Theorem addscom 28385

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 7453	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 +s 𝑦) = (𝑥𝑂 +s 𝑦))
2		oveq2 7454	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (𝑦 +s 𝑥) = (𝑦 +s 𝑥𝑂))
3	1, 2	eqeq12d 2779	. 2 ⊢ (𝑥 = 𝑥𝑂 → ((𝑥 +s 𝑦) = (𝑦 +s 𝑥) ↔ (𝑥𝑂 +s 𝑦) = (𝑦 +s 𝑥𝑂)))
4		oveq2 7454	. . 3 ⊢ (𝑦 = 𝑦𝑂 → (𝑥𝑂 +s 𝑦) = (𝑥𝑂 +s 𝑦𝑂))
5		oveq1 7453	. . 3 ⊢ (𝑦 = 𝑦𝑂 → (𝑦 +s 𝑥𝑂) = (𝑦𝑂 +s 𝑥𝑂))
6	4, 5	eqeq12d 2779	. 2 ⊢ (𝑦 = 𝑦𝑂 → ((𝑥𝑂 +s 𝑦) = (𝑦 +s 𝑥𝑂) ↔ (𝑥𝑂 +s 𝑦𝑂) = (𝑦𝑂 +s 𝑥𝑂)))
7		oveq1 7453	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 +s 𝑦𝑂) = (𝑥𝑂 +s 𝑦𝑂))
8		oveq2 7454	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (𝑦𝑂 +s 𝑥) = (𝑦𝑂 +s 𝑥𝑂))
9	7, 8	eqeq12d 2779	. 2 ⊢ (𝑥 = 𝑥𝑂 → ((𝑥 +s 𝑦𝑂) = (𝑦𝑂 +s 𝑥) ↔ (𝑥𝑂 +s 𝑦𝑂) = (𝑦𝑂 +s 𝑥𝑂)))
10		oveq1 7453	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 +s 𝑦) = (𝐴 +s 𝑦))
11		oveq2 7454	. . 3 ⊢ (𝑥 = 𝐴 → (𝑦 +s 𝑥) = (𝑦 +s 𝐴))
12	10, 11	eqeq12d 2779	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 +s 𝑦) = (𝑦 +s 𝑥) ↔ (𝐴 +s 𝑦) = (𝑦 +s 𝐴)))
13		oveq2 7454	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 +s 𝑦) = (𝐴 +s 𝐵))
14		oveq1 7453	. . 3 ⊢ (𝑦 = 𝐵 → (𝑦 +s 𝐴) = (𝐵 +s 𝐴))
15	13, 14	eqeq12d 2779	. 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 4183	. . . . . . . . . . 11 ⊢ (𝑙 ∈ (L‘𝑥) → 𝑙 ∈ ((L‘𝑥) ∪ (R‘𝑥)))
18		oveq1 7453	. . . . . . . . . . . . 13 ⊢ (𝑥𝑂 = 𝑙 → (𝑥𝑂 +s 𝑦) = (𝑙 +s 𝑦))
19		oveq2 7454	. . . . . . . . . . . . 13 ⊢ (𝑥𝑂 = 𝑙 → (𝑦 +s 𝑥𝑂) = (𝑦 +s 𝑙))
20	18, 19	eqeq12d 2779	. . . . . . . . . . . 12 ⊢ (𝑥𝑂 = 𝑙 → ((𝑥𝑂 +s 𝑦) = (𝑦 +s 𝑥𝑂) ↔ (𝑙 +s 𝑦) = (𝑦 +s 𝑙)))
21	20	rspccva 3579	. . . . . . . . . . 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 2774	. . . . . . . . 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 3188	. . . . . . . 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 2829	. . . . . . 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 4183	. . . . . . . . . . 11 ⊢ (𝑙 ∈ (L‘𝑦) → 𝑙 ∈ ((L‘𝑦) ∪ (R‘𝑦)))
28		oveq2 7454	. . . . . . . . . . . . 13 ⊢ (𝑦𝑂 = 𝑙 → (𝑥 +s 𝑦𝑂) = (𝑥 +s 𝑙))
29		oveq1 7453	. . . . . . . . . . . . 13 ⊢ (𝑦𝑂 = 𝑙 → (𝑦𝑂 +s 𝑥) = (𝑙 +s 𝑥))
30	28, 29	eqeq12d 2779	. . . . . . . . . . . 12 ⊢ (𝑦𝑂 = 𝑙 → ((𝑥 +s 𝑦𝑂) = (𝑦𝑂 +s 𝑥) ↔ (𝑥 +s 𝑙) = (𝑙 +s 𝑥)))
31	30	rspccva 3579	. . . . . . . . . . 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 2774	. . . . . . . . 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 3188	. . . . . . . 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 2829	. . . . . . 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 4171	. . . . . 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 4160	. . . . . 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 2814	. . . . 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 4184	. . . . . . . . . . 11 ⊢ (𝑟 ∈ (R‘𝑥) → 𝑟 ∈ ((L‘𝑥) ∪ (R‘𝑥)))
40		oveq1 7453	. . . . . . . . . . . . 13 ⊢ (𝑥𝑂 = 𝑟 → (𝑥𝑂 +s 𝑦) = (𝑟 +s 𝑦))
41		oveq2 7454	. . . . . . . . . . . . 13 ⊢ (𝑥𝑂 = 𝑟 → (𝑦 +s 𝑥𝑂) = (𝑦 +s 𝑟))
42	40, 41	eqeq12d 2779	. . . . . . . . . . . 12 ⊢ (𝑥𝑂 = 𝑟 → ((𝑥𝑂 +s 𝑦) = (𝑦 +s 𝑥𝑂) ↔ (𝑟 +s 𝑦) = (𝑦 +s 𝑟)))
43	42	rspccva 3579	. . . . . . . . . . 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 2774	. . . . . . . . 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 3188	. . . . . . . 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 2829	. . . . . . 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 4184	. . . . . . . . . . 11 ⊢ (𝑟 ∈ (R‘𝑦) → 𝑟 ∈ ((L‘𝑦) ∪ (R‘𝑦)))
49		oveq2 7454	. . . . . . . . . . . . 13 ⊢ (𝑦𝑂 = 𝑟 → (𝑥 +s 𝑦𝑂) = (𝑥 +s 𝑟))
50		oveq1 7453	. . . . . . . . . . . . 13 ⊢ (𝑦𝑂 = 𝑟 → (𝑦𝑂 +s 𝑥) = (𝑟 +s 𝑥))
51	49, 50	eqeq12d 2779	. . . . . . . . . . . 12 ⊢ (𝑦𝑂 = 𝑟 → ((𝑥 +s 𝑦𝑂) = (𝑦𝑂 +s 𝑥) ↔ (𝑥 +s 𝑟) = (𝑟 +s 𝑥)))
52	51	rspccva 3579	. . . . . . . . . . 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 2774	. . . . . . . . 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 3188	. . . . . . . 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 2829	. . . . . . 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 4171	. . . . . 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 4160	. . . . . 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 2814	. . . . 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 7464	. . . 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 28381	. . . . 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 28381	. . . . . 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 2808	. . 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 28374	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (𝐴 +s 𝐵) = (𝐵 +s 𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  {cab 2741  ∀wral 3080  ∃wrex 3090   ∪ cun 4152  ‘cfv 6561  (class class class)co 7446  Nocsur 28030   |s ccuts 28178  Lcleft 28244  Rcright 28245   +_s cadds 28378

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-se 5610  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-1st 8028  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-1o 8507  df-2o 8508  df-no 28033  df-lts 28034  df-bday 28035  df-slts 28177  df-cuts 28179  df-made 28246  df-old 28247  df-left 28249  df-right 28250  df-norec2 28368  df-adds 28379

This theorem is used by:  addscomd  28386  ltadds2im  28405  leadds2im  28407  leadds2  28409  ltadds1  28411  addscan1  28413  pncans  28491  twocut  28842