Theorem nmulcom 36549

Description: Natural multiplication commutes. (Contributed by Scott Fenton, 10-Jun-2026.)

Assertion

Ref	Expression
nmulcom	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·no 𝐵) = (𝐵 ·no 𝐴))

Proof of Theorem nmulcom

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7403	. . 3 ⊢ (𝑎 = 𝑐 → (𝑎 ·no 𝑏) = (𝑐 ·no 𝑏))
2		oveq2 7404	. . 3 ⊢ (𝑎 = 𝑐 → (𝑏 ·no 𝑎) = (𝑏 ·no 𝑐))
3	1, 2	eqeq12d 2779	. 2 ⊢ (𝑎 = 𝑐 → ((𝑎 ·no 𝑏) = (𝑏 ·no 𝑎) ↔ (𝑐 ·no 𝑏) = (𝑏 ·no 𝑐)))
4		oveq2 7404	. . 3 ⊢ (𝑏 = 𝑑 → (𝑐 ·no 𝑏) = (𝑐 ·no 𝑑))
5		oveq1 7403	. . 3 ⊢ (𝑏 = 𝑑 → (𝑏 ·no 𝑐) = (𝑑 ·no 𝑐))
6	4, 5	eqeq12d 2779	. 2 ⊢ (𝑏 = 𝑑 → ((𝑐 ·no 𝑏) = (𝑏 ·no 𝑐) ↔ (𝑐 ·no 𝑑) = (𝑑 ·no 𝑐)))
7		oveq1 7403	. . 3 ⊢ (𝑎 = 𝑐 → (𝑎 ·no 𝑑) = (𝑐 ·no 𝑑))
8		oveq2 7404	. . 3 ⊢ (𝑎 = 𝑐 → (𝑑 ·no 𝑎) = (𝑑 ·no 𝑐))
9	7, 8	eqeq12d 2779	. 2 ⊢ (𝑎 = 𝑐 → ((𝑎 ·no 𝑑) = (𝑑 ·no 𝑎) ↔ (𝑐 ·no 𝑑) = (𝑑 ·no 𝑐)))
10		oveq1 7403	. . 3 ⊢ (𝑎 = 𝐴 → (𝑎 ·no 𝑏) = (𝐴 ·no 𝑏))
11		oveq2 7404	. . 3 ⊢ (𝑎 = 𝐴 → (𝑏 ·no 𝑎) = (𝑏 ·no 𝐴))
12	10, 11	eqeq12d 2779	. 2 ⊢ (𝑎 = 𝐴 → ((𝑎 ·no 𝑏) = (𝑏 ·no 𝑎) ↔ (𝐴 ·no 𝑏) = (𝑏 ·no 𝐴)))
13		oveq2 7404	. . 3 ⊢ (𝑏 = 𝐵 → (𝐴 ·no 𝑏) = (𝐴 ·no 𝐵))
14		oveq1 7403	. . 3 ⊢ (𝑏 = 𝐵 → (𝑏 ·no 𝐴) = (𝐵 ·no 𝐴))
15	13, 14	eqeq12d 2779	. 2 ⊢ (𝑏 = 𝐵 → ((𝐴 ·no 𝑏) = (𝑏 ·no 𝐴) ↔ (𝐴 ·no 𝐵) = (𝐵 ·no 𝐴)))
16		oveq1 7403	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑧 → (𝑐 ·no 𝑏) = (𝑧 ·no 𝑏))
17		oveq2 7404	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑧 → (𝑏 ·no 𝑐) = (𝑏 ·no 𝑧))
18	16, 17	eqeq12d 2779	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑧 → ((𝑐 ·no 𝑏) = (𝑏 ·no 𝑐) ↔ (𝑧 ·no 𝑏) = (𝑏 ·no 𝑧)))
19		simplr2 1231	. . . . . . . . . . . 12 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 ·no 𝑑) = (𝑑 ·no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 ·no 𝑏) = (𝑏 ·no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 ·no 𝑑) = (𝑑 ·no 𝑎))) ∧ (𝑧 ∈ 𝑎 ∧ 𝑦 ∈ 𝑏)) → ∀𝑐 ∈ 𝑎 (𝑐 ·no 𝑏) = (𝑏 ·no 𝑐))
20		simprl 780	. . . . . . . . . . . 12 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 ·no 𝑑) = (𝑑 ·no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 ·no 𝑏) = (𝑏 ·no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 ·no 𝑑) = (𝑑 ·no 𝑎))) ∧ (𝑧 ∈ 𝑎 ∧ 𝑦 ∈ 𝑏)) → 𝑧 ∈ 𝑎)
21	18, 19, 20	rspcdva 3583	. . . . . . . . . . 11 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 ·no 𝑑) = (𝑑 ·no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 ·no 𝑏) = (𝑏 ·no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 ·no 𝑑) = (𝑑 ·no 𝑎))) ∧ (𝑧 ∈ 𝑎 ∧ 𝑦 ∈ 𝑏)) → (𝑧 ·no 𝑏) = (𝑏 ·no 𝑧))
22		oveq2 7404	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑦 → (𝑎 ·no 𝑑) = (𝑎 ·no 𝑦))
23		oveq1 7403	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑦 → (𝑑 ·no 𝑎) = (𝑦 ·no 𝑎))
24	22, 23	eqeq12d 2779	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑦 → ((𝑎 ·no 𝑑) = (𝑑 ·no 𝑎) ↔ (𝑎 ·no 𝑦) = (𝑦 ·no 𝑎)))
25		simplr3 1232	. . . . . . . . . . . 12 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 ·no 𝑑) = (𝑑 ·no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 ·no 𝑏) = (𝑏 ·no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 ·no 𝑑) = (𝑑 ·no 𝑎))) ∧ (𝑧 ∈ 𝑎 ∧ 𝑦 ∈ 𝑏)) → ∀𝑑 ∈ 𝑏 (𝑎 ·no 𝑑) = (𝑑 ·no 𝑎))
26		simprr 782	. . . . . . . . . . . 12 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 ·no 𝑑) = (𝑑 ·no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 ·no 𝑏) = (𝑏 ·no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 ·no 𝑑) = (𝑑 ·no 𝑎))) ∧ (𝑧 ∈ 𝑎 ∧ 𝑦 ∈ 𝑏)) → 𝑦 ∈ 𝑏)
27	24, 25, 26	rspcdva 3583	. . . . . . . . . . 11 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 ·no 𝑑) = (𝑑 ·no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 ·no 𝑏) = (𝑏 ·no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 ·no 𝑑) = (𝑑 ·no 𝑎))) ∧ (𝑧 ∈ 𝑎 ∧ 𝑦 ∈ 𝑏)) → (𝑎 ·no 𝑦) = (𝑦 ·no 𝑎))
28	21, 27	oveq12d 7414	. . . . . . . . . 10 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 ·no 𝑑) = (𝑑 ·no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 ·no 𝑏) = (𝑏 ·no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 ·no 𝑑) = (𝑑 ·no 𝑎))) ∧ (𝑧 ∈ 𝑎 ∧ 𝑦 ∈ 𝑏)) → ((𝑧 ·no 𝑏) +no (𝑎 ·no 𝑦)) = ((𝑏 ·no 𝑧) +no (𝑦 ·no 𝑎)))
29		simpllr 785	. . . . . . . . . . . 12 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 ·no 𝑑) = (𝑑 ·no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 ·no 𝑏) = (𝑏 ·no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 ·no 𝑑) = (𝑑 ·no 𝑎))) ∧ (𝑧 ∈ 𝑎 ∧ 𝑦 ∈ 𝑏)) → 𝑏 ∈ On)
30		simplll 784	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 ·no 𝑑) = (𝑑 ·no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 ·no 𝑏) = (𝑏 ·no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 ·no 𝑑) = (𝑑 ·no 𝑎))) ∧ (𝑧 ∈ 𝑎 ∧ 𝑦 ∈ 𝑏)) → 𝑎 ∈ On)
31		onelon 6371	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ On ∧ 𝑧 ∈ 𝑎) → 𝑧 ∈ On)
32	30, 20, 31	syl2anc 593	. . . . . . . . . . . 12 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 ·no 𝑑) = (𝑑 ·no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 ·no 𝑏) = (𝑏 ·no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 ·no 𝑑) = (𝑑 ·no 𝑎))) ∧ (𝑧 ∈ 𝑎 ∧ 𝑦 ∈ 𝑏)) → 𝑧 ∈ On)
33		nmulcl 36546	. . . . . . . . . . . 12 ⊢ ((𝑏 ∈ On ∧ 𝑧 ∈ On) → (𝑏 ·no 𝑧) ∈ On)
34	29, 32, 33	syl2anc 593	. . . . . . . . . . 11 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 ·no 𝑑) = (𝑑 ·no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 ·no 𝑏) = (𝑏 ·no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 ·no 𝑑) = (𝑑 ·no 𝑎))) ∧ (𝑧 ∈ 𝑎 ∧ 𝑦 ∈ 𝑏)) → (𝑏 ·no 𝑧) ∈ On)
35		onelon 6371	. . . . . . . . . . . . 13 ⊢ ((𝑏 ∈ On ∧ 𝑦 ∈ 𝑏) → 𝑦 ∈ On)
36	29, 26, 35	syl2anc 593	. . . . . . . . . . . 12 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 ·no 𝑑) = (𝑑 ·no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 ·no 𝑏) = (𝑏 ·no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 ·no 𝑑) = (𝑑 ·no 𝑎))) ∧ (𝑧 ∈ 𝑎 ∧ 𝑦 ∈ 𝑏)) → 𝑦 ∈ On)
37		nmulcl 36546	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ On ∧ 𝑎 ∈ On) → (𝑦 ·no 𝑎) ∈ On)
38	36, 30, 37	syl2anc 593	. . . . . . . . . . 11 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 ·no 𝑑) = (𝑑 ·no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 ·no 𝑏) = (𝑏 ·no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 ·no 𝑑) = (𝑑 ·no 𝑎))) ∧ (𝑧 ∈ 𝑎 ∧ 𝑦 ∈ 𝑏)) → (𝑦 ·no 𝑎) ∈ On)
39		naddcom 8653	. . . . . . . . . . 11 ⊢ (((𝑏 ·no 𝑧) ∈ On ∧ (𝑦 ·no 𝑎) ∈ On) → ((𝑏 ·no 𝑧) +no (𝑦 ·no 𝑎)) = ((𝑦 ·no 𝑎) +no (𝑏 ·no 𝑧)))
40	34, 38, 39	syl2anc 593	. . . . . . . . . 10 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 ·no 𝑑) = (𝑑 ·no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 ·no 𝑏) = (𝑏 ·no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 ·no 𝑑) = (𝑑 ·no 𝑎))) ∧ (𝑧 ∈ 𝑎 ∧ 𝑦 ∈ 𝑏)) → ((𝑏 ·no 𝑧) +no (𝑦 ·no 𝑎)) = ((𝑦 ·no 𝑎) +no (𝑏 ·no 𝑧)))
41	28, 40	eqtrd 2798	. . . . . . . . 9 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 ·no 𝑑) = (𝑑 ·no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 ·no 𝑏) = (𝑏 ·no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 ·no 𝑑) = (𝑑 ·no 𝑎))) ∧ (𝑧 ∈ 𝑎 ∧ 𝑦 ∈ 𝑏)) → ((𝑧 ·no 𝑏) +no (𝑎 ·no 𝑦)) = ((𝑦 ·no 𝑎) +no (𝑏 ·no 𝑧)))
42		oveq1 7403	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑧 → (𝑐 ·no 𝑑) = (𝑧 ·no 𝑑))
43		oveq2 7404	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑧 → (𝑑 ·no 𝑐) = (𝑑 ·no 𝑧))
44	42, 43	eqeq12d 2779	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑧 → ((𝑐 ·no 𝑑) = (𝑑 ·no 𝑐) ↔ (𝑧 ·no 𝑑) = (𝑑 ·no 𝑧)))
45		oveq2 7404	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑦 → (𝑧 ·no 𝑑) = (𝑧 ·no 𝑦))
46		oveq1 7403	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑦 → (𝑑 ·no 𝑧) = (𝑦 ·no 𝑧))
47	45, 46	eqeq12d 2779	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑦 → ((𝑧 ·no 𝑑) = (𝑑 ·no 𝑧) ↔ (𝑧 ·no 𝑦) = (𝑦 ·no 𝑧)))
48		simplr1 1230	. . . . . . . . . . 11 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 ·no 𝑑) = (𝑑 ·no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 ·no 𝑏) = (𝑏 ·no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 ·no 𝑑) = (𝑑 ·no 𝑎))) ∧ (𝑧 ∈ 𝑎 ∧ 𝑦 ∈ 𝑏)) → ∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 ·no 𝑑) = (𝑑 ·no 𝑐))
49	44, 47, 48, 20, 26	rspc2dv 3597	. . . . . . . . . 10 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 ·no 𝑑) = (𝑑 ·no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 ·no 𝑏) = (𝑏 ·no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 ·no 𝑑) = (𝑑 ·no 𝑎))) ∧ (𝑧 ∈ 𝑎 ∧ 𝑦 ∈ 𝑏)) → (𝑧 ·no 𝑦) = (𝑦 ·no 𝑧))
50	49	oveq2d 7412	. . . . . . . . 9 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 ·no 𝑑) = (𝑑 ·no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 ·no 𝑏) = (𝑏 ·no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 ·no 𝑑) = (𝑑 ·no 𝑎))) ∧ (𝑧 ∈ 𝑎 ∧ 𝑦 ∈ 𝑏)) → (𝑥 +no (𝑧 ·no 𝑦)) = (𝑥 +no (𝑦 ·no 𝑧)))
51	41, 50	eleq12d 2857	. . . . . . . 8 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 ·no 𝑑) = (𝑑 ·no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 ·no 𝑏) = (𝑏 ·no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 ·no 𝑑) = (𝑑 ·no 𝑎))) ∧ (𝑧 ∈ 𝑎 ∧ 𝑦 ∈ 𝑏)) → (((𝑧 ·no 𝑏) +no (𝑎 ·no 𝑦)) ∈ (𝑥 +no (𝑧 ·no 𝑦)) ↔ ((𝑦 ·no 𝑎) +no (𝑏 ·no 𝑧)) ∈ (𝑥 +no (𝑦 ·no 𝑧))))
52	51	2ralbidva 3225	. . . . . . 7 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 ·no 𝑑) = (𝑑 ·no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 ·no 𝑏) = (𝑏 ·no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 ·no 𝑑) = (𝑑 ·no 𝑎))) → (∀𝑧 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑧 ·no 𝑏) +no (𝑎 ·no 𝑦)) ∈ (𝑥 +no (𝑧 ·no 𝑦)) ↔ ∀𝑧 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑦 ·no 𝑎) +no (𝑏 ·no 𝑧)) ∈ (𝑥 +no (𝑦 ·no 𝑧))))
53		ralcom 3291	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑦 ·no 𝑎) +no (𝑏 ·no 𝑧)) ∈ (𝑥 +no (𝑦 ·no 𝑧)) ↔ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑎 ((𝑦 ·no 𝑎) +no (𝑏 ·no 𝑧)) ∈ (𝑥 +no (𝑦 ·no 𝑧)))
54	52, 53	bitrdi 289	. . . . . 6 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 ·no 𝑑) = (𝑑 ·no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 ·no 𝑏) = (𝑏 ·no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 ·no 𝑑) = (𝑑 ·no 𝑎))) → (∀𝑧 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑧 ·no 𝑏) +no (𝑎 ·no 𝑦)) ∈ (𝑥 +no (𝑧 ·no 𝑦)) ↔ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑎 ((𝑦 ·no 𝑎) +no (𝑏 ·no 𝑧)) ∈ (𝑥 +no (𝑦 ·no 𝑧))))
55	54	rabbidv 3422	. . . . 5 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 ·no 𝑑) = (𝑑 ·no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 ·no 𝑏) = (𝑏 ·no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 ·no 𝑑) = (𝑑 ·no 𝑎))) → {𝑥 ∈ On ∣ ∀𝑧 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑧 ·no 𝑏) +no (𝑎 ·no 𝑦)) ∈ (𝑥 +no (𝑧 ·no 𝑦))} = {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑎 ((𝑦 ·no 𝑎) +no (𝑏 ·no 𝑧)) ∈ (𝑥 +no (𝑦 ·no 𝑧))})
56	55	inteqd 4911	. . . 4 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 ·no 𝑑) = (𝑑 ·no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 ·no 𝑏) = (𝑏 ·no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 ·no 𝑑) = (𝑑 ·no 𝑎))) → ∩ {𝑥 ∈ On ∣ ∀𝑧 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑧 ·no 𝑏) +no (𝑎 ·no 𝑦)) ∈ (𝑥 +no (𝑧 ·no 𝑦))} = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑎 ((𝑦 ·no 𝑎) +no (𝑏 ·no 𝑧)) ∈ (𝑥 +no (𝑦 ·no 𝑧))})
57		nmulval 36547	. . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 ·no 𝑏) = ∩ {𝑥 ∈ On ∣ ∀𝑧 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑧 ·no 𝑏) +no (𝑎 ·no 𝑦)) ∈ (𝑥 +no (𝑧 ·no 𝑦))})
58	57	adantr 484	. . . 4 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 ·no 𝑑) = (𝑑 ·no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 ·no 𝑏) = (𝑏 ·no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 ·no 𝑑) = (𝑑 ·no 𝑎))) → (𝑎 ·no 𝑏) = ∩ {𝑥 ∈ On ∣ ∀𝑧 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑧 ·no 𝑏) +no (𝑎 ·no 𝑦)) ∈ (𝑥 +no (𝑧 ·no 𝑦))})
59		nmulval 36547	. . . . . 6 ⊢ ((𝑏 ∈ On ∧ 𝑎 ∈ On) → (𝑏 ·no 𝑎) = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑎 ((𝑦 ·no 𝑎) +no (𝑏 ·no 𝑧)) ∈ (𝑥 +no (𝑦 ·no 𝑧))})
60	59	ancoms 462	. . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑏 ·no 𝑎) = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑎 ((𝑦 ·no 𝑎) +no (𝑏 ·no 𝑧)) ∈ (𝑥 +no (𝑦 ·no 𝑧))})
61	60	adantr 484	. . . 4 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 ·no 𝑑) = (𝑑 ·no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 ·no 𝑏) = (𝑏 ·no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 ·no 𝑑) = (𝑑 ·no 𝑎))) → (𝑏 ·no 𝑎) = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑎 ((𝑦 ·no 𝑎) +no (𝑏 ·no 𝑧)) ∈ (𝑥 +no (𝑦 ·no 𝑧))})
62	56, 58, 61	3eqtr4d 2808	. . 3 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 ·no 𝑑) = (𝑑 ·no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 ·no 𝑏) = (𝑏 ·no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 ·no 𝑑) = (𝑑 ·no 𝑎))) → (𝑎 ·no 𝑏) = (𝑏 ·no 𝑎))
63	62	ex 416	. 2 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 ·no 𝑑) = (𝑑 ·no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 ·no 𝑏) = (𝑏 ·no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 ·no 𝑑) = (𝑑 ·no 𝑎)) → (𝑎 ·no 𝑏) = (𝑏 ·no 𝑎)))
64	3, 6, 9, 12, 15, 63	on2ind 8639	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·no 𝐵) = (𝐵 ·no 𝐴))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1099   = wceq 1561   ∈ wcel 2143  ∀wral 3077  {crab 3415  ∩ cint 4906  Oncon0 6346  (class class class)co 7396   +no cnadd 8635   ·_no cnmul 36542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-se 5602  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6288  df-ord 6349  df-on 6350  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7970  df-2nd 7971  df-frecs 8262  df-nadd 8636  df-nmul 36543

This theorem is referenced by:  nmull0  36551