Theorem nnaass 6064

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 5520	. . . . . 6 |- ( x = C -> ( ( A +o B ) +o x ) = ( ( A +o B ) +o C ) )
2		oveq2 5520	. . . . . . 7 |- ( x = C -> ( B +o x ) = ( B +o C ) )
3	2	oveq2d 5528	. . . . . 6 |- ( x = C -> ( A +o ( B +o x ) ) = ( A +o ( B +o C ) ) )
4	1, 3	eqeq12d 2054	. . . . 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 219	. . . 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 5520	. . . . . 6 |- ( x = (/) -> ( ( A +o B ) +o x ) = ( ( A +o B ) +o (/) ) )
7		oveq2 5520	. . . . . . 7 |- ( x = (/) -> ( B +o x ) = ( B +o (/) ) )
8	7	oveq2d 5528	. . . . . 6 |- ( x = (/) -> ( A +o ( B +o x ) ) = ( A +o ( B +o (/) ) ) )
9	6, 8	eqeq12d 2054	. . . . 5 |- ( x = (/) -> ( ( ( A +o B ) +o x ) = ( A +o ( B +o x ) ) <-> ( ( A +o B ) +o (/) ) = ( A +o ( B +o (/) ) ) ) )
10		oveq2 5520	. . . . . 6 |- ( x = y -> ( ( A +o B ) +o x ) = ( ( A +o B ) +o y ) )
11		oveq2 5520	. . . . . . 7 |- ( x = y -> ( B +o x ) = ( B +o y ) )
12	11	oveq2d 5528	. . . . . 6 |- ( x = y -> ( A +o ( B +o x ) ) = ( A +o ( B +o y ) ) )
13	10, 12	eqeq12d 2054	. . . . 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 5520	. . . . . 6 |- ( x = suc y -> ( ( A +o B ) +o x ) = ( ( A +o B ) +o suc y ) )
15		oveq2 5520	. . . . . . 7 |- ( x = suc y -> ( B +o x ) = ( B +o suc y ) )
16	15	oveq2d 5528	. . . . . 6 |- ( x = suc y -> ( A +o ( B +o x ) ) = ( A +o ( B +o suc y ) ) )
17	14, 16	eqeq12d 2054	. . . . 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 6059	. . . . . . 7 |- ( ( A e. om /\ B e. om ) -> ( A +o B ) e. om )
19		nna0 6053	. . . . . . 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 6053	. . . . . . . 8 |- ( B e. om -> ( B +o (/) ) = B )
22	21	oveq2d 5528	. . . . . . 7 |- ( B e. om -> ( A +o ( B +o (/) ) ) = ( A +o B ) )
23	22	adantl 262	. . . . . 6 |- ( ( A e. om /\ B e. om ) -> ( A +o ( B +o (/) ) ) = ( A +o B ) )
24	20, 23	eqtr4d 2075	. . . . 5 |- ( ( A e. om /\ B e. om ) -> ( ( A +o B ) +o (/) ) = ( A +o ( B +o (/) ) ) )
25		suceq 4139	. . . . . . 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 6055	. . . . . . . . 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 267	. . . . . . . 8 |- ( ( ( A e. om /\ B e. om ) /\ y e. om ) -> ( ( A +o B ) +o suc y ) = suc ( ( A +o B ) +o y ) )
28		nnasuc 6055	. . . . . . . . . . . 12 |- ( ( B e. om /\ y e. om ) -> ( B +o suc y ) = suc ( B +o y ) )
29	28	oveq2d 5528	. . . . . . . . . . 11 |- ( ( B e. om /\ y e. om ) -> ( A +o ( B +o suc y ) ) = ( A +o suc ( B +o y ) ) )
30	29	adantl 262	. . . . . . . . . 10 |- ( ( A e. om /\ ( B e. om /\ y e. om ) ) -> ( A +o ( B +o suc y ) ) = ( A +o suc ( B +o y ) ) )
31		nnacl 6059	. . . . . . . . . . 11 |- ( ( B e. om /\ y e. om ) -> ( B +o y ) e. om )
32		nnasuc 6055	. . . . . . . . . . 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 270	. . . . . . . . . 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 2072	. . . . . . . . 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 380	. . . . . . . 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 2054	. . . . . . 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 145	. . . . . 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 109	. . . . 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 4324	. . . 4 |- ( x e. om -> ( ( A e. om /\ B e. om ) -> ( ( A +o B ) +o x ) = ( A +o ( B +o x ) ) ) )
40	5, 39	vtoclga 2619	. . 3 |- ( C e. om -> ( ( A e. om /\ B e. om ) -> ( ( A +o B ) +o C ) = ( A +o ( B +o C ) ) ) )
41	40	com12 27	. 2 |- ( ( A e. om /\ B e. om ) -> ( C e. om -> ( ( A +o B ) +o C ) = ( A +o ( B +o C ) ) ) )
42	41	3impia 1101	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 97   /\ w3a 885   = wceq 1243   e. wcel 1393   (/)c0 3224   suc csuc 4102   omcom 4313  (class class class)co 5512   +o coa 5998

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-oadd 6005

This theorem is referenced by:  nndi  6065  nnmsucr  6067  addasspig  6428  addassnq0  6560  prarloclemlo  6592