Theorem nnaass 6629

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 6008	. . . . . 6 |- ( x = C -> ( ( A +o B ) +o x ) = ( ( A +o B ) +o C ) )
2		oveq2 6008	. . . . . . 7 |- ( x = C -> ( B +o x ) = ( B +o C ) )
3	2	oveq2d 6016	. . . . . 6 |- ( x = C -> ( A +o ( B +o x ) ) = ( A +o ( B +o C ) ) )
4	1, 3	eqeq12d 2244	. . . . 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 230	. . . 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 6008	. . . . . 6 |- ( x = (/) -> ( ( A +o B ) +o x ) = ( ( A +o B ) +o (/) ) )
7		oveq2 6008	. . . . . . 7 |- ( x = (/) -> ( B +o x ) = ( B +o (/) ) )
8	7	oveq2d 6016	. . . . . 6 |- ( x = (/) -> ( A +o ( B +o x ) ) = ( A +o ( B +o (/) ) ) )
9	6, 8	eqeq12d 2244	. . . . 5 |- ( x = (/) -> ( ( ( A +o B ) +o x ) = ( A +o ( B +o x ) ) <-> ( ( A +o B ) +o (/) ) = ( A +o ( B +o (/) ) ) ) )
10		oveq2 6008	. . . . . 6 |- ( x = y -> ( ( A +o B ) +o x ) = ( ( A +o B ) +o y ) )
11		oveq2 6008	. . . . . . 7 |- ( x = y -> ( B +o x ) = ( B +o y ) )
12	11	oveq2d 6016	. . . . . 6 |- ( x = y -> ( A +o ( B +o x ) ) = ( A +o ( B +o y ) ) )
13	10, 12	eqeq12d 2244	. . . . 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 6008	. . . . . 6 |- ( x = suc y -> ( ( A +o B ) +o x ) = ( ( A +o B ) +o suc y ) )
15		oveq2 6008	. . . . . . 7 |- ( x = suc y -> ( B +o x ) = ( B +o suc y ) )
16	15	oveq2d 6016	. . . . . 6 |- ( x = suc y -> ( A +o ( B +o x ) ) = ( A +o ( B +o suc y ) ) )
17	14, 16	eqeq12d 2244	. . . . 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 6624	. . . . . . 7 |- ( ( A e. om /\ B e. om ) -> ( A +o B ) e. om )
19		nna0 6618	. . . . . . 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 6618	. . . . . . . 8 |- ( B e. om -> ( B +o (/) ) = B )
22	21	oveq2d 6016	. . . . . . 7 |- ( B e. om -> ( A +o ( B +o (/) ) ) = ( A +o B ) )
23	22	adantl 277	. . . . . 6 |- ( ( A e. om /\ B e. om ) -> ( A +o ( B +o (/) ) ) = ( A +o B ) )
24	20, 23	eqtr4d 2265	. . . . 5 |- ( ( A e. om /\ B e. om ) -> ( ( A +o B ) +o (/) ) = ( A +o ( B +o (/) ) ) )
25		suceq 4492	. . . . . . 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 6620	. . . . . . . . 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 283	. . . . . . . 8 |- ( ( ( A e. om /\ B e. om ) /\ y e. om ) -> ( ( A +o B ) +o suc y ) = suc ( ( A +o B ) +o y ) )
28		nnasuc 6620	. . . . . . . . . . . 12 |- ( ( B e. om /\ y e. om ) -> ( B +o suc y ) = suc ( B +o y ) )
29	28	oveq2d 6016	. . . . . . . . . . 11 |- ( ( B e. om /\ y e. om ) -> ( A +o ( B +o suc y ) ) = ( A +o suc ( B +o y ) ) )
30	29	adantl 277	. . . . . . . . . 10 |- ( ( A e. om /\ ( B e. om /\ y e. om ) ) -> ( A +o ( B +o suc y ) ) = ( A +o suc ( B +o y ) ) )
31		nnacl 6624	. . . . . . . . . . 11 |- ( ( B e. om /\ y e. om ) -> ( B +o y ) e. om )
32		nnasuc 6620	. . . . . . . . . . 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 286	. . . . . . . . . 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 2262	. . . . . . . . 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 400	. . . . . . . 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 2244	. . . . . . 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	imbitrrid 156	. . . . . 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 116	. . . . 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 4692	. . . 4 |- ( x e. om -> ( ( A e. om /\ B e. om ) -> ( ( A +o B ) +o x ) = ( A +o ( B +o x ) ) ) )
40	5, 39	vtoclga 2867	. . 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 1224	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 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   (/)c0 3491   suc csuc 4455   omcom 4681  (class class class)co 6000   +o coa 6557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-irdg 6514  df-oadd 6564

This theorem is referenced by:  nndi  6630  nnmsucr  6632  addasspig  7513  addassnq0  7645  prarloclemlo  7677