Theorem nnaass 8607

Description: Addition of natural numbers is associative. Theorem 4K(1) of [Enderton] p. 81. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)

Assertion

Ref	Expression
nnaass	⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 +o 𝐵) +o 𝐶) = (𝐴 +o (𝐵 +o 𝐶)))

Proof of Theorem nnaass

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

Step	Hyp	Ref	Expression
1		oveq2 7418	. . . . . 6 ⊢ (𝑥 = 𝐶 → ((𝐴 +o 𝐵) +o 𝑥) = ((𝐴 +o 𝐵) +o 𝐶))
2		oveq2 7418	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝐵 +o 𝑥) = (𝐵 +o 𝐶))
3	2	oveq2d 7426	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐴 +o (𝐵 +o 𝑥)) = (𝐴 +o (𝐵 +o 𝐶)))
4	1, 3	eqeq12d 2777	. . . . 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 7418	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝐴 +o 𝐵) +o 𝑥) = ((𝐴 +o 𝐵) +o ∅))
7		oveq2 7418	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐵 +o 𝑥) = (𝐵 +o ∅))
8	7	oveq2d 7426	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐴 +o (𝐵 +o 𝑥)) = (𝐴 +o (𝐵 +o ∅)))
9	6, 8	eqeq12d 2777	. . . . 5 ⊢ (𝑥 = ∅ → (((𝐴 +o 𝐵) +o 𝑥) = (𝐴 +o (𝐵 +o 𝑥)) ↔ ((𝐴 +o 𝐵) +o ∅) = (𝐴 +o (𝐵 +o ∅))))
10		oveq2 7418	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 +o 𝐵) +o 𝑥) = ((𝐴 +o 𝐵) +o 𝑦))
11		oveq2 7418	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o 𝑦))
12	11	oveq2d 7426	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 +o (𝐵 +o 𝑥)) = (𝐴 +o (𝐵 +o 𝑦)))
13	10, 12	eqeq12d 2777	. . . . 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 7418	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → ((𝐴 +o 𝐵) +o 𝑥) = ((𝐴 +o 𝐵) +o suc 𝑦))
15		oveq2 7418	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o suc 𝑦))
16	15	oveq2d 7426	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝐴 +o (𝐵 +o 𝑥)) = (𝐴 +o (𝐵 +o suc 𝑦)))
17	14, 16	eqeq12d 2777	. . . . 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		nnacl 8596	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +o 𝐵) ∈ ω)
19		nna0 8589	. . . . . . 7 ⊢ ((𝐴 +o 𝐵) ∈ ω → ((𝐴 +o 𝐵) +o ∅) = (𝐴 +o 𝐵))
20	18, 19	syl 18	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 +o 𝐵) +o ∅) = (𝐴 +o 𝐵))
21		nna0 8589	. . . . . . . 8 ⊢ (𝐵 ∈ ω → (𝐵 +o ∅) = 𝐵)
22	21	oveq2d 7426	. . . . . . 7 ⊢ (𝐵 ∈ ω → (𝐴 +o (𝐵 +o ∅)) = (𝐴 +o 𝐵))
23	22	adantl 486	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +o (𝐵 +o ∅)) = (𝐴 +o 𝐵))
24	20, 23	eqtr4d 2799	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 +o 𝐵) +o ∅) = (𝐴 +o (𝐵 +o ∅)))
25		suceq 6429	. . . . . . 7 ⊢ (((𝐴 +o 𝐵) +o 𝑦) = (𝐴 +o (𝐵 +o 𝑦)) → suc ((𝐴 +o 𝐵) +o 𝑦) = suc (𝐴 +o (𝐵 +o 𝑦)))
26		nnasuc 8591	. . . . . . . . 9 ⊢ (((𝐴 +o 𝐵) ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 +o 𝐵) +o suc 𝑦) = suc ((𝐴 +o 𝐵) +o 𝑦))
27	18, 26	sylan 591	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑦 ∈ ω) → ((𝐴 +o 𝐵) +o suc 𝑦) = suc ((𝐴 +o 𝐵) +o 𝑦))
28		nnasuc 8591	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
29	28	oveq2d 7426	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o (𝐵 +o suc 𝑦)) = (𝐴 +o suc (𝐵 +o 𝑦)))
30	29	adantl 486	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝑦 ∈ ω)) → (𝐴 +o (𝐵 +o suc 𝑦)) = (𝐴 +o suc (𝐵 +o 𝑦)))
31		nnacl 8596	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o 𝑦) ∈ ω)
32		nnasuc 8591	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ (𝐵 +o 𝑦) ∈ ω) → (𝐴 +o suc (𝐵 +o 𝑦)) = suc (𝐴 +o (𝐵 +o 𝑦)))
33	31, 32	sylan2 604	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝑦 ∈ ω)) → (𝐴 +o suc (𝐵 +o 𝑦)) = suc (𝐴 +o (𝐵 +o 𝑦)))
34	30, 33	eqtrd 2796	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝑦 ∈ ω)) → (𝐴 +o (𝐵 +o suc 𝑦)) = suc (𝐴 +o (𝐵 +o 𝑦)))
35	34	anassrs 472	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑦 ∈ ω) → (𝐴 +o (𝐵 +o suc 𝑦)) = suc (𝐴 +o (𝐵 +o 𝑦)))
36	27, 35	eqeq12d 2777	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑦 ∈ ω) → (((𝐴 +o 𝐵) +o suc 𝑦) = (𝐴 +o (𝐵 +o suc 𝑦)) ↔ suc ((𝐴 +o 𝐵) +o 𝑦) = suc (𝐴 +o (𝐵 +o 𝑦))))
37	25, 36	imbitrrid 249	. . . . . 6 ⊢ (((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))))
38	37	expcom 418	. . . . 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)))))
39	9, 13, 17, 24, 38	finds2 7894	. . . 4 ⊢ (𝑥 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 +o 𝐵) +o 𝑥) = (𝐴 +o (𝐵 +o 𝑥))))
40	5, 39	vtoclga 3540	. . 3 ⊢ (𝐶 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 +o 𝐵) +o 𝐶) = (𝐴 +o (𝐵 +o 𝐶))))
41	40	com12 33	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐶 ∈ ω → ((𝐴 +o 𝐵) +o 𝐶) = (𝐴 +o (𝐵 +o 𝐶))))
42	41	3impia 1133	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 +o 𝐵) +o 𝐶) = (𝐴 +o (𝐵 +o 𝐶)))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1568   ∈ wcel 2141  ∅c0 4285  suc csuc 6362  (class class class)co 7410  ωcom 7861   +_o coa 8449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  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-om 7862  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-oadd 8456

This theorem is referenced by:  nndi  8608  nnmsucr  8610  omopthlem1  8644  omopthlem2  8645  addasspi  10879  naddcnfass  44044