Theorem nnmass 8613

Description: Multiplication of natural numbers is associative. Theorem 4K(4) of [Enderton] p. 81. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)

Assertion

Ref	Expression
nnmass	⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 ·o 𝐵) ·o 𝐶) = (𝐴 ·o (𝐵 ·o 𝐶)))

Proof of Theorem nnmass

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7422	. . . . . 6 ⊢ (𝑥 = 𝐶 → ((𝐴 ·o 𝐵) ·o 𝑥) = ((𝐴 ·o 𝐵) ·o 𝐶))
2		oveq2 7422	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝐵 ·o 𝑥) = (𝐵 ·o 𝐶))
3	2	oveq2d 7430	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐴 ·o (𝐵 ·o 𝑥)) = (𝐴 ·o (𝐵 ·o 𝐶)))
4	1, 3	eqeq12d 2776	. . . . 5 ⊢ (𝑥 = 𝐶 → (((𝐴 ·o 𝐵) ·o 𝑥) = (𝐴 ·o (𝐵 ·o 𝑥)) ↔ ((𝐴 ·o 𝐵) ·o 𝐶) = (𝐴 ·o (𝐵 ·o 𝐶))))
5	4	imbi2d 343	. . . 4 ⊢ (𝑥 = 𝐶 → (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ·o 𝐵) ·o 𝑥) = (𝐴 ·o (𝐵 ·o 𝑥))) ↔ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ·o 𝐵) ·o 𝐶) = (𝐴 ·o (𝐵 ·o 𝐶)))))
6		oveq2 7422	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝐴 ·o 𝐵) ·o 𝑥) = ((𝐴 ·o 𝐵) ·o ∅))
7		oveq2 7422	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐵 ·o 𝑥) = (𝐵 ·o ∅))
8	7	oveq2d 7430	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐴 ·o (𝐵 ·o 𝑥)) = (𝐴 ·o (𝐵 ·o ∅)))
9	6, 8	eqeq12d 2776	. . . . 5 ⊢ (𝑥 = ∅ → (((𝐴 ·o 𝐵) ·o 𝑥) = (𝐴 ·o (𝐵 ·o 𝑥)) ↔ ((𝐴 ·o 𝐵) ·o ∅) = (𝐴 ·o (𝐵 ·o ∅))))
10		oveq2 7422	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 ·o 𝐵) ·o 𝑥) = ((𝐴 ·o 𝐵) ·o 𝑦))
11		oveq2 7422	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐵 ·o 𝑥) = (𝐵 ·o 𝑦))
12	11	oveq2d 7430	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 ·o (𝐵 ·o 𝑥)) = (𝐴 ·o (𝐵 ·o 𝑦)))
13	10, 12	eqeq12d 2776	. . . . 5 ⊢ (𝑥 = 𝑦 → (((𝐴 ·o 𝐵) ·o 𝑥) = (𝐴 ·o (𝐵 ·o 𝑥)) ↔ ((𝐴 ·o 𝐵) ·o 𝑦) = (𝐴 ·o (𝐵 ·o 𝑦))))
14		oveq2 7422	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ·o 𝐵) ·o 𝑥) = ((𝐴 ·o 𝐵) ·o suc 𝑦))
15		oveq2 7422	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝐵 ·o 𝑥) = (𝐵 ·o suc 𝑦))
16	15	oveq2d 7430	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝐴 ·o (𝐵 ·o 𝑥)) = (𝐴 ·o (𝐵 ·o suc 𝑦)))
17	14, 16	eqeq12d 2776	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (((𝐴 ·o 𝐵) ·o 𝑥) = (𝐴 ·o (𝐵 ·o 𝑥)) ↔ ((𝐴 ·o 𝐵) ·o suc 𝑦) = (𝐴 ·o (𝐵 ·o suc 𝑦))))
18		nnmcl 8601	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o 𝐵) ∈ ω)
19		nnm0 8594	. . . . . . 7 ⊢ ((𝐴 ·o 𝐵) ∈ ω → ((𝐴 ·o 𝐵) ·o ∅) = ∅)
20	18, 19	syl 18	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ·o 𝐵) ·o ∅) = ∅)
21		nnm0 8594	. . . . . . . 8 ⊢ (𝐵 ∈ ω → (𝐵 ·o ∅) = ∅)
22	21	oveq2d 7430	. . . . . . 7 ⊢ (𝐵 ∈ ω → (𝐴 ·o (𝐵 ·o ∅)) = (𝐴 ·o ∅))
23		nnm0 8594	. . . . . . 7 ⊢ (𝐴 ∈ ω → (𝐴 ·o ∅) = ∅)
24	22, 23	sylan9eqr 2817	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o (𝐵 ·o ∅)) = ∅)
25	20, 24	eqtr4d 2798	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ·o 𝐵) ·o ∅) = (𝐴 ·o (𝐵 ·o ∅)))
26		oveq1 7421	. . . . . . . . 9 ⊢ (((𝐴 ·o 𝐵) ·o 𝑦) = (𝐴 ·o (𝐵 ·o 𝑦)) → (((𝐴 ·o 𝐵) ·o 𝑦) +o (𝐴 ·o 𝐵)) = ((𝐴 ·o (𝐵 ·o 𝑦)) +o (𝐴 ·o 𝐵)))
27		nnmsuc 8596	. . . . . . . . . . 11 ⊢ (((𝐴 ·o 𝐵) ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·o 𝐵) ·o suc 𝑦) = (((𝐴 ·o 𝐵) ·o 𝑦) +o (𝐴 ·o 𝐵)))
28	18, 27	stoic3 1809	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·o 𝐵) ·o suc 𝑦) = (((𝐴 ·o 𝐵) ·o 𝑦) +o (𝐴 ·o 𝐵)))
29		nnmsuc 8596	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 ·o suc 𝑦) = ((𝐵 ·o 𝑦) +o 𝐵))
30	29	3adant1 1148	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 ·o suc 𝑦) = ((𝐵 ·o 𝑦) +o 𝐵))
31	30	oveq2d 7430	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·o (𝐵 ·o suc 𝑦)) = (𝐴 ·o ((𝐵 ·o 𝑦) +o 𝐵)))
32		nnmcl 8601	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 ·o 𝑦) ∈ ω)
33		nndi 8612	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ω ∧ (𝐵 ·o 𝑦) ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o ((𝐵 ·o 𝑦) +o 𝐵)) = ((𝐴 ·o (𝐵 ·o 𝑦)) +o (𝐴 ·o 𝐵)))
34	32, 33	syl3an2 1182	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝑦 ∈ ω) ∧ 𝐵 ∈ ω) → (𝐴 ·o ((𝐵 ·o 𝑦) +o 𝐵)) = ((𝐴 ·o (𝐵 ·o 𝑦)) +o (𝐴 ·o 𝐵)))
35	34	3exp 1137	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ω → ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 ∈ ω → (𝐴 ·o ((𝐵 ·o 𝑦) +o 𝐵)) = ((𝐴 ·o (𝐵 ·o 𝑦)) +o (𝐴 ·o 𝐵)))))
36	35	expd 421	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ω → (𝐵 ∈ ω → (𝑦 ∈ ω → (𝐵 ∈ ω → (𝐴 ·o ((𝐵 ·o 𝑦) +o 𝐵)) = ((𝐴 ·o (𝐵 ·o 𝑦)) +o (𝐴 ·o 𝐵))))))
37	36	com34 92	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ω → (𝐵 ∈ ω → (𝐵 ∈ ω → (𝑦 ∈ ω → (𝐴 ·o ((𝐵 ·o 𝑦) +o 𝐵)) = ((𝐴 ·o (𝐵 ·o 𝑦)) +o (𝐴 ·o 𝐵))))))
38	37	pm2.43d 54	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ω → (𝐵 ∈ ω → (𝑦 ∈ ω → (𝐴 ·o ((𝐵 ·o 𝑦) +o 𝐵)) = ((𝐴 ·o (𝐵 ·o 𝑦)) +o (𝐴 ·o 𝐵)))))
39	38	3imp 1128	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·o ((𝐵 ·o 𝑦) +o 𝐵)) = ((𝐴 ·o (𝐵 ·o 𝑦)) +o (𝐴 ·o 𝐵)))
40	31, 39	eqtrd 2795	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·o (𝐵 ·o suc 𝑦)) = ((𝐴 ·o (𝐵 ·o 𝑦)) +o (𝐴 ·o 𝐵)))
41	28, 40	eqeq12d 2776	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (((𝐴 ·o 𝐵) ·o suc 𝑦) = (𝐴 ·o (𝐵 ·o suc 𝑦)) ↔ (((𝐴 ·o 𝐵) ·o 𝑦) +o (𝐴 ·o 𝐵)) = ((𝐴 ·o (𝐵 ·o 𝑦)) +o (𝐴 ·o 𝐵))))
42	26, 41	imbitrrid 249	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (((𝐴 ·o 𝐵) ·o 𝑦) = (𝐴 ·o (𝐵 ·o 𝑦)) → ((𝐴 ·o 𝐵) ·o suc 𝑦) = (𝐴 ·o (𝐵 ·o suc 𝑦))))
43	42	3exp 1137	. . . . . . 7 ⊢ (𝐴 ∈ ω → (𝐵 ∈ ω → (𝑦 ∈ ω → (((𝐴 ·o 𝐵) ·o 𝑦) = (𝐴 ·o (𝐵 ·o 𝑦)) → ((𝐴 ·o 𝐵) ·o suc 𝑦) = (𝐴 ·o (𝐵 ·o suc 𝑦))))))
44	43	com3r 88	. . . . . 6 ⊢ (𝑦 ∈ ω → (𝐴 ∈ ω → (𝐵 ∈ ω → (((𝐴 ·o 𝐵) ·o 𝑦) = (𝐴 ·o (𝐵 ·o 𝑦)) → ((𝐴 ·o 𝐵) ·o suc 𝑦) = (𝐴 ·o (𝐵 ·o suc 𝑦))))))
45	44	impd 416	. . . . 5 ⊢ (𝑦 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (((𝐴 ·o 𝐵) ·o 𝑦) = (𝐴 ·o (𝐵 ·o 𝑦)) → ((𝐴 ·o 𝐵) ·o suc 𝑦) = (𝐴 ·o (𝐵 ·o suc 𝑦)))))
46	9, 13, 17, 25, 45	finds2 7896	. . . 4 ⊢ (𝑥 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ·o 𝐵) ·o 𝑥) = (𝐴 ·o (𝐵 ·o 𝑥))))
47	5, 46	vtoclga 3536	. . 3 ⊢ (𝐶 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ·o 𝐵) ·o 𝐶) = (𝐴 ·o (𝐵 ·o 𝐶))))
48	47	expdcom 420	. 2 ⊢ (𝐴 ∈ ω → (𝐵 ∈ ω → (𝐶 ∈ ω → ((𝐴 ·o 𝐵) ·o 𝐶) = (𝐴 ·o (𝐵 ·o 𝐶)))))
49	48	3imp 1128	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 ·o 𝐵) ·o 𝐶) = (𝐴 ·o (𝐵 ·o 𝐶)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∅c0 4279  suc csuc 6359  (class class class)co 7414  ωcom 7863   +_o coa 8453   ·_o comu 8454

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7737

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 2100  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-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7864  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-oadd 8460  df-omul 8461

This theorem is used by:  mulasspi  10907