Theorem nnaass 6374

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 5775	. . . . . 6 |- ( x = C -> ( ( A +o B ) +o x ) = ( ( A +o B ) +o C ) )
2		oveq2 5775	. . . . . . 7 |- ( x = C -> ( B +o x ) = ( B +o C ) )
3	2	oveq2d 5783	. . . . . 6 |- ( x = C -> ( A +o ( B +o x ) ) = ( A +o ( B +o C ) ) )
4	1, 3	eqeq12d 2152	. . . . 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 5775	. . . . . 6 |- ( x = (/) -> ( ( A +o B ) +o x ) = ( ( A +o B ) +o (/) ) )
7		oveq2 5775	. . . . . . 7 |- ( x = (/) -> ( B +o x ) = ( B +o (/) ) )
8	7	oveq2d 5783	. . . . . 6 |- ( x = (/) -> ( A +o ( B +o x ) ) = ( A +o ( B +o (/) ) ) )
9	6, 8	eqeq12d 2152	. . . . 5 |- ( x = (/) -> ( ( ( A +o B ) +o x ) = ( A +o ( B +o x ) ) <-> ( ( A +o B ) +o (/) ) = ( A +o ( B +o (/) ) ) ) )
10		oveq2 5775	. . . . . 6 |- ( x = y -> ( ( A +o B ) +o x ) = ( ( A +o B ) +o y ) )
11		oveq2 5775	. . . . . . 7 |- ( x = y -> ( B +o x ) = ( B +o y ) )
12	11	oveq2d 5783	. . . . . 6 |- ( x = y -> ( A +o ( B +o x ) ) = ( A +o ( B +o y ) ) )
13	10, 12	eqeq12d 2152	. . . . 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 5775	. . . . . 6 |- ( x = suc y -> ( ( A +o B ) +o x ) = ( ( A +o B ) +o suc y ) )
15		oveq2 5775	. . . . . . 7 |- ( x = suc y -> ( B +o x ) = ( B +o suc y ) )
16	15	oveq2d 5783	. . . . . 6 |- ( x = suc y -> ( A +o ( B +o x ) ) = ( A +o ( B +o suc y ) ) )
17	14, 16	eqeq12d 2152	. . . . 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 6369	. . . . . . 7 |- ( ( A e. om /\ B e. om ) -> ( A +o B ) e. om )
19		nna0 6363	. . . . . . 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 6363	. . . . . . . 8 |- ( B e. om -> ( B +o (/) ) = B )
22	21	oveq2d 5783	. . . . . . 7 |- ( B e. om -> ( A +o ( B +o (/) ) ) = ( A +o B ) )
23	22	adantl 275	. . . . . 6 |- ( ( A e. om /\ B e. om ) -> ( A +o ( B +o (/) ) ) = ( A +o B ) )
24	20, 23	eqtr4d 2173	. . . . 5 |- ( ( A e. om /\ B e. om ) -> ( ( A +o B ) +o (/) ) = ( A +o ( B +o (/) ) ) )
25		suceq 4319	. . . . . . 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 6365	. . . . . . . . 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 281	. . . . . . . 8 |- ( ( ( A e. om /\ B e. om ) /\ y e. om ) -> ( ( A +o B ) +o suc y ) = suc ( ( A +o B ) +o y ) )
28		nnasuc 6365	. . . . . . . . . . . 12 |- ( ( B e. om /\ y e. om ) -> ( B +o suc y ) = suc ( B +o y ) )
29	28	oveq2d 5783	. . . . . . . . . . 11 |- ( ( B e. om /\ y e. om ) -> ( A +o ( B +o suc y ) ) = ( A +o suc ( B +o y ) ) )
30	29	adantl 275	. . . . . . . . . 10 |- ( ( A e. om /\ ( B e. om /\ y e. om ) ) -> ( A +o ( B +o suc y ) ) = ( A +o suc ( B +o y ) ) )
31		nnacl 6369	. . . . . . . . . . 11 |- ( ( B e. om /\ y e. om ) -> ( B +o y ) e. om )
32		nnasuc 6365	. . . . . . . . . . 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 284	. . . . . . . . . 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 2170	. . . . . . . . 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 397	. . . . . . . 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 2152	. . . . . . 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 4510	. . . 4 |- ( x e. om -> ( ( A e. om /\ B e. om ) -> ( ( A +o B ) +o x ) = ( A +o ( B +o x ) ) ) )
40	5, 39	vtoclga 2747	. . 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 1178	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 962   = wceq 1331   e. wcel 1480   (/)c0 3358   suc csuc 4282   omcom 4499  (class class class)co 5767   +o coa 6303

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-oadd 6310

This theorem is referenced by:  nndi  6375  nnmsucr  6377  addasspig  7131  addassnq0  7263  prarloclemlo  7295