Theorem nnaass 8543

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 7360	. . . . . 6 ⊢ (𝑥 = 𝐶 → ((𝐴 +o 𝐵) +o 𝑥) = ((𝐴 +o 𝐵) +o 𝐶))
2		oveq2 7360	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝐵 +o 𝑥) = (𝐵 +o 𝐶))
3	2	oveq2d 7368	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐴 +o (𝐵 +o 𝑥)) = (𝐴 +o (𝐵 +o 𝐶)))
4	1, 3	eqeq12d 2747	. . . . 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 340	. . . 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 7360	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝐴 +o 𝐵) +o 𝑥) = ((𝐴 +o 𝐵) +o ∅))
7		oveq2 7360	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐵 +o 𝑥) = (𝐵 +o ∅))
8	7	oveq2d 7368	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐴 +o (𝐵 +o 𝑥)) = (𝐴 +o (𝐵 +o ∅)))
9	6, 8	eqeq12d 2747	. . . . 5 ⊢ (𝑥 = ∅ → (((𝐴 +o 𝐵) +o 𝑥) = (𝐴 +o (𝐵 +o 𝑥)) ↔ ((𝐴 +o 𝐵) +o ∅) = (𝐴 +o (𝐵 +o ∅))))
10		oveq2 7360	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 +o 𝐵) +o 𝑥) = ((𝐴 +o 𝐵) +o 𝑦))
11		oveq2 7360	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o 𝑦))
12	11	oveq2d 7368	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 +o (𝐵 +o 𝑥)) = (𝐴 +o (𝐵 +o 𝑦)))
13	10, 12	eqeq12d 2747	. . . . 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 7360	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → ((𝐴 +o 𝐵) +o 𝑥) = ((𝐴 +o 𝐵) +o suc 𝑦))
15		oveq2 7360	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o suc 𝑦))
16	15	oveq2d 7368	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝐴 +o (𝐵 +o 𝑥)) = (𝐴 +o (𝐵 +o suc 𝑦)))
17	14, 16	eqeq12d 2747	. . . . 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 8532	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +o 𝐵) ∈ ω)
19		nna0 8525	. . . . . . 7 ⊢ ((𝐴 +o 𝐵) ∈ ω → ((𝐴 +o 𝐵) +o ∅) = (𝐴 +o 𝐵))
20	18, 19	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 +o 𝐵) +o ∅) = (𝐴 +o 𝐵))
21		nna0 8525	. . . . . . . 8 ⊢ (𝐵 ∈ ω → (𝐵 +o ∅) = 𝐵)
22	21	oveq2d 7368	. . . . . . 7 ⊢ (𝐵 ∈ ω → (𝐴 +o (𝐵 +o ∅)) = (𝐴 +o 𝐵))
23	22	adantl 481	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +o (𝐵 +o ∅)) = (𝐴 +o 𝐵))
24	20, 23	eqtr4d 2769	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 +o 𝐵) +o ∅) = (𝐴 +o (𝐵 +o ∅)))
25		suceq 6380	. . . . . . 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 8527	. . . . . . . . 9 ⊢ (((𝐴 +o 𝐵) ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 +o 𝐵) +o suc 𝑦) = suc ((𝐴 +o 𝐵) +o 𝑦))
27	18, 26	sylan 580	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑦 ∈ ω) → ((𝐴 +o 𝐵) +o suc 𝑦) = suc ((𝐴 +o 𝐵) +o 𝑦))
28		nnasuc 8527	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
29	28	oveq2d 7368	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o (𝐵 +o suc 𝑦)) = (𝐴 +o suc (𝐵 +o 𝑦)))
30	29	adantl 481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝑦 ∈ ω)) → (𝐴 +o (𝐵 +o suc 𝑦)) = (𝐴 +o suc (𝐵 +o 𝑦)))
31		nnacl 8532	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o 𝑦) ∈ ω)
32		nnasuc 8527	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ (𝐵 +o 𝑦) ∈ ω) → (𝐴 +o suc (𝐵 +o 𝑦)) = suc (𝐴 +o (𝐵 +o 𝑦)))
33	31, 32	sylan2 593	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝑦 ∈ ω)) → (𝐴 +o suc (𝐵 +o 𝑦)) = suc (𝐴 +o (𝐵 +o 𝑦)))
34	30, 33	eqtrd 2766	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝑦 ∈ ω)) → (𝐴 +o (𝐵 +o suc 𝑦)) = suc (𝐴 +o (𝐵 +o 𝑦)))
35	34	anassrs 467	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑦 ∈ ω) → (𝐴 +o (𝐵 +o suc 𝑦)) = suc (𝐴 +o (𝐵 +o 𝑦)))
36	27, 35	eqeq12d 2747	. . . . . . 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 246	. . . . . 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 413	. . . . 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 7834	. . . 4 ⊢ (𝑥 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 +o 𝐵) +o 𝑥) = (𝐴 +o (𝐵 +o 𝑥))))
40	5, 39	vtoclga 3528	. . 3 ⊢ (𝐶 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 +o 𝐵) +o 𝐶) = (𝐴 +o (𝐵 +o 𝐶))))
41	40	com12 32	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐶 ∈ ω → ((𝐴 +o 𝐵) +o 𝐶) = (𝐴 +o (𝐵 +o 𝐶))))
42	41	3impia 1117	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 +o 𝐵) +o 𝐶) = (𝐴 +o (𝐵 +o 𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∅c0 4282  suc csuc 6314  (class class class)co 7352  ωcom 7802   +_o coa 8388

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5236  ax-nul 5246  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-oadd 8395

This theorem is referenced by:  nndi  8544  nnmsucr  8546  omopthlem1  8580  omopthlem2  8581  addasspi  10792  naddcnfass  43467