Theorem nnaass 6540

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 5927	. . . . . 6 |- ( x = C -> ( ( A +o B ) +o x ) = ( ( A +o B ) +o C ) )
2		oveq2 5927	. . . . . . 7 |- ( x = C -> ( B +o x ) = ( B +o C ) )
3	2	oveq2d 5935	. . . . . 6 |- ( x = C -> ( A +o ( B +o x ) ) = ( A +o ( B +o C ) ) )
4	1, 3	eqeq12d 2208	. . . . 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 5927	. . . . . 6 |- ( x = (/) -> ( ( A +o B ) +o x ) = ( ( A +o B ) +o (/) ) )
7		oveq2 5927	. . . . . . 7 |- ( x = (/) -> ( B +o x ) = ( B +o (/) ) )
8	7	oveq2d 5935	. . . . . 6 |- ( x = (/) -> ( A +o ( B +o x ) ) = ( A +o ( B +o (/) ) ) )
9	6, 8	eqeq12d 2208	. . . . 5 |- ( x = (/) -> ( ( ( A +o B ) +o x ) = ( A +o ( B +o x ) ) <-> ( ( A +o B ) +o (/) ) = ( A +o ( B +o (/) ) ) ) )
10		oveq2 5927	. . . . . 6 |- ( x = y -> ( ( A +o B ) +o x ) = ( ( A +o B ) +o y ) )
11		oveq2 5927	. . . . . . 7 |- ( x = y -> ( B +o x ) = ( B +o y ) )
12	11	oveq2d 5935	. . . . . 6 |- ( x = y -> ( A +o ( B +o x ) ) = ( A +o ( B +o y ) ) )
13	10, 12	eqeq12d 2208	. . . . 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 5927	. . . . . 6 |- ( x = suc y -> ( ( A +o B ) +o x ) = ( ( A +o B ) +o suc y ) )
15		oveq2 5927	. . . . . . 7 |- ( x = suc y -> ( B +o x ) = ( B +o suc y ) )
16	15	oveq2d 5935	. . . . . 6 |- ( x = suc y -> ( A +o ( B +o x ) ) = ( A +o ( B +o suc y ) ) )
17	14, 16	eqeq12d 2208	. . . . 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 6535	. . . . . . 7 |- ( ( A e. om /\ B e. om ) -> ( A +o B ) e. om )
19		nna0 6529	. . . . . . 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 6529	. . . . . . . 8 |- ( B e. om -> ( B +o (/) ) = B )
22	21	oveq2d 5935	. . . . . . 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 2229	. . . . 5 |- ( ( A e. om /\ B e. om ) -> ( ( A +o B ) +o (/) ) = ( A +o ( B +o (/) ) ) )
25		suceq 4434	. . . . . . 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 6531	. . . . . . . . 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 6531	. . . . . . . . . . . 12 |- ( ( B e. om /\ y e. om ) -> ( B +o suc y ) = suc ( B +o y ) )
29	28	oveq2d 5935	. . . . . . . . . . 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 6535	. . . . . . . . . . 11 |- ( ( B e. om /\ y e. om ) -> ( B +o y ) e. om )
32		nnasuc 6531	. . . . . . . . . . 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 2226	. . . . . . . . 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 2208	. . . . . . 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 4634	. . . 4 |- ( x e. om -> ( ( A e. om /\ B e. om ) -> ( ( A +o B ) +o x ) = ( A +o ( B +o x ) ) ) )
40	5, 39	vtoclga 2827	. . 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 1202	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 980   = wceq 1364   e. wcel 2164   (/)c0 3447   suc csuc 4397   omcom 4623  (class class class)co 5919   +o coa 6468

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-iord 4398  df-on 4400  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-irdg 6425  df-oadd 6475

This theorem is referenced by:  nndi  6541  nnmsucr  6543  addasspig  7392  addassnq0  7524  prarloclemlo  7556