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Theorem nnaass 6514
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 5908 . . . . . 6  |-  ( x  =  C  ->  (
( A  +o  B
)  +o  x )  =  ( ( A  +o  B )  +o  C ) )
2 oveq2 5908 . . . . . . 7  |-  ( x  =  C  ->  ( B  +o  x )  =  ( B  +o  C
) )
32oveq2d 5916 . . . . . 6  |-  ( x  =  C  ->  ( A  +o  ( B  +o  x ) )  =  ( A  +o  ( B  +o  C ) ) )
41, 3eqeq12d 2204 . . . . 5  |-  ( x  =  C  ->  (
( ( A  +o  B )  +o  x
)  =  ( A  +o  ( B  +o  x ) )  <->  ( ( A  +o  B )  +o  C )  =  ( A  +o  ( B  +o  C ) ) ) )
54imbi2d 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 5908 . . . . . 6  |-  ( x  =  (/)  ->  ( ( A  +o  B )  +o  x )  =  ( ( A  +o  B )  +o  (/) ) )
7 oveq2 5908 . . . . . . 7  |-  ( x  =  (/)  ->  ( B  +o  x )  =  ( B  +o  (/) ) )
87oveq2d 5916 . . . . . 6  |-  ( x  =  (/)  ->  ( A  +o  ( B  +o  x ) )  =  ( A  +o  ( B  +o  (/) ) ) )
96, 8eqeq12d 2204 . . . . 5  |-  ( x  =  (/)  ->  ( ( ( A  +o  B
)  +o  x )  =  ( A  +o  ( B  +o  x
) )  <->  ( ( A  +o  B )  +o  (/) )  =  ( A  +o  ( B  +o  (/) ) ) ) )
10 oveq2 5908 . . . . . 6  |-  ( x  =  y  ->  (
( A  +o  B
)  +o  x )  =  ( ( A  +o  B )  +o  y ) )
11 oveq2 5908 . . . . . . 7  |-  ( x  =  y  ->  ( B  +o  x )  =  ( B  +o  y
) )
1211oveq2d 5916 . . . . . 6  |-  ( x  =  y  ->  ( A  +o  ( B  +o  x ) )  =  ( A  +o  ( B  +o  y ) ) )
1310, 12eqeq12d 2204 . . . . 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 5908 . . . . . 6  |-  ( x  =  suc  y  -> 
( ( A  +o  B )  +o  x
)  =  ( ( A  +o  B )  +o  suc  y ) )
15 oveq2 5908 . . . . . . 7  |-  ( x  =  suc  y  -> 
( B  +o  x
)  =  ( B  +o  suc  y ) )
1615oveq2d 5916 . . . . . 6  |-  ( x  =  suc  y  -> 
( A  +o  ( B  +o  x ) )  =  ( A  +o  ( B  +o  suc  y
) ) )
1714, 16eqeq12d 2204 . . . . 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 6509 . . . . . . 7  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  +o  B
)  e.  om )
19 nna0 6503 . . . . . . 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 6503 . . . . . . . 8  |-  ( B  e.  om  ->  ( B  +o  (/) )  =  B )
2221oveq2d 5916 . . . . . . 7  |-  ( B  e.  om  ->  ( A  +o  ( B  +o  (/) ) )  =  ( A  +o  B ) )
2322adantl 277 . . . . . 6  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  +o  ( B  +o  (/) ) )  =  ( A  +o  B
) )
2420, 23eqtr4d 2225 . . . . 5  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( ( A  +o  B )  +o  (/) )  =  ( A  +o  ( B  +o  (/) ) ) )
25 suceq 4423 . . . . . . 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 6505 . . . . . . . . 9  |-  ( ( ( A  +o  B
)  e.  om  /\  y  e.  om )  ->  ( ( A  +o  B )  +o  suc  y )  =  suc  ( ( A  +o  B )  +o  y
) )
2718, 26sylan 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 6505 . . . . . . . . . . . 12  |-  ( ( B  e.  om  /\  y  e.  om )  ->  ( B  +o  suc  y )  =  suc  ( B  +o  y
) )
2928oveq2d 5916 . . . . . . . . . . 11  |-  ( ( B  e.  om  /\  y  e.  om )  ->  ( A  +o  ( B  +o  suc  y ) )  =  ( A  +o  suc  ( B  +o  y ) ) )
3029adantl 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 6509 . . . . . . . . . . 11  |-  ( ( B  e.  om  /\  y  e.  om )  ->  ( B  +o  y
)  e.  om )
32 nnasuc 6505 . . . . . . . . . . 11  |-  ( ( A  e.  om  /\  ( B  +o  y
)  e.  om )  ->  ( A  +o  suc  ( B  +o  y
) )  =  suc  ( A  +o  ( B  +o  y ) ) )
3331, 32sylan2 286 . . . . . . . . . 10  |-  ( ( A  e.  om  /\  ( B  e.  om  /\  y  e.  om )
)  ->  ( A  +o  suc  ( B  +o  y ) )  =  suc  ( A  +o  ( B  +o  y
) ) )
3430, 33eqtrd 2222 . . . . . . . . 9  |-  ( ( A  e.  om  /\  ( B  e.  om  /\  y  e.  om )
)  ->  ( A  +o  ( B  +o  suc  y ) )  =  suc  ( A  +o  ( B  +o  y
) ) )
3534anassrs 400 . . . . . . . 8  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  y  e.  om )  ->  ( A  +o  ( B  +o  suc  y
) )  =  suc  ( A  +o  ( B  +o  y ) ) )
3627, 35eqeq12d 2204 . . . . . . 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, 36imbitrrid 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 ) ) ) )
3837expcom 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 ) ) ) ) )
399, 13, 17, 24, 38finds2 4621 . . . 4  |-  ( x  e.  om  ->  (
( A  e.  om  /\  B  e.  om )  ->  ( ( A  +o  B )  +o  x
)  =  ( A  +o  ( B  +o  x ) ) ) )
405, 39vtoclga 2818 . . 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 1202 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 104    /\ w3a 980    = wceq 1364    e. wcel 2160   (/)c0 3437   suc csuc 4386   omcom 4610  (class class class)co 5900    +o coa 6442
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 2162  ax-14 2163  ax-ext 2171  ax-coll 4136  ax-sep 4139  ax-nul 4147  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-setind 4557  ax-iinf 4608
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 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-int 3863  df-iun 3906  df-br 4022  df-opab 4083  df-mpt 4084  df-tr 4120  df-id 4314  df-iord 4387  df-on 4389  df-suc 4392  df-iom 4611  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-f1 5243  df-fo 5244  df-f1o 5245  df-fv 5246  df-ov 5903  df-oprab 5904  df-mpo 5905  df-1st 6169  df-2nd 6170  df-recs 6334  df-irdg 6399  df-oadd 6449
This theorem is referenced by:  nndi  6515  nnmsucr  6517  addasspig  7364  addassnq0  7496  prarloclemlo  7528
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