Theorem addscom 28357

Description: Surreal addition is commutative. 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 7427	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 +s 𝑦) = (𝑥𝑂 +s 𝑦))
2		oveq2 7428	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (𝑦 +s 𝑥) = (𝑦 +s 𝑥𝑂))
3	1, 2	eqeq12d 2776	. 2 ⊢ (𝑥 = 𝑥𝑂 → ((𝑥 +s 𝑦) = (𝑦 +s 𝑥) ↔ (𝑥𝑂 +s 𝑦) = (𝑦 +s 𝑥𝑂)))
4		oveq2 7428	. . 3 ⊢ (𝑦 = 𝑦𝑂 → (𝑥𝑂 +s 𝑦) = (𝑥𝑂 +s 𝑦𝑂))
5		oveq1 7427	. . 3 ⊢ (𝑦 = 𝑦𝑂 → (𝑦 +s 𝑥𝑂) = (𝑦𝑂 +s 𝑥𝑂))
6	4, 5	eqeq12d 2776	. 2 ⊢ (𝑦 = 𝑦𝑂 → ((𝑥𝑂 +s 𝑦) = (𝑦 +s 𝑥𝑂) ↔ (𝑥𝑂 +s 𝑦𝑂) = (𝑦𝑂 +s 𝑥𝑂)))
7		oveq1 7427	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 +s 𝑦𝑂) = (𝑥𝑂 +s 𝑦𝑂))
8		oveq2 7428	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (𝑦𝑂 +s 𝑥) = (𝑦𝑂 +s 𝑥𝑂))
9	7, 8	eqeq12d 2776	. 2 ⊢ (𝑥 = 𝑥𝑂 → ((𝑥 +s 𝑦𝑂) = (𝑦𝑂 +s 𝑥) ↔ (𝑥𝑂 +s 𝑦𝑂) = (𝑦𝑂 +s 𝑥𝑂)))
10		oveq1 7427	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 +s 𝑦) = (𝐴 +s 𝑦))
11		oveq2 7428	. . 3 ⊢ (𝑥 = 𝐴 → (𝑦 +s 𝑥) = (𝑦 +s 𝐴))
12	10, 11	eqeq12d 2776	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 +s 𝑦) = (𝑦 +s 𝑥) ↔ (𝐴 +s 𝑦) = (𝑦 +s 𝐴)))
13		oveq2 7428	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 +s 𝑦) = (𝐴 +s 𝐵))
14		oveq1 7427	. . 3 ⊢ (𝑦 = 𝐵 → (𝑦 +s 𝐴) = (𝐵 +s 𝐴))
15	13, 14	eqeq12d 2776	. 2 ⊢ (𝑦 = 𝐵 → ((𝐴 +s 𝑦) = (𝑦 +s 𝐴) ↔ (𝐴 +s 𝐵) = (𝐵 +s 𝐴)))
16		simpr2 1214	. . . . . . . . . . 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 4180	. . . . . . . . . . 11 ⊢ (𝑙 ∈ (L‘𝑥) → 𝑙 ∈ ((L‘𝑥) ∪ (R‘𝑥)))
18		oveq1 7427	. . . . . . . . . . . . 13 ⊢ (𝑥𝑂 = 𝑙 → (𝑥𝑂 +s 𝑦) = (𝑙 +s 𝑦))
19		oveq2 7428	. . . . . . . . . . . . 13 ⊢ (𝑥𝑂 = 𝑙 → (𝑦 +s 𝑥𝑂) = (𝑦 +s 𝑙))
20	18, 19	eqeq12d 2776	. . . . . . . . . . . 12 ⊢ (𝑥𝑂 = 𝑙 → ((𝑥𝑂 +s 𝑦) = (𝑦 +s 𝑥𝑂) ↔ (𝑙 +s 𝑦) = (𝑦 +s 𝑙)))
21	20	rspccva 3575	. . . . . . . . . . 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 608	. . . . . . . . . 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 2771	. . . . . . . . 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 3184	. . . . . . . 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 2826	. . . . . . 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 1215	. . . . . . . . . . 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 4180	. . . . . . . . . . 11 ⊢ (𝑙 ∈ (L‘𝑦) → 𝑙 ∈ ((L‘𝑦) ∪ (R‘𝑦)))
28		oveq2 7428	. . . . . . . . . . . . 13 ⊢ (𝑦𝑂 = 𝑙 → (𝑥 +s 𝑦𝑂) = (𝑥 +s 𝑙))
29		oveq1 7427	. . . . . . . . . . . . 13 ⊢ (𝑦𝑂 = 𝑙 → (𝑦𝑂 +s 𝑥) = (𝑙 +s 𝑥))
30	28, 29	eqeq12d 2776	. . . . . . . . . . . 12 ⊢ (𝑦𝑂 = 𝑙 → ((𝑥 +s 𝑦𝑂) = (𝑦𝑂 +s 𝑥) ↔ (𝑥 +s 𝑙) = (𝑙 +s 𝑥)))
31	30	rspccva 3575	. . . . . . . . . . 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 608	. . . . . . . . . 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 2771	. . . . . . . . 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 3184	. . . . . . . 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 2826	. . . . . . 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 4168	. . . . . 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 4157	. . . . . 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 2811	. . . . 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 4181	. . . . . . . . . . 11 ⊢ (𝑟 ∈ (R‘𝑥) → 𝑟 ∈ ((L‘𝑥) ∪ (R‘𝑥)))
40		oveq1 7427	. . . . . . . . . . . . 13 ⊢ (𝑥𝑂 = 𝑟 → (𝑥𝑂 +s 𝑦) = (𝑟 +s 𝑦))
41		oveq2 7428	. . . . . . . . . . . . 13 ⊢ (𝑥𝑂 = 𝑟 → (𝑦 +s 𝑥𝑂) = (𝑦 +s 𝑟))
42	40, 41	eqeq12d 2776	. . . . . . . . . . . 12 ⊢ (𝑥𝑂 = 𝑟 → ((𝑥𝑂 +s 𝑦) = (𝑦 +s 𝑥𝑂) ↔ (𝑟 +s 𝑦) = (𝑦 +s 𝑟)))
43	42	rspccva 3575	. . . . . . . . . . 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 608	. . . . . . . . . 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 2771	. . . . . . . . 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 3184	. . . . . . . 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 2826	. . . . . . 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 4181	. . . . . . . . . . 11 ⊢ (𝑟 ∈ (R‘𝑦) → 𝑟 ∈ ((L‘𝑦) ∪ (R‘𝑦)))
49		oveq2 7428	. . . . . . . . . . . . 13 ⊢ (𝑦𝑂 = 𝑟 → (𝑥 +s 𝑦𝑂) = (𝑥 +s 𝑟))
50		oveq1 7427	. . . . . . . . . . . . 13 ⊢ (𝑦𝑂 = 𝑟 → (𝑦𝑂 +s 𝑥) = (𝑟 +s 𝑥))
51	49, 50	eqeq12d 2776	. . . . . . . . . . . 12 ⊢ (𝑦𝑂 = 𝑟 → ((𝑥 +s 𝑦𝑂) = (𝑦𝑂 +s 𝑥) ↔ (𝑥 +s 𝑟) = (𝑟 +s 𝑥)))
52	51	rspccva 3575	. . . . . . . . . . 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 608	. . . . . . . . . 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 2771	. . . . . . . . 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 3184	. . . . . . . 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 2826	. . . . . . 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 4168	. . . . . 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 4157	. . . . . 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 2811	. . . . 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 7438	. . . 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 28353	. . . . 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 486	. . . 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 28353	. . . . . 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 464	. . . . 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 486	. . . 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 2805	. . 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 418	. 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 28346	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (𝐴 +s 𝐵) = (𝐵 +s 𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  {cab 2738  ∀wral 3076  ∃wrex 3086   ∪ cun 4149  ‘cfv 6538  (class class class)co 7420  Nocsur 28002   |s ccuts 28150  Lcleft 28216  Rcright 28217   +_s cadds 28350

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-1o 8479  df-2o 8480  df-no 28005  df-lts 28006  df-bday 28007  df-slts 28149  df-cuts 28151  df-made 28218  df-old 28219  df-left 28221  df-right 28222  df-norec2 28340  df-adds 28351

This theorem is used by:  addscomd  28358  ltadds2im  28377  leadds2im  28379  leadds2  28381  ltadds1  28383  addscan1  28385  pncans  28463  twocut  28814