Theorem addscom 27958

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 7374	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 +s 𝑦) = (𝑥𝑂 +s 𝑦))
2		oveq2 7375	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (𝑦 +s 𝑥) = (𝑦 +s 𝑥𝑂))
3	1, 2	eqeq12d 2752	. 2 ⊢ (𝑥 = 𝑥𝑂 → ((𝑥 +s 𝑦) = (𝑦 +s 𝑥) ↔ (𝑥𝑂 +s 𝑦) = (𝑦 +s 𝑥𝑂)))
4		oveq2 7375	. . 3 ⊢ (𝑦 = 𝑦𝑂 → (𝑥𝑂 +s 𝑦) = (𝑥𝑂 +s 𝑦𝑂))
5		oveq1 7374	. . 3 ⊢ (𝑦 = 𝑦𝑂 → (𝑦 +s 𝑥𝑂) = (𝑦𝑂 +s 𝑥𝑂))
6	4, 5	eqeq12d 2752	. 2 ⊢ (𝑦 = 𝑦𝑂 → ((𝑥𝑂 +s 𝑦) = (𝑦 +s 𝑥𝑂) ↔ (𝑥𝑂 +s 𝑦𝑂) = (𝑦𝑂 +s 𝑥𝑂)))
7		oveq1 7374	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 +s 𝑦𝑂) = (𝑥𝑂 +s 𝑦𝑂))
8		oveq2 7375	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (𝑦𝑂 +s 𝑥) = (𝑦𝑂 +s 𝑥𝑂))
9	7, 8	eqeq12d 2752	. 2 ⊢ (𝑥 = 𝑥𝑂 → ((𝑥 +s 𝑦𝑂) = (𝑦𝑂 +s 𝑥) ↔ (𝑥𝑂 +s 𝑦𝑂) = (𝑦𝑂 +s 𝑥𝑂)))
10		oveq1 7374	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 +s 𝑦) = (𝐴 +s 𝑦))
11		oveq2 7375	. . 3 ⊢ (𝑥 = 𝐴 → (𝑦 +s 𝑥) = (𝑦 +s 𝐴))
12	10, 11	eqeq12d 2752	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 +s 𝑦) = (𝑦 +s 𝑥) ↔ (𝐴 +s 𝑦) = (𝑦 +s 𝐴)))
13		oveq2 7375	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 +s 𝑦) = (𝐴 +s 𝐵))
14		oveq1 7374	. . 3 ⊢ (𝑦 = 𝐵 → (𝑦 +s 𝐴) = (𝐵 +s 𝐴))
15	13, 14	eqeq12d 2752	. 2 ⊢ (𝑦 = 𝐵 → ((𝐴 +s 𝑦) = (𝑦 +s 𝐴) ↔ (𝐴 +s 𝐵) = (𝐵 +s 𝐴)))
16		simpr2 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))) → ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 +s 𝑦) = (𝑦 +s 𝑥𝑂))
17		elun1 4122	. . . . . . . . . . 11 ⊢ (𝑙 ∈ ( L ‘𝑥) → 𝑙 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
18		oveq1 7374	. . . . . . . . . . . . 13 ⊢ (𝑥𝑂 = 𝑙 → (𝑥𝑂 +s 𝑦) = (𝑙 +s 𝑦))
19		oveq2 7375	. . . . . . . . . . . . 13 ⊢ (𝑥𝑂 = 𝑙 → (𝑦 +s 𝑥𝑂) = (𝑦 +s 𝑙))
20	18, 19	eqeq12d 2752	. . . . . . . . . . . 12 ⊢ (𝑥𝑂 = 𝑙 → ((𝑥𝑂 +s 𝑦) = (𝑦 +s 𝑥𝑂) ↔ (𝑙 +s 𝑦) = (𝑦 +s 𝑙)))
21	20	rspccva 3563	. . . . . . . . . . 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 597	. . . . . . . . . 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 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))) ∧ 𝑙 ∈ ( L ‘𝑥)) → (𝑤 = (𝑙 +s 𝑦) ↔ 𝑤 = (𝑦 +s 𝑙)))
24	23	rexbidva 3159	. . . . . . . 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 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))) → {𝑤 ∣ ∃𝑙 ∈ ( L ‘𝑥)𝑤 = (𝑙 +s 𝑦)} = {𝑤 ∣ ∃𝑙 ∈ ( L ‘𝑥)𝑤 = (𝑦 +s 𝑙)})
26		simpr3 1198	. . . . . . . . . . 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 4122	. . . . . . . . . . 11 ⊢ (𝑙 ∈ ( L ‘𝑦) → 𝑙 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
28		oveq2 7375	. . . . . . . . . . . . 13 ⊢ (𝑦𝑂 = 𝑙 → (𝑥 +s 𝑦𝑂) = (𝑥 +s 𝑙))
29		oveq1 7374	. . . . . . . . . . . . 13 ⊢ (𝑦𝑂 = 𝑙 → (𝑦𝑂 +s 𝑥) = (𝑙 +s 𝑥))
30	28, 29	eqeq12d 2752	. . . . . . . . . . . 12 ⊢ (𝑦𝑂 = 𝑙 → ((𝑥 +s 𝑦𝑂) = (𝑦𝑂 +s 𝑥) ↔ (𝑥 +s 𝑙) = (𝑙 +s 𝑥)))
31	30	rspccva 3563	. . . . . . . . . . 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 597	. . . . . . . . . 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 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))) ∧ 𝑙 ∈ ( L ‘𝑦)) → (𝑧 = (𝑥 +s 𝑙) ↔ 𝑧 = (𝑙 +s 𝑥)))
34	33	rexbidva 3159	. . . . . . . 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 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))) → {𝑧 ∣ ∃𝑙 ∈ ( L ‘𝑦)𝑧 = (𝑥 +s 𝑙)} = {𝑧 ∣ ∃𝑙 ∈ ( L ‘𝑦)𝑧 = (𝑙 +s 𝑥)})
36	25, 35	uneq12d 4109	. . . . . 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 4098	. . . . . 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 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))) → ({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 4123	. . . . . . . . . . 11 ⊢ (𝑟 ∈ ( R ‘𝑥) → 𝑟 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
40		oveq1 7374	. . . . . . . . . . . . 13 ⊢ (𝑥𝑂 = 𝑟 → (𝑥𝑂 +s 𝑦) = (𝑟 +s 𝑦))
41		oveq2 7375	. . . . . . . . . . . . 13 ⊢ (𝑥𝑂 = 𝑟 → (𝑦 +s 𝑥𝑂) = (𝑦 +s 𝑟))
42	40, 41	eqeq12d 2752	. . . . . . . . . . . 12 ⊢ (𝑥𝑂 = 𝑟 → ((𝑥𝑂 +s 𝑦) = (𝑦 +s 𝑥𝑂) ↔ (𝑟 +s 𝑦) = (𝑦 +s 𝑟)))
43	42	rspccva 3563	. . . . . . . . . . 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 597	. . . . . . . . . 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 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)) → (𝑤 = (𝑟 +s 𝑦) ↔ 𝑤 = (𝑦 +s 𝑟)))
46	45	rexbidva 3159	. . . . . . . 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 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))) → {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑥)𝑤 = (𝑟 +s 𝑦)} = {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑥)𝑤 = (𝑦 +s 𝑟)})
48		elun2 4123	. . . . . . . . . . 11 ⊢ (𝑟 ∈ ( R ‘𝑦) → 𝑟 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
49		oveq2 7375	. . . . . . . . . . . . 13 ⊢ (𝑦𝑂 = 𝑟 → (𝑥 +s 𝑦𝑂) = (𝑥 +s 𝑟))
50		oveq1 7374	. . . . . . . . . . . . 13 ⊢ (𝑦𝑂 = 𝑟 → (𝑦𝑂 +s 𝑥) = (𝑟 +s 𝑥))
51	49, 50	eqeq12d 2752	. . . . . . . . . . . 12 ⊢ (𝑦𝑂 = 𝑟 → ((𝑥 +s 𝑦𝑂) = (𝑦𝑂 +s 𝑥) ↔ (𝑥 +s 𝑟) = (𝑟 +s 𝑥)))
52	51	rspccva 3563	. . . . . . . . . . 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 597	. . . . . . . . . 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 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 ‘𝑦)) → (𝑧 = (𝑥 +s 𝑟) ↔ 𝑧 = (𝑟 +s 𝑥)))
55	54	rexbidva 3159	. . . . . . . 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 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))) → {𝑧 ∣ ∃𝑟 ∈ ( R ‘𝑦)𝑧 = (𝑥 +s 𝑟)} = {𝑧 ∣ ∃𝑟 ∈ ( R ‘𝑦)𝑧 = (𝑟 +s 𝑥)})
57	47, 56	uneq12d 4109	. . . . . 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 4098	. . . . . 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 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))) → ({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 7385	. . . 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 27954	. . . . 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 27954	. . . . . 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 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))
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 27947	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +s 𝐵) = (𝐵 +s 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  {cab 2714  ∀wral 3051  ∃wrex 3061   ∪ cun 3887  ‘cfv 6498  (class class class)co 7367   No csur 27603   |s ccuts 27751   L cleft 27817   R cright 27818   +_s cadds 27951

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-1o 8405  df-2o 8406  df-no 27606  df-lts 27607  df-bday 27608  df-slts 27750  df-cuts 27752  df-made 27819  df-old 27820  df-left 27822  df-right 27823  df-norec2 27941  df-adds 27952

This theorem is referenced by:  addscomd  27959  ltadds2im  27978  leadds2im  27980  leadds2  27982  ltadds1  27984  addscan1  27986  pncans  28064  twocut  28415