Theorem naddass 8713

Description: Natural ordinal addition is associative. (Contributed by Scott Fenton, 20-Jan-2025.)

Assertion

Ref	Expression
naddass	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 +no 𝐵) +no 𝐶) = (𝐴 +no (𝐵 +no 𝐶)))

Proof of Theorem naddass

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

Step	Hyp	Ref	Expression
1		oveq1 7417	. . . 4 ⊢ (𝑎 = 𝑥 → (𝑎 +no 𝑏) = (𝑥 +no 𝑏))
2	1	oveq1d 7425	. . 3 ⊢ (𝑎 = 𝑥 → ((𝑎 +no 𝑏) +no 𝑐) = ((𝑥 +no 𝑏) +no 𝑐))
3		oveq1 7417	. . 3 ⊢ (𝑎 = 𝑥 → (𝑎 +no (𝑏 +no 𝑐)) = (𝑥 +no (𝑏 +no 𝑐)))
4	2, 3	eqeq12d 2752	. 2 ⊢ (𝑎 = 𝑥 → (((𝑎 +no 𝑏) +no 𝑐) = (𝑎 +no (𝑏 +no 𝑐)) ↔ ((𝑥 +no 𝑏) +no 𝑐) = (𝑥 +no (𝑏 +no 𝑐))))
5		oveq2 7418	. . . 4 ⊢ (𝑏 = 𝑦 → (𝑥 +no 𝑏) = (𝑥 +no 𝑦))
6	5	oveq1d 7425	. . 3 ⊢ (𝑏 = 𝑦 → ((𝑥 +no 𝑏) +no 𝑐) = ((𝑥 +no 𝑦) +no 𝑐))
7		oveq1 7417	. . . 4 ⊢ (𝑏 = 𝑦 → (𝑏 +no 𝑐) = (𝑦 +no 𝑐))
8	7	oveq2d 7426	. . 3 ⊢ (𝑏 = 𝑦 → (𝑥 +no (𝑏 +no 𝑐)) = (𝑥 +no (𝑦 +no 𝑐)))
9	6, 8	eqeq12d 2752	. 2 ⊢ (𝑏 = 𝑦 → (((𝑥 +no 𝑏) +no 𝑐) = (𝑥 +no (𝑏 +no 𝑐)) ↔ ((𝑥 +no 𝑦) +no 𝑐) = (𝑥 +no (𝑦 +no 𝑐))))
10		oveq2 7418	. . 3 ⊢ (𝑐 = 𝑧 → ((𝑥 +no 𝑦) +no 𝑐) = ((𝑥 +no 𝑦) +no 𝑧))
11		oveq2 7418	. . . 4 ⊢ (𝑐 = 𝑧 → (𝑦 +no 𝑐) = (𝑦 +no 𝑧))
12	11	oveq2d 7426	. . 3 ⊢ (𝑐 = 𝑧 → (𝑥 +no (𝑦 +no 𝑐)) = (𝑥 +no (𝑦 +no 𝑧)))
13	10, 12	eqeq12d 2752	. 2 ⊢ (𝑐 = 𝑧 → (((𝑥 +no 𝑦) +no 𝑐) = (𝑥 +no (𝑦 +no 𝑐)) ↔ ((𝑥 +no 𝑦) +no 𝑧) = (𝑥 +no (𝑦 +no 𝑧))))
14		oveq1 7417	. . . 4 ⊢ (𝑎 = 𝑥 → (𝑎 +no 𝑦) = (𝑥 +no 𝑦))
15	14	oveq1d 7425	. . 3 ⊢ (𝑎 = 𝑥 → ((𝑎 +no 𝑦) +no 𝑧) = ((𝑥 +no 𝑦) +no 𝑧))
16		oveq1 7417	. . 3 ⊢ (𝑎 = 𝑥 → (𝑎 +no (𝑦 +no 𝑧)) = (𝑥 +no (𝑦 +no 𝑧)))
17	15, 16	eqeq12d 2752	. 2 ⊢ (𝑎 = 𝑥 → (((𝑎 +no 𝑦) +no 𝑧) = (𝑎 +no (𝑦 +no 𝑧)) ↔ ((𝑥 +no 𝑦) +no 𝑧) = (𝑥 +no (𝑦 +no 𝑧))))
18		oveq2 7418	. . . 4 ⊢ (𝑏 = 𝑦 → (𝑎 +no 𝑏) = (𝑎 +no 𝑦))
19	18	oveq1d 7425	. . 3 ⊢ (𝑏 = 𝑦 → ((𝑎 +no 𝑏) +no 𝑧) = ((𝑎 +no 𝑦) +no 𝑧))
20		oveq1 7417	. . . 4 ⊢ (𝑏 = 𝑦 → (𝑏 +no 𝑧) = (𝑦 +no 𝑧))
21	20	oveq2d 7426	. . 3 ⊢ (𝑏 = 𝑦 → (𝑎 +no (𝑏 +no 𝑧)) = (𝑎 +no (𝑦 +no 𝑧)))
22	19, 21	eqeq12d 2752	. 2 ⊢ (𝑏 = 𝑦 → (((𝑎 +no 𝑏) +no 𝑧) = (𝑎 +no (𝑏 +no 𝑧)) ↔ ((𝑎 +no 𝑦) +no 𝑧) = (𝑎 +no (𝑦 +no 𝑧))))
23	5	oveq1d 7425	. . 3 ⊢ (𝑏 = 𝑦 → ((𝑥 +no 𝑏) +no 𝑧) = ((𝑥 +no 𝑦) +no 𝑧))
24	20	oveq2d 7426	. . 3 ⊢ (𝑏 = 𝑦 → (𝑥 +no (𝑏 +no 𝑧)) = (𝑥 +no (𝑦 +no 𝑧)))
25	23, 24	eqeq12d 2752	. 2 ⊢ (𝑏 = 𝑦 → (((𝑥 +no 𝑏) +no 𝑧) = (𝑥 +no (𝑏 +no 𝑧)) ↔ ((𝑥 +no 𝑦) +no 𝑧) = (𝑥 +no (𝑦 +no 𝑧))))
26		oveq2 7418	. . 3 ⊢ (𝑐 = 𝑧 → ((𝑎 +no 𝑦) +no 𝑐) = ((𝑎 +no 𝑦) +no 𝑧))
27	11	oveq2d 7426	. . 3 ⊢ (𝑐 = 𝑧 → (𝑎 +no (𝑦 +no 𝑐)) = (𝑎 +no (𝑦 +no 𝑧)))
28	26, 27	eqeq12d 2752	. 2 ⊢ (𝑐 = 𝑧 → (((𝑎 +no 𝑦) +no 𝑐) = (𝑎 +no (𝑦 +no 𝑐)) ↔ ((𝑎 +no 𝑦) +no 𝑧) = (𝑎 +no (𝑦 +no 𝑧))))
29		oveq1 7417	. . . 4 ⊢ (𝑎 = 𝐴 → (𝑎 +no 𝑏) = (𝐴 +no 𝑏))
30	29	oveq1d 7425	. . 3 ⊢ (𝑎 = 𝐴 → ((𝑎 +no 𝑏) +no 𝑐) = ((𝐴 +no 𝑏) +no 𝑐))
31		oveq1 7417	. . 3 ⊢ (𝑎 = 𝐴 → (𝑎 +no (𝑏 +no 𝑐)) = (𝐴 +no (𝑏 +no 𝑐)))
32	30, 31	eqeq12d 2752	. 2 ⊢ (𝑎 = 𝐴 → (((𝑎 +no 𝑏) +no 𝑐) = (𝑎 +no (𝑏 +no 𝑐)) ↔ ((𝐴 +no 𝑏) +no 𝑐) = (𝐴 +no (𝑏 +no 𝑐))))
33		oveq2 7418	. . . 4 ⊢ (𝑏 = 𝐵 → (𝐴 +no 𝑏) = (𝐴 +no 𝐵))
34	33	oveq1d 7425	. . 3 ⊢ (𝑏 = 𝐵 → ((𝐴 +no 𝑏) +no 𝑐) = ((𝐴 +no 𝐵) +no 𝑐))
35		oveq1 7417	. . . 4 ⊢ (𝑏 = 𝐵 → (𝑏 +no 𝑐) = (𝐵 +no 𝑐))
36	35	oveq2d 7426	. . 3 ⊢ (𝑏 = 𝐵 → (𝐴 +no (𝑏 +no 𝑐)) = (𝐴 +no (𝐵 +no 𝑐)))
37	34, 36	eqeq12d 2752	. 2 ⊢ (𝑏 = 𝐵 → (((𝐴 +no 𝑏) +no 𝑐) = (𝐴 +no (𝑏 +no 𝑐)) ↔ ((𝐴 +no 𝐵) +no 𝑐) = (𝐴 +no (𝐵 +no 𝑐))))
38		oveq2 7418	. . 3 ⊢ (𝑐 = 𝐶 → ((𝐴 +no 𝐵) +no 𝑐) = ((𝐴 +no 𝐵) +no 𝐶))
39		oveq2 7418	. . . 4 ⊢ (𝑐 = 𝐶 → (𝐵 +no 𝑐) = (𝐵 +no 𝐶))
40	39	oveq2d 7426	. . 3 ⊢ (𝑐 = 𝐶 → (𝐴 +no (𝐵 +no 𝑐)) = (𝐴 +no (𝐵 +no 𝐶)))
41	38, 40	eqeq12d 2752	. 2 ⊢ (𝑐 = 𝐶 → (((𝐴 +no 𝐵) +no 𝑐) = (𝐴 +no (𝐵 +no 𝑐)) ↔ ((𝐴 +no 𝐵) +no 𝐶) = (𝐴 +no (𝐵 +no 𝐶))))
42		simpr21 1261	. . . . . . . 8 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) ∧ ((∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑦) +no 𝑧) = (𝑥 +no (𝑦 +no 𝑧)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑥 +no 𝑦) +no 𝑐) = (𝑥 +no (𝑦 +no 𝑐)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑏) +no 𝑧) = (𝑥 +no (𝑏 +no 𝑧))) ∧ (∀𝑥 ∈ 𝑎 ((𝑥 +no 𝑏) +no 𝑐) = (𝑥 +no (𝑏 +no 𝑐)) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑦) +no 𝑧) = (𝑎 +no (𝑦 +no 𝑧)) ∧ ∀𝑦 ∈ 𝑏 ((𝑎 +no 𝑦) +no 𝑐) = (𝑎 +no (𝑦 +no 𝑐))) ∧ ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑏) +no 𝑧) = (𝑎 +no (𝑏 +no 𝑧)))) → ∀𝑥 ∈ 𝑎 ((𝑥 +no 𝑏) +no 𝑐) = (𝑥 +no (𝑏 +no 𝑐)))
43		eleq1 2823	. . . . . . . . 9 ⊢ (((𝑥 +no 𝑏) +no 𝑐) = (𝑥 +no (𝑏 +no 𝑐)) → (((𝑥 +no 𝑏) +no 𝑐) ∈ 𝑤 ↔ (𝑥 +no (𝑏 +no 𝑐)) ∈ 𝑤))
44	43	ralimi 3074	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑎 ((𝑥 +no 𝑏) +no 𝑐) = (𝑥 +no (𝑏 +no 𝑐)) → ∀𝑥 ∈ 𝑎 (((𝑥 +no 𝑏) +no 𝑐) ∈ 𝑤 ↔ (𝑥 +no (𝑏 +no 𝑐)) ∈ 𝑤))
45		ralbi 3093	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑎 (((𝑥 +no 𝑏) +no 𝑐) ∈ 𝑤 ↔ (𝑥 +no (𝑏 +no 𝑐)) ∈ 𝑤) → (∀𝑥 ∈ 𝑎 ((𝑥 +no 𝑏) +no 𝑐) ∈ 𝑤 ↔ ∀𝑥 ∈ 𝑎 (𝑥 +no (𝑏 +no 𝑐)) ∈ 𝑤))
46	42, 44, 45	3syl 18	. . . . . . 7 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) ∧ ((∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑦) +no 𝑧) = (𝑥 +no (𝑦 +no 𝑧)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑥 +no 𝑦) +no 𝑐) = (𝑥 +no (𝑦 +no 𝑐)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑏) +no 𝑧) = (𝑥 +no (𝑏 +no 𝑧))) ∧ (∀𝑥 ∈ 𝑎 ((𝑥 +no 𝑏) +no 𝑐) = (𝑥 +no (𝑏 +no 𝑐)) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑦) +no 𝑧) = (𝑎 +no (𝑦 +no 𝑧)) ∧ ∀𝑦 ∈ 𝑏 ((𝑎 +no 𝑦) +no 𝑐) = (𝑎 +no (𝑦 +no 𝑐))) ∧ ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑏) +no 𝑧) = (𝑎 +no (𝑏 +no 𝑧)))) → (∀𝑥 ∈ 𝑎 ((𝑥 +no 𝑏) +no 𝑐) ∈ 𝑤 ↔ ∀𝑥 ∈ 𝑎 (𝑥 +no (𝑏 +no 𝑐)) ∈ 𝑤))
47		simpr23 1263	. . . . . . . 8 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) ∧ ((∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑦) +no 𝑧) = (𝑥 +no (𝑦 +no 𝑧)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑥 +no 𝑦) +no 𝑐) = (𝑥 +no (𝑦 +no 𝑐)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑏) +no 𝑧) = (𝑥 +no (𝑏 +no 𝑧))) ∧ (∀𝑥 ∈ 𝑎 ((𝑥 +no 𝑏) +no 𝑐) = (𝑥 +no (𝑏 +no 𝑐)) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑦) +no 𝑧) = (𝑎 +no (𝑦 +no 𝑧)) ∧ ∀𝑦 ∈ 𝑏 ((𝑎 +no 𝑦) +no 𝑐) = (𝑎 +no (𝑦 +no 𝑐))) ∧ ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑏) +no 𝑧) = (𝑎 +no (𝑏 +no 𝑧)))) → ∀𝑦 ∈ 𝑏 ((𝑎 +no 𝑦) +no 𝑐) = (𝑎 +no (𝑦 +no 𝑐)))
48		eleq1 2823	. . . . . . . . 9 ⊢ (((𝑎 +no 𝑦) +no 𝑐) = (𝑎 +no (𝑦 +no 𝑐)) → (((𝑎 +no 𝑦) +no 𝑐) ∈ 𝑤 ↔ (𝑎 +no (𝑦 +no 𝑐)) ∈ 𝑤))
49	48	ralimi 3074	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑏 ((𝑎 +no 𝑦) +no 𝑐) = (𝑎 +no (𝑦 +no 𝑐)) → ∀𝑦 ∈ 𝑏 (((𝑎 +no 𝑦) +no 𝑐) ∈ 𝑤 ↔ (𝑎 +no (𝑦 +no 𝑐)) ∈ 𝑤))
50		ralbi 3093	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑏 (((𝑎 +no 𝑦) +no 𝑐) ∈ 𝑤 ↔ (𝑎 +no (𝑦 +no 𝑐)) ∈ 𝑤) → (∀𝑦 ∈ 𝑏 ((𝑎 +no 𝑦) +no 𝑐) ∈ 𝑤 ↔ ∀𝑦 ∈ 𝑏 (𝑎 +no (𝑦 +no 𝑐)) ∈ 𝑤))
51	47, 49, 50	3syl 18	. . . . . . 7 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) ∧ ((∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑦) +no 𝑧) = (𝑥 +no (𝑦 +no 𝑧)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑥 +no 𝑦) +no 𝑐) = (𝑥 +no (𝑦 +no 𝑐)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑏) +no 𝑧) = (𝑥 +no (𝑏 +no 𝑧))) ∧ (∀𝑥 ∈ 𝑎 ((𝑥 +no 𝑏) +no 𝑐) = (𝑥 +no (𝑏 +no 𝑐)) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑦) +no 𝑧) = (𝑎 +no (𝑦 +no 𝑧)) ∧ ∀𝑦 ∈ 𝑏 ((𝑎 +no 𝑦) +no 𝑐) = (𝑎 +no (𝑦 +no 𝑐))) ∧ ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑏) +no 𝑧) = (𝑎 +no (𝑏 +no 𝑧)))) → (∀𝑦 ∈ 𝑏 ((𝑎 +no 𝑦) +no 𝑐) ∈ 𝑤 ↔ ∀𝑦 ∈ 𝑏 (𝑎 +no (𝑦 +no 𝑐)) ∈ 𝑤))
52		simpr3 1197	. . . . . . . 8 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) ∧ ((∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑦) +no 𝑧) = (𝑥 +no (𝑦 +no 𝑧)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑥 +no 𝑦) +no 𝑐) = (𝑥 +no (𝑦 +no 𝑐)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑏) +no 𝑧) = (𝑥 +no (𝑏 +no 𝑧))) ∧ (∀𝑥 ∈ 𝑎 ((𝑥 +no 𝑏) +no 𝑐) = (𝑥 +no (𝑏 +no 𝑐)) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑦) +no 𝑧) = (𝑎 +no (𝑦 +no 𝑧)) ∧ ∀𝑦 ∈ 𝑏 ((𝑎 +no 𝑦) +no 𝑐) = (𝑎 +no (𝑦 +no 𝑐))) ∧ ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑏) +no 𝑧) = (𝑎 +no (𝑏 +no 𝑧)))) → ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑏) +no 𝑧) = (𝑎 +no (𝑏 +no 𝑧)))
53		eleq1 2823	. . . . . . . . 9 ⊢ (((𝑎 +no 𝑏) +no 𝑧) = (𝑎 +no (𝑏 +no 𝑧)) → (((𝑎 +no 𝑏) +no 𝑧) ∈ 𝑤 ↔ (𝑎 +no (𝑏 +no 𝑧)) ∈ 𝑤))
54	53	ralimi 3074	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑏) +no 𝑧) = (𝑎 +no (𝑏 +no 𝑧)) → ∀𝑧 ∈ 𝑐 (((𝑎 +no 𝑏) +no 𝑧) ∈ 𝑤 ↔ (𝑎 +no (𝑏 +no 𝑧)) ∈ 𝑤))
55		ralbi 3093	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝑐 (((𝑎 +no 𝑏) +no 𝑧) ∈ 𝑤 ↔ (𝑎 +no (𝑏 +no 𝑧)) ∈ 𝑤) → (∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑏) +no 𝑧) ∈ 𝑤 ↔ ∀𝑧 ∈ 𝑐 (𝑎 +no (𝑏 +no 𝑧)) ∈ 𝑤))
56	52, 54, 55	3syl 18	. . . . . . 7 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) ∧ ((∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑦) +no 𝑧) = (𝑥 +no (𝑦 +no 𝑧)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑥 +no 𝑦) +no 𝑐) = (𝑥 +no (𝑦 +no 𝑐)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑏) +no 𝑧) = (𝑥 +no (𝑏 +no 𝑧))) ∧ (∀𝑥 ∈ 𝑎 ((𝑥 +no 𝑏) +no 𝑐) = (𝑥 +no (𝑏 +no 𝑐)) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑦) +no 𝑧) = (𝑎 +no (𝑦 +no 𝑧)) ∧ ∀𝑦 ∈ 𝑏 ((𝑎 +no 𝑦) +no 𝑐) = (𝑎 +no (𝑦 +no 𝑐))) ∧ ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑏) +no 𝑧) = (𝑎 +no (𝑏 +no 𝑧)))) → (∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑏) +no 𝑧) ∈ 𝑤 ↔ ∀𝑧 ∈ 𝑐 (𝑎 +no (𝑏 +no 𝑧)) ∈ 𝑤))
57	46, 51, 56	3anbi123d 1438	. . . . . 6 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) ∧ ((∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑦) +no 𝑧) = (𝑥 +no (𝑦 +no 𝑧)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑥 +no 𝑦) +no 𝑐) = (𝑥 +no (𝑦 +no 𝑐)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑏) +no 𝑧) = (𝑥 +no (𝑏 +no 𝑧))) ∧ (∀𝑥 ∈ 𝑎 ((𝑥 +no 𝑏) +no 𝑐) = (𝑥 +no (𝑏 +no 𝑐)) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑦) +no 𝑧) = (𝑎 +no (𝑦 +no 𝑧)) ∧ ∀𝑦 ∈ 𝑏 ((𝑎 +no 𝑦) +no 𝑐) = (𝑎 +no (𝑦 +no 𝑐))) ∧ ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑏) +no 𝑧) = (𝑎 +no (𝑏 +no 𝑧)))) → ((∀𝑥 ∈ 𝑎 ((𝑥 +no 𝑏) +no 𝑐) ∈ 𝑤 ∧ ∀𝑦 ∈ 𝑏 ((𝑎 +no 𝑦) +no 𝑐) ∈ 𝑤 ∧ ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑏) +no 𝑧) ∈ 𝑤) ↔ (∀𝑥 ∈ 𝑎 (𝑥 +no (𝑏 +no 𝑐)) ∈ 𝑤 ∧ ∀𝑦 ∈ 𝑏 (𝑎 +no (𝑦 +no 𝑐)) ∈ 𝑤 ∧ ∀𝑧 ∈ 𝑐 (𝑎 +no (𝑏 +no 𝑧)) ∈ 𝑤)))
58	57	rabbidv 3428	. . . . 5 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) ∧ ((∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑦) +no 𝑧) = (𝑥 +no (𝑦 +no 𝑧)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑥 +no 𝑦) +no 𝑐) = (𝑥 +no (𝑦 +no 𝑐)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑏) +no 𝑧) = (𝑥 +no (𝑏 +no 𝑧))) ∧ (∀𝑥 ∈ 𝑎 ((𝑥 +no 𝑏) +no 𝑐) = (𝑥 +no (𝑏 +no 𝑐)) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑦) +no 𝑧) = (𝑎 +no (𝑦 +no 𝑧)) ∧ ∀𝑦 ∈ 𝑏 ((𝑎 +no 𝑦) +no 𝑐) = (𝑎 +no (𝑦 +no 𝑐))) ∧ ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑏) +no 𝑧) = (𝑎 +no (𝑏 +no 𝑧)))) → {𝑤 ∈ On ∣ (∀𝑥 ∈ 𝑎 ((𝑥 +no 𝑏) +no 𝑐) ∈ 𝑤 ∧ ∀𝑦 ∈ 𝑏 ((𝑎 +no 𝑦) +no 𝑐) ∈ 𝑤 ∧ ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑏) +no 𝑧) ∈ 𝑤)} = {𝑤 ∈ On ∣ (∀𝑥 ∈ 𝑎 (𝑥 +no (𝑏 +no 𝑐)) ∈ 𝑤 ∧ ∀𝑦 ∈ 𝑏 (𝑎 +no (𝑦 +no 𝑐)) ∈ 𝑤 ∧ ∀𝑧 ∈ 𝑐 (𝑎 +no (𝑏 +no 𝑧)) ∈ 𝑤)})
59	58	inteqd 4932	. . . 4 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) ∧ ((∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑦) +no 𝑧) = (𝑥 +no (𝑦 +no 𝑧)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑥 +no 𝑦) +no 𝑐) = (𝑥 +no (𝑦 +no 𝑐)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑏) +no 𝑧) = (𝑥 +no (𝑏 +no 𝑧))) ∧ (∀𝑥 ∈ 𝑎 ((𝑥 +no 𝑏) +no 𝑐) = (𝑥 +no (𝑏 +no 𝑐)) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑦) +no 𝑧) = (𝑎 +no (𝑦 +no 𝑧)) ∧ ∀𝑦 ∈ 𝑏 ((𝑎 +no 𝑦) +no 𝑐) = (𝑎 +no (𝑦 +no 𝑐))) ∧ ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑏) +no 𝑧) = (𝑎 +no (𝑏 +no 𝑧)))) → ∩ {𝑤 ∈ On ∣ (∀𝑥 ∈ 𝑎 ((𝑥 +no 𝑏) +no 𝑐) ∈ 𝑤 ∧ ∀𝑦 ∈ 𝑏 ((𝑎 +no 𝑦) +no 𝑐) ∈ 𝑤 ∧ ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑏) +no 𝑧) ∈ 𝑤)} = ∩ {𝑤 ∈ On ∣ (∀𝑥 ∈ 𝑎 (𝑥 +no (𝑏 +no 𝑐)) ∈ 𝑤 ∧ ∀𝑦 ∈ 𝑏 (𝑎 +no (𝑦 +no 𝑐)) ∈ 𝑤 ∧ ∀𝑧 ∈ 𝑐 (𝑎 +no (𝑏 +no 𝑧)) ∈ 𝑤)})
60		naddasslem1 8711	. . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → ((𝑎 +no 𝑏) +no 𝑐) = ∩ {𝑤 ∈ On ∣ (∀𝑥 ∈ 𝑎 ((𝑥 +no 𝑏) +no 𝑐) ∈ 𝑤 ∧ ∀𝑦 ∈ 𝑏 ((𝑎 +no 𝑦) +no 𝑐) ∈ 𝑤 ∧ ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑏) +no 𝑧) ∈ 𝑤)})
61	60	adantr 480	. . . 4 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) ∧ ((∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑦) +no 𝑧) = (𝑥 +no (𝑦 +no 𝑧)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑥 +no 𝑦) +no 𝑐) = (𝑥 +no (𝑦 +no 𝑐)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑏) +no 𝑧) = (𝑥 +no (𝑏 +no 𝑧))) ∧ (∀𝑥 ∈ 𝑎 ((𝑥 +no 𝑏) +no 𝑐) = (𝑥 +no (𝑏 +no 𝑐)) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑦) +no 𝑧) = (𝑎 +no (𝑦 +no 𝑧)) ∧ ∀𝑦 ∈ 𝑏 ((𝑎 +no 𝑦) +no 𝑐) = (𝑎 +no (𝑦 +no 𝑐))) ∧ ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑏) +no 𝑧) = (𝑎 +no (𝑏 +no 𝑧)))) → ((𝑎 +no 𝑏) +no 𝑐) = ∩ {𝑤 ∈ On ∣ (∀𝑥 ∈ 𝑎 ((𝑥 +no 𝑏) +no 𝑐) ∈ 𝑤 ∧ ∀𝑦 ∈ 𝑏 ((𝑎 +no 𝑦) +no 𝑐) ∈ 𝑤 ∧ ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑏) +no 𝑧) ∈ 𝑤)})
62		naddasslem2 8712	. . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (𝑎 +no (𝑏 +no 𝑐)) = ∩ {𝑤 ∈ On ∣ (∀𝑥 ∈ 𝑎 (𝑥 +no (𝑏 +no 𝑐)) ∈ 𝑤 ∧ ∀𝑦 ∈ 𝑏 (𝑎 +no (𝑦 +no 𝑐)) ∈ 𝑤 ∧ ∀𝑧 ∈ 𝑐 (𝑎 +no (𝑏 +no 𝑧)) ∈ 𝑤)})
63	62	adantr 480	. . . 4 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) ∧ ((∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑦) +no 𝑧) = (𝑥 +no (𝑦 +no 𝑧)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑥 +no 𝑦) +no 𝑐) = (𝑥 +no (𝑦 +no 𝑐)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑏) +no 𝑧) = (𝑥 +no (𝑏 +no 𝑧))) ∧ (∀𝑥 ∈ 𝑎 ((𝑥 +no 𝑏) +no 𝑐) = (𝑥 +no (𝑏 +no 𝑐)) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑦) +no 𝑧) = (𝑎 +no (𝑦 +no 𝑧)) ∧ ∀𝑦 ∈ 𝑏 ((𝑎 +no 𝑦) +no 𝑐) = (𝑎 +no (𝑦 +no 𝑐))) ∧ ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑏) +no 𝑧) = (𝑎 +no (𝑏 +no 𝑧)))) → (𝑎 +no (𝑏 +no 𝑐)) = ∩ {𝑤 ∈ On ∣ (∀𝑥 ∈ 𝑎 (𝑥 +no (𝑏 +no 𝑐)) ∈ 𝑤 ∧ ∀𝑦 ∈ 𝑏 (𝑎 +no (𝑦 +no 𝑐)) ∈ 𝑤 ∧ ∀𝑧 ∈ 𝑐 (𝑎 +no (𝑏 +no 𝑧)) ∈ 𝑤)})
64	59, 61, 63	3eqtr4d 2781	. . 3 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) ∧ ((∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑦) +no 𝑧) = (𝑥 +no (𝑦 +no 𝑧)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑥 +no 𝑦) +no 𝑐) = (𝑥 +no (𝑦 +no 𝑐)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑏) +no 𝑧) = (𝑥 +no (𝑏 +no 𝑧))) ∧ (∀𝑥 ∈ 𝑎 ((𝑥 +no 𝑏) +no 𝑐) = (𝑥 +no (𝑏 +no 𝑐)) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑦) +no 𝑧) = (𝑎 +no (𝑦 +no 𝑧)) ∧ ∀𝑦 ∈ 𝑏 ((𝑎 +no 𝑦) +no 𝑐) = (𝑎 +no (𝑦 +no 𝑐))) ∧ ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑏) +no 𝑧) = (𝑎 +no (𝑏 +no 𝑧)))) → ((𝑎 +no 𝑏) +no 𝑐) = (𝑎 +no (𝑏 +no 𝑐)))
65	64	ex 412	. 2 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (((∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑦) +no 𝑧) = (𝑥 +no (𝑦 +no 𝑧)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑥 +no 𝑦) +no 𝑐) = (𝑥 +no (𝑦 +no 𝑐)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑏) +no 𝑧) = (𝑥 +no (𝑏 +no 𝑧))) ∧ (∀𝑥 ∈ 𝑎 ((𝑥 +no 𝑏) +no 𝑐) = (𝑥 +no (𝑏 +no 𝑐)) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑦) +no 𝑧) = (𝑎 +no (𝑦 +no 𝑧)) ∧ ∀𝑦 ∈ 𝑏 ((𝑎 +no 𝑦) +no 𝑐) = (𝑎 +no (𝑦 +no 𝑐))) ∧ ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑏) +no 𝑧) = (𝑎 +no (𝑏 +no 𝑧))) → ((𝑎 +no 𝑏) +no 𝑐) = (𝑎 +no (𝑏 +no 𝑐))))
66	4, 9, 13, 17, 22, 25, 28, 32, 37, 41, 65	on3ind 8687	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 +no 𝐵) +no 𝐶) = (𝐴 +no (𝐵 +no 𝐶)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3052  {crab 3420  ∩ cint 4927  Oncon0 6357  (class class class)co 7410   +no cnadd 8682

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 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-ot 4615  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7993  df-2nd 7994  df-frecs 8285  df-nadd 8683

This theorem is referenced by:  nadd32  8714  nadd4  8715  naddwordnexlem4  43392