 𝑦) ↔ 𝑤 = (𝑦 +s 𝑟)))
4645rexbidva 3175 . . . . . . . 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)𝑤 = (𝑟 +s 𝑦) ↔ ∃𝑟 ∈ ( R ‘𝑥)𝑤 = (𝑦 +s 𝑟)))
4746abbidv 2806 . . . . . . 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))) → {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑥)𝑤 = (𝑟 +s 𝑦)} = {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑥)𝑤 = (𝑦 +s 𝑟)})
48 elun2 4193 . . . . . . . . . . 11 (𝑟 ∈ ( R ‘𝑦) → 𝑟 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
49 oveq2 7439 . . . . . . . . . . . . 13 (𝑦𝑂 = 𝑟 → (𝑥 +s 𝑦𝑂) = (𝑥 +s 𝑟))
50 oveq1 7438 . . . . . . . . . . . . 13 (𝑦𝑂 = 𝑟 → (𝑦𝑂 +s 𝑥) = (𝑟 +s 𝑥))
5149, 50eqeq12d 2751 . . . . . . . . . . . 12 (𝑦𝑂 = 𝑟 → ((𝑥 +s 𝑦𝑂) = (𝑦𝑂 +s 𝑥) ↔ (𝑥 +s 𝑟) = (𝑟 +s 𝑥)))
5251rspccva 3621 . . . . . . . . . . 11 ((∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 +s 𝑦𝑂) = (𝑦𝑂 +s 𝑥) ∧ 𝑟 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))) → (𝑥 +s 𝑟) = (𝑟 +s 𝑥))
5326, 48, 52syl2an 596 . . . . . . . . . 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 ‘𝑦)) → (𝑥 +s 𝑟) = (𝑟 +s 𝑥))
5453eqeq2d 2746 . . . . . . . . 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 ‘𝑦)) → (𝑧 = (𝑥 +s 𝑟) ↔ 𝑧 = (𝑟 +s 𝑥)))
5554rexbidva 3175 . . . . . . . 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 ‘𝑦)𝑧 = (𝑥 +s 𝑟) ↔ ∃𝑟 ∈ ( R ‘𝑦)𝑧 = (𝑟 +s 𝑥)))
5655abbidv 2806 . . . . . . 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))) → {𝑧 ∣ ∃𝑟 ∈ ( R ‘𝑦)𝑧 = (𝑥 +s 𝑟)} = {𝑧 ∣ ∃𝑟 ∈ ( R ‘𝑦)𝑧 = (𝑟 +s 𝑥)})
5747, 56uneq12d 4179 . . . . . 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))) → ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑥)𝑤 = (𝑟 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑟 ∈ ( R ‘𝑦)𝑧 = (𝑥 +s 𝑟)}) = ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑥)𝑤 = (𝑦 +s 𝑟)} ∪ {𝑧 ∣ ∃𝑟 ∈ ( R ‘𝑦)𝑧 = (𝑟 +s 𝑥)}))
58 uncom 4168 . . . . . 6 ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑥)𝑤 = (𝑦 +s 𝑟)} ∪ {𝑧 ∣ ∃𝑟 ∈ ( R ‘𝑦)𝑧 = (𝑟 +s 𝑥)}) = ({𝑧 ∣ ∃𝑟 ∈ ( R ‘𝑦)𝑧 = (𝑟 +s 𝑥)} ∪ {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑥)𝑤 = (𝑦 +s 𝑟)})
5957, 58eqtrdi 2791 . . . . 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))) → ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑥)𝑤 = (𝑟 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑟 ∈ ( R ‘𝑦)𝑧 = (𝑥 +s 𝑟)}) = ({𝑧 ∣ ∃𝑟 ∈ ( R ‘𝑦)𝑧 = (𝑟 +s 𝑥)} ∪ {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑥)𝑤 = (𝑦 +s 𝑟)}))
6038, 59oveq12d 7449 . . . 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))) → (({𝑤 ∣ ∃𝑙 ∈ ( L ‘𝑥)𝑤 = (𝑙 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑙 ∈ ( L ‘𝑦)𝑧 = (𝑥 +s 𝑙)}) |s ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑥)𝑤 = (𝑟 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑟 ∈ ( R ‘𝑦)𝑧 = (𝑥 +s 𝑟)})) = (({𝑧 ∣ ∃𝑙 ∈ ( L ‘𝑦)𝑧 = (𝑙 +s 𝑥)} ∪ {𝑤 ∣ ∃𝑙 ∈ ( L ‘𝑥)𝑤 = (𝑦 +s 𝑙)}) |s ({𝑧 ∣ ∃𝑟 ∈ ( R ‘𝑦)𝑧 = (𝑟 +s 𝑥)} ∪ {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑥)𝑤 = (𝑦 +s 𝑟)})))
61 addsval 28010 . . . . 5 ((𝑥 No 𝑦 No ) → (𝑥 +s 𝑦) = (({𝑤 ∣ ∃𝑙 ∈ ( L ‘𝑥)𝑤 = (𝑙 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑙 ∈ ( L ‘𝑦)𝑧 = (𝑥 +s 𝑙)}) |s ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑥)𝑤 = (𝑟 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑟 ∈ ( R ‘𝑦)𝑧 = (𝑥 +s 𝑟)})))
6261adantr 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) = (({𝑤 ∣ ∃𝑙 ∈ ( L ‘𝑥)𝑤 = (𝑙 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑙 ∈ ( L ‘𝑦)𝑧 = (𝑥 +s 𝑙)}) |s ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑥)𝑤 = (𝑟 +s 𝑦)} ∪ {𝑧 ∣ ∃𝑟 ∈ ( R ‘𝑦)𝑧 = (𝑥 +s 𝑟)})))
63 addsval 28010 . . . . . 6 ((𝑦 No 𝑥 No ) → (𝑦 +s 𝑥) = (({𝑧 ∣ ∃𝑙 ∈ ( L ‘𝑦)𝑧 = (𝑙 +s 𝑥)} ∪ {𝑤 ∣ ∃𝑙 ∈ ( L ‘𝑥)𝑤 = (𝑦 +s 𝑙)}) |s ({𝑧 ∣ ∃𝑟 ∈ ( R ‘𝑦)𝑧 = (𝑟 +s 𝑥)} ∪ {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑥)𝑤 = (𝑦 +s 𝑟)})))
6463ancoms 458 . . . . 5 ((𝑥 No 𝑦 No ) → (𝑦 +s 𝑥) = (({𝑧 ∣ ∃𝑙 ∈ ( L ‘𝑦)𝑧 = (𝑙 +s 𝑥)} ∪ {𝑤 ∣ ∃𝑙 ∈ ( L ‘𝑥)𝑤 = (𝑦 +s 𝑙)}) |s ({𝑧 ∣ ∃𝑟 ∈ ( R ‘𝑦)𝑧 = (𝑟 +s 𝑥)} ∪ {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑥)𝑤 = (𝑦 +s 𝑟)})))
6564adantr 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) = (({𝑧 ∣ ∃𝑙 ∈ ( L ‘𝑦)𝑧 = (𝑙 +s 𝑥)} ∪ {𝑤 ∣ ∃𝑙 ∈ ( L ‘𝑥)𝑤 = (𝑦 +s 𝑙)}) |s ({𝑧 ∣ ∃𝑟 ∈ ( R ‘𝑦)𝑧 = (𝑟 +s 𝑥)} ∪ {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑥)𝑤 = (𝑦 +s 𝑟)})))
6660, 62, 653eqtr4d 2785 . . 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))
6766ex 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)))
683, 6, 9, 12, 15, 67no2inds 28003 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 1537  wcel 2106  {cab 2712  wral 3059  wrex 3068  cun 3961  cfv 6563  (class class class)co 7431   No csur 27699   |s cscut 27842   L cleft 27899   R cright 27900   +s cadds 28007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-1o 8505  df-2o 8506  df-no 27702  df-slt 27703  df-bday 27704  df-sslt 27841  df-scut 27843  df-made 27901  df-old 27902  df-left 27904  df-right 27905  df-norec2 27997  df-adds 28008
This theorem is referenced by:  addscomd  28015  sltadd2im  28034  sleadd2im  28036  sleadd2  28038  sltadd1  28040  addscan1  28042  pncans  28117  nohalf  28422
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