Theorem nnaass 6260

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 e. om /\ B e. om /\ C e. om ) -> ( ( 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 5674	. . . . . 6 |- ( x = C -> ( ( A +o B ) +o x ) = ( ( A +o B ) +o C ) )
2		oveq2 5674	. . . . . . 7 |- ( x = C -> ( B +o x ) = ( B +o C ) )
3	2	oveq2d 5682	. . . . . 6 |- ( x = C -> ( A +o ( B +o x ) ) = ( A +o ( B +o C ) ) )
4	1, 3	eqeq12d 2103	. . . . 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 229	. . . 4 |- ( x = C -> ( ( ( A e. om /\ B e. om ) -> ( ( A +o B ) +o x ) = ( A +o ( B +o x ) ) ) <-> ( ( A e. om /\ B e. om ) -> ( ( A +o B ) +o C ) = ( A +o ( B +o C ) ) ) ) )
6		oveq2 5674	. . . . . 6 |- ( x = (/) -> ( ( A +o B ) +o x ) = ( ( A +o B ) +o (/) ) )
7		oveq2 5674	. . . . . . 7 |- ( x = (/) -> ( B +o x ) = ( B +o (/) ) )
8	7	oveq2d 5682	. . . . . 6 |- ( x = (/) -> ( A +o ( B +o x ) ) = ( A +o ( B +o (/) ) ) )
9	6, 8	eqeq12d 2103	. . . . 5 |- ( x = (/) -> ( ( ( A +o B ) +o x ) = ( A +o ( B +o x ) ) <-> ( ( A +o B ) +o (/) ) = ( A +o ( B +o (/) ) ) ) )
10		oveq2 5674	. . . . . 6 |- ( x = y -> ( ( A +o B ) +o x ) = ( ( A +o B ) +o y ) )
11		oveq2 5674	. . . . . . 7 |- ( x = y -> ( B +o x ) = ( B +o y ) )
12	11	oveq2d 5682	. . . . . 6 |- ( x = y -> ( A +o ( B +o x ) ) = ( A +o ( B +o y ) ) )
13	10, 12	eqeq12d 2103	. . . . 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 5674	. . . . . 6 |- ( x = suc y -> ( ( A +o B ) +o x ) = ( ( A +o B ) +o suc y ) )
15		oveq2 5674	. . . . . . 7 |- ( x = suc y -> ( B +o x ) = ( B +o suc y ) )
16	15	oveq2d 5682	. . . . . 6 |- ( x = suc y -> ( A +o ( B +o x ) ) = ( A +o ( B +o suc y ) ) )
17	14, 16	eqeq12d 2103	. . . . 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 6255	. . . . . . 7 |- ( ( A e. om /\ B e. om ) -> ( A +o B ) e. om )
19		nna0 6249	. . . . . . 7 |- ( ( A +o B ) e. om -> ( ( A +o B ) +o (/) ) = ( A +o B ) )
20	18, 19	syl 14	. . . . . 6 |- ( ( A e. om /\ B e. om ) -> ( ( A +o B ) +o (/) ) = ( A +o B ) )
21		nna0 6249	. . . . . . . 8 |- ( B e. om -> ( B +o (/) ) = B )
22	21	oveq2d 5682	. . . . . . 7 |- ( B e. om -> ( A +o ( B +o (/) ) ) = ( A +o B ) )
23	22	adantl 272	. . . . . 6 |- ( ( A e. om /\ B e. om ) -> ( A +o ( B +o (/) ) ) = ( A +o B ) )
24	20, 23	eqtr4d 2124	. . . . 5 |- ( ( A e. om /\ B e. om ) -> ( ( A +o B ) +o (/) ) = ( A +o ( B +o (/) ) ) )
25		suceq 4238	. . . . . . 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 6251	. . . . . . . . 9 |- ( ( ( A +o B ) e. om /\ y e. om ) -> ( ( A +o B ) +o suc y ) = suc ( ( A +o B ) +o y ) )
27	18, 26	sylan 278	. . . . . . . 8 |- ( ( ( A e. om /\ B e. om ) /\ y e. om ) -> ( ( A +o B ) +o suc y ) = suc ( ( A +o B ) +o y ) )
28		nnasuc 6251	. . . . . . . . . . . 12 |- ( ( B e. om /\ y e. om ) -> ( B +o suc y ) = suc ( B +o y ) )
29	28	oveq2d 5682	. . . . . . . . . . 11 |- ( ( B e. om /\ y e. om ) -> ( A +o ( B +o suc y ) ) = ( A +o suc ( B +o y ) ) )
30	29	adantl 272	. . . . . . . . . 10 |- ( ( A e. om /\ ( B e. om /\ y e. om ) ) -> ( A +o ( B +o suc y ) ) = ( A +o suc ( B +o y ) ) )
31		nnacl 6255	. . . . . . . . . . 11 |- ( ( B e. om /\ y e. om ) -> ( B +o y ) e. om )
32		nnasuc 6251	. . . . . . . . . . 11 |- ( ( A e. om /\ ( B +o y ) e. om ) -> ( A +o suc ( B +o y ) ) = suc ( A +o ( B +o y ) ) )
33	31, 32	sylan2 281	. . . . . . . . . 10 |- ( ( A e. om /\ ( B e. om /\ y e. om ) ) -> ( A +o suc ( B +o y ) ) = suc ( A +o ( B +o y ) ) )
34	30, 33	eqtrd 2121	. . . . . . . . 9 |- ( ( A e. om /\ ( B e. om /\ y e. om ) ) -> ( A +o ( B +o suc y ) ) = suc ( A +o ( B +o y ) ) )
35	34	anassrs 393	. . . . . . . 8 |- ( ( ( A e. om /\ B e. om ) /\ y e. om ) -> ( A +o ( B +o suc y ) ) = suc ( A +o ( B +o y ) ) )
36	27, 35	eqeq12d 2103	. . . . . . 7 |- ( ( ( A e. om /\ B e. om ) /\ y e. om ) -> ( ( ( 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	syl5ibr 155	. . . . . 6 |- ( ( ( A e. om /\ B e. om ) /\ y e. om ) -> ( ( ( 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 115	. . . . 5 |- ( y e. om -> ( ( A e. om /\ B e. om ) -> ( ( ( 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 4429	. . . 4 |- ( x e. om -> ( ( A e. om /\ B e. om ) -> ( ( A +o B ) +o x ) = ( A +o ( B +o x ) ) ) )
40	5, 39	vtoclga 2686	. . 3 |- ( C e. om -> ( ( A e. om /\ B e. om ) -> ( ( A +o B ) +o C ) = ( A +o ( B +o C ) ) ) )
41	40	com12 30	. 2 |- ( ( A e. om /\ B e. om ) -> ( C e. om -> ( ( A +o B ) +o C ) = ( A +o ( B +o C ) ) ) )
42	41	3impia 1141	1 |- ( ( A e. om /\ B e. om /\ C e. om ) -> ( ( A +o B ) +o C ) = ( A +o ( B +o C ) ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 925   = wceq 1290   e. wcel 1439   (/)c0 3287   suc csuc 4201   omcom 4418  (class class class)co 5666   +o coa 6192

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-id 4129  df-iord 4202  df-on 4204  df-suc 4207  df-iom 4419  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-irdg 6149  df-oadd 6199

This theorem is referenced by:  nndi  6261  nnmsucr  6263  addasspig  6950  addassnq0  7082  prarloclemlo  7114