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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.
StepHypRef 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
) )
32oveq2d 5682 . . . . . 6  |-  ( x  =  C  ->  ( A  +o  ( B  +o  x ) )  =  ( A  +o  ( B  +o  C ) ) )
41, 3eqeq12d 2103 . . . . 5  |-  ( x  =  C  ->  (
( ( A  +o  B )  +o  x
)  =  ( A  +o  ( B  +o  x ) )  <->  ( ( A  +o  B )  +o  C )  =  ( A  +o  ( B  +o  C ) ) ) )
54imbi2d 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  (/) ) )
87oveq2d 5682 . . . . . 6  |-  ( x  =  (/)  ->  ( A  +o  ( B  +o  x ) )  =  ( A  +o  ( B  +o  (/) ) ) )
96, 8eqeq12d 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
) )
1211oveq2d 5682 . . . . . 6  |-  ( x  =  y  ->  ( A  +o  ( B  +o  x ) )  =  ( A  +o  ( B  +o  y ) ) )
1310, 12eqeq12d 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 ) )
1615oveq2d 5682 . . . . . 6  |-  ( x  =  suc  y  -> 
( A  +o  ( B  +o  x ) )  =  ( A  +o  ( B  +o  suc  y
) ) )
1714, 16eqeq12d 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
) )
2018, 19syl 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 )
2221oveq2d 5682 . . . . . . 7  |-  ( B  e.  om  ->  ( A  +o  ( B  +o  (/) ) )  =  ( A  +o  B ) )
2322adantl 272 . . . . . 6  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  +o  ( B  +o  (/) ) )  =  ( A  +o  B
) )
2420, 23eqtr4d 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
) )
2718, 26sylan 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
) )
2928oveq2d 5682 . . . . . . . . . . 11  |-  ( ( B  e.  om  /\  y  e.  om )  ->  ( A  +o  ( B  +o  suc  y ) )  =  ( A  +o  suc  ( B  +o  y ) ) )
3029adantl 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 ) ) )
3331, 32sylan2 281 . . . . . . . . . 10  |-  ( ( A  e.  om  /\  ( B  e.  om  /\  y  e.  om )
)  ->  ( A  +o  suc  ( B  +o  y ) )  =  suc  ( A  +o  ( B  +o  y
) ) )
3430, 33eqtrd 2121 . . . . . . . . 9  |-  ( ( A  e.  om  /\  ( B  e.  om  /\  y  e.  om )
)  ->  ( A  +o  ( B  +o  suc  y ) )  =  suc  ( A  +o  ( B  +o  y
) ) )
3534anassrs 393 . . . . . . . 8  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  y  e.  om )  ->  ( A  +o  ( B  +o  suc  y
) )  =  suc  ( A  +o  ( B  +o  y ) ) )
3627, 35eqeq12d 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
) ) ) )
3725, 36syl5ibr 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 ) ) ) )
3837expcom 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 ) ) ) ) )
399, 13, 17, 24, 38finds2 4429 . . . 4  |-  ( x  e.  om  ->  (
( A  e.  om  /\  B  e.  om )  ->  ( ( A  +o  B )  +o  x
)  =  ( A  +o  ( B  +o  x ) ) ) )
405, 39vtoclga 2686 . . 3  |-  ( C  e.  om  ->  (
( A  e.  om  /\  B  e.  om )  ->  ( ( A  +o  B )  +o  C
)  =  ( A  +o  ( B  +o  C ) ) ) )
4140com12 30 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( C  e.  om  ->  ( ( A  +o  B )  +o  C
)  =  ( A  +o  ( B  +o  C ) ) ) )
42413impia 1141 1  |-  ( ( A  e.  om  /\  B  e.  om  /\  C  e.  om )  ->  (
( A  +o  B
)  +o  C )  =  ( A  +o  ( B  +o  C
) ) )
Colors of variables: wff set class
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
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