Theorem addscom 27880

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 𝑥_𝑂 𝑦_𝑂 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7397	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 +s 𝑦) = (𝑥𝑂 +s 𝑦))
2		oveq2 7398	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (𝑦 +s 𝑥) = (𝑦 +s 𝑥𝑂))
3	1, 2	eqeq12d 2746	. 2 ⊢ (𝑥 = 𝑥𝑂 → ((𝑥 +s 𝑦) = (𝑦 +s 𝑥) ↔ (𝑥𝑂 +s 𝑦) = (𝑦 +s 𝑥𝑂)))
4		oveq2 7398	. . 3 ⊢ (𝑦 = 𝑦𝑂 → (𝑥𝑂 +s 𝑦) = (𝑥𝑂 +s 𝑦𝑂))
5		oveq1 7397	. . 3 ⊢ (𝑦 = 𝑦𝑂 → (𝑦 +s 𝑥𝑂) = (𝑦𝑂 +s 𝑥𝑂))
6	4, 5	eqeq12d 2746	. 2 ⊢ (𝑦 = 𝑦𝑂 → ((𝑥𝑂 +s 𝑦) = (𝑦 +s 𝑥𝑂) ↔ (𝑥𝑂 +s 𝑦𝑂) = (𝑦𝑂 +s 𝑥𝑂)))
7		oveq1 7397	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 +s 𝑦𝑂) = (𝑥𝑂 +s 𝑦𝑂))
8		oveq2 7398	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (𝑦𝑂 +s 𝑥) = (𝑦𝑂 +s 𝑥𝑂))
9	7, 8	eqeq12d 2746	. 2 ⊢ (𝑥 = 𝑥𝑂 → ((𝑥 +s 𝑦𝑂) = (𝑦𝑂 +s 𝑥) ↔ (𝑥𝑂 +s 𝑦𝑂) = (𝑦𝑂 +s 𝑥𝑂)))
10		oveq1 7397	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 +s 𝑦) = (𝐴 +s 𝑦))
11		oveq2 7398	. . 3 ⊢ (𝑥 = 𝐴 → (𝑦 +s 𝑥) = (𝑦 +s 𝐴))
12	10, 11	eqeq12d 2746	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 +s 𝑦) = (𝑦 +s 𝑥) ↔ (𝐴 +s 𝑦) = (𝑦 +s 𝐴)))
13		oveq2 7398	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 +s 𝑦) = (𝐴 +s 𝐵))
14		oveq1 7397	. . 3 ⊢ (𝑦 = 𝐵 → (𝑦 +s 𝐴) = (𝐵 +s 𝐴))
15	13, 14	eqeq12d 2746	. 2 ⊢ (𝑦 = 𝐵 → ((𝐴 +s 𝑦) = (𝑦 +s 𝐴) ↔ (𝐴 +s 𝐵) = (𝐵 +s 𝐴)))
16		simpr2 1196	. . . . . . . . . . 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 4148	. . . . . . . . . . 11 ⊢ (𝑙 ∈ ( L ‘𝑥) → 𝑙 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
18		oveq1 7397	. . . . . . . . . . . . 13 ⊢ (𝑥𝑂 = 𝑙 → (𝑥𝑂 +s 𝑦) = (𝑙 +s 𝑦))
19		oveq2 7398	. . . . . . . . . . . . 13 ⊢ (𝑥𝑂 = 𝑙 → (𝑦 +s 𝑥𝑂) = (𝑦 +s 𝑙))
20	18, 19	eqeq12d 2746	. . . . . . . . . . . 12 ⊢ (𝑥𝑂 = 𝑙 → ((𝑥𝑂 +s 𝑦) = (𝑦 +s 𝑥𝑂) ↔ (𝑙 +s 𝑦) = (𝑦 +s 𝑙)))
21	20	rspccva 3590	. . . . . . . . . . 11 ⊢ ((∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s 𝑦) = (𝑦 +s 𝑥𝑂) ∧ 𝑙 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))) → (𝑙 +s 𝑦) = (𝑦 +s 𝑙))
22	16, 17, 21	syl2an 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))
23	22	eqeq2d 2741	. . . . . . . . 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)))
24	23	rexbidva 3156	. . . . . . . 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)))
25	24	abbidv 2796	. . . . . . 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 1197	. . . . . . . . . . 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 4148	. . . . . . . . . . 11 ⊢ (𝑙 ∈ ( L ‘𝑦) → 𝑙 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
28		oveq2 7398	. . . . . . . . . . . . 13 ⊢ (𝑦𝑂 = 𝑙 → (𝑥 +s 𝑦𝑂) = (𝑥 +s 𝑙))
29		oveq1 7397	. . . . . . . . . . . . 13 ⊢ (𝑦𝑂 = 𝑙 → (𝑦𝑂 +s 𝑥) = (𝑙 +s 𝑥))
30	28, 29	eqeq12d 2746	. . . . . . . . . . . 12 ⊢ (𝑦𝑂 = 𝑙 → ((𝑥 +s 𝑦𝑂) = (𝑦𝑂 +s 𝑥) ↔ (𝑥 +s 𝑙) = (𝑙 +s 𝑥)))
31	30	rspccva 3590	. . . . . . . . . . 11 ⊢ ((∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 +s 𝑦𝑂) = (𝑦𝑂 +s 𝑥) ∧ 𝑙 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))) → (𝑥 +s 𝑙) = (𝑙 +s 𝑥))
32	26, 27, 31	syl2an 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))
33	32	eqeq2d 2741	. . . . . . . . 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)))
34	33	rexbidva 3156	. . . . . . . 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)))
35	34	abbidv 2796	. . . . . . 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)})
36	25, 35	uneq12d 4135	. . . . . 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 4124	. . . . . 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)})
38	36, 37	eqtrdi 2781	. . . . 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 4149	. . . . . . . . . . 11 ⊢ (𝑟 ∈ ( R ‘𝑥) → 𝑟 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
40		oveq1 7397	. . . . . . . . . . . . 13 ⊢ (𝑥𝑂 = 𝑟 → (𝑥𝑂 +s 𝑦) = (𝑟 +s 𝑦))
41		oveq2 7398	. . . . . . . . . . . . 13 ⊢ (𝑥𝑂 = 𝑟 → (𝑦 +s 𝑥𝑂) = (𝑦 +s 𝑟))
42	40, 41	eqeq12d 2746	. . . . . . . . . . . 12 ⊢ (𝑥𝑂 = 𝑟 → ((𝑥𝑂 +s 𝑦) = (𝑦 +s 𝑥𝑂) ↔ (𝑟 +s 𝑦) = (𝑦 +s 𝑟)))
43	42	rspccva 3590	. . . . . . . . . . 11 ⊢ ((∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s 𝑦) = (𝑦 +s 𝑥𝑂) ∧ 𝑟 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))) → (𝑟 +s 𝑦) = (𝑦 +s 𝑟))
44	16, 39, 43	syl2an 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))
45	44	eqeq2d 2741	. . . . . . . . 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)))
46	45	rexbidva 3156	. . . . . . . 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)))
47	46	abbidv 2796	. . . . . . 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 4149	. . . . . . . . . . 11 ⊢ (𝑟 ∈ ( R ‘𝑦) → 𝑟 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
49		oveq2 7398	. . . . . . . . . . . . 13 ⊢ (𝑦𝑂 = 𝑟 → (𝑥 +s 𝑦𝑂) = (𝑥 +s 𝑟))
50		oveq1 7397	. . . . . . . . . . . . 13 ⊢ (𝑦𝑂 = 𝑟 → (𝑦𝑂 +s 𝑥) = (𝑟 +s 𝑥))
51	49, 50	eqeq12d 2746	. . . . . . . . . . . 12 ⊢ (𝑦𝑂 = 𝑟 → ((𝑥 +s 𝑦𝑂) = (𝑦𝑂 +s 𝑥) ↔ (𝑥 +s 𝑟) = (𝑟 +s 𝑥)))
52	51	rspccva 3590	. . . . . . . . . . 11 ⊢ ((∀𝑦𝑂 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))(𝑥 +s 𝑦𝑂) = (𝑦𝑂 +s 𝑥) ∧ 𝑟 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))) → (𝑥 +s 𝑟) = (𝑟 +s 𝑥))
53	26, 48, 52	syl2an 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))
54	53	eqeq2d 2741	. . . . . . . . 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)))
55	54	rexbidva 3156	. . . . . . . 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)))
56	55	abbidv 2796	. . . . . . 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)})
57	47, 56	uneq12d 4135	. . . . . 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 4124	. . . . . 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)})
59	57, 58	eqtrdi 2781	. . . . 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)}))
60	38, 59	oveq12d 7408	. . . 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 27876	. . . . 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)})))
62	61	adantr 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 27876	. . . . . 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)})))
64	63	ancoms 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)})))
65	64	adantr 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)})))
66	60, 62, 65	3eqtr4d 2775	. . 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))
67	66	ex 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)))
68	3, 6, 9, 12, 15, 67	no2inds 27869	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +s 𝐵) = (𝐵 +s 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  {cab 2708  ∀wral 3045  ∃wrex 3054   ∪ cun 3915  ‘cfv 6514  (class class class)co 7390   No csur 27558   |s cscut 27701   L cleft 27760   R cright 27761   +_s cadds 27873

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-1o 8437  df-2o 8438  df-no 27561  df-slt 27562  df-bday 27563  df-sslt 27700  df-scut 27702  df-made 27762  df-old 27763  df-left 27765  df-right 27766  df-norec2 27863  df-adds 27874

This theorem is referenced by:  addscomd  27881  sltadd2im  27900  sleadd2im  27902  sleadd2  27904  sltadd1  27906  addscan1  27908  pncans  27983  twocut  28316