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Theorem nnaass 6832
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 6056 . . . . . 6  |-  ( x  =  C  ->  (
( A  +o  B
)  +o  x )  =  ( ( A  +o  B )  +o  C ) )
2 oveq2 6056 . . . . . . 7  |-  ( x  =  C  ->  ( B  +o  x )  =  ( B  +o  C
) )
32oveq2d 6064 . . . . . 6  |-  ( x  =  C  ->  ( A  +o  ( B  +o  x ) )  =  ( A  +o  ( B  +o  C ) ) )
41, 3eqeq12d 2426 . . . . 5  |-  ( x  =  C  ->  (
( ( A  +o  B )  +o  x
)  =  ( A  +o  ( B  +o  x ) )  <->  ( ( A  +o  B )  +o  C )  =  ( A  +o  ( B  +o  C ) ) ) )
54imbi2d 308 . . . 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 6056 . . . . . 6  |-  ( x  =  (/)  ->  ( ( A  +o  B )  +o  x )  =  ( ( A  +o  B )  +o  (/) ) )
7 oveq2 6056 . . . . . . 7  |-  ( x  =  (/)  ->  ( B  +o  x )  =  ( B  +o  (/) ) )
87oveq2d 6064 . . . . . 6  |-  ( x  =  (/)  ->  ( A  +o  ( B  +o  x ) )  =  ( A  +o  ( B  +o  (/) ) ) )
96, 8eqeq12d 2426 . . . . 5  |-  ( x  =  (/)  ->  ( ( ( A  +o  B
)  +o  x )  =  ( A  +o  ( B  +o  x
) )  <->  ( ( A  +o  B )  +o  (/) )  =  ( A  +o  ( B  +o  (/) ) ) ) )
10 oveq2 6056 . . . . . 6  |-  ( x  =  y  ->  (
( A  +o  B
)  +o  x )  =  ( ( A  +o  B )  +o  y ) )
11 oveq2 6056 . . . . . . 7  |-  ( x  =  y  ->  ( B  +o  x )  =  ( B  +o  y
) )
1211oveq2d 6064 . . . . . 6  |-  ( x  =  y  ->  ( A  +o  ( B  +o  x ) )  =  ( A  +o  ( B  +o  y ) ) )
1310, 12eqeq12d 2426 . . . . 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 6056 . . . . . 6  |-  ( x  =  suc  y  -> 
( ( A  +o  B )  +o  x
)  =  ( ( A  +o  B )  +o  suc  y ) )
15 oveq2 6056 . . . . . . 7  |-  ( x  =  suc  y  -> 
( B  +o  x
)  =  ( B  +o  suc  y ) )
1615oveq2d 6064 . . . . . 6  |-  ( x  =  suc  y  -> 
( A  +o  ( B  +o  x ) )  =  ( A  +o  ( B  +o  suc  y
) ) )
1714, 16eqeq12d 2426 . . . . 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 6821 . . . . . . 7  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  +o  B
)  e.  om )
19 nna0 6814 . . . . . . 7  |-  ( ( A  +o  B )  e.  om  ->  (
( A  +o  B
)  +o  (/) )  =  ( A  +o  B
) )
2018, 19syl 16 . . . . . 6  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( ( A  +o  B )  +o  (/) )  =  ( A  +o  B
) )
21 nna0 6814 . . . . . . . 8  |-  ( B  e.  om  ->  ( B  +o  (/) )  =  B )
2221oveq2d 6064 . . . . . . 7  |-  ( B  e.  om  ->  ( A  +o  ( B  +o  (/) ) )  =  ( A  +o  B ) )
2322adantl 453 . . . . . 6  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  +o  ( B  +o  (/) ) )  =  ( A  +o  B
) )
2420, 23eqtr4d 2447 . . . . 5  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( ( A  +o  B )  +o  (/) )  =  ( A  +o  ( B  +o  (/) ) ) )
25 suceq 4614 . . . . . . 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 6816 . . . . . . . . 9  |-  ( ( ( A  +o  B
)  e.  om  /\  y  e.  om )  ->  ( ( A  +o  B )  +o  suc  y )  =  suc  ( ( A  +o  B )  +o  y
) )
2718, 26sylan 458 . . . . . . . 8  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  y  e.  om )  ->  ( ( A  +o  B )  +o 
suc  y )  =  suc  ( ( A  +o  B )  +o  y ) )
28 nnasuc 6816 . . . . . . . . . . . 12  |-  ( ( B  e.  om  /\  y  e.  om )  ->  ( B  +o  suc  y )  =  suc  ( B  +o  y
) )
2928oveq2d 6064 . . . . . . . . . . 11  |-  ( ( B  e.  om  /\  y  e.  om )  ->  ( A  +o  ( B  +o  suc  y ) )  =  ( A  +o  suc  ( B  +o  y ) ) )
3029adantl 453 . . . . . . . . . 10  |-  ( ( A  e.  om  /\  ( B  e.  om  /\  y  e.  om )
)  ->  ( A  +o  ( B  +o  suc  y ) )  =  ( A  +o  suc  ( B  +o  y
) ) )
31 nnacl 6821 . . . . . . . . . . 11  |-  ( ( B  e.  om  /\  y  e.  om )  ->  ( B  +o  y
)  e.  om )
32 nnasuc 6816 . . . . . . . . . . 11  |-  ( ( A  e.  om  /\  ( B  +o  y
)  e.  om )  ->  ( A  +o  suc  ( B  +o  y
) )  =  suc  ( A  +o  ( B  +o  y ) ) )
3331, 32sylan2 461 . . . . . . . . . 10  |-  ( ( A  e.  om  /\  ( B  e.  om  /\  y  e.  om )
)  ->  ( A  +o  suc  ( B  +o  y ) )  =  suc  ( A  +o  ( B  +o  y
) ) )
3430, 33eqtrd 2444 . . . . . . . . 9  |-  ( ( A  e.  om  /\  ( B  e.  om  /\  y  e.  om )
)  ->  ( A  +o  ( B  +o  suc  y ) )  =  suc  ( A  +o  ( B  +o  y
) ) )
3534anassrs 630 . . . . . . . 8  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  y  e.  om )  ->  ( A  +o  ( B  +o  suc  y
) )  =  suc  ( A  +o  ( B  +o  y ) ) )
3627, 35eqeq12d 2426 . . . . . . 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 213 . . . . . 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 425 . . . . 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 4840 . . . 4  |-  ( x  e.  om  ->  (
( A  e.  om  /\  B  e.  om )  ->  ( ( A  +o  B )  +o  x
)  =  ( A  +o  ( B  +o  x ) ) ) )
405, 39vtoclga 2985 . . 3  |-  ( C  e.  om  ->  (
( A  e.  om  /\  B  e.  om )  ->  ( ( A  +o  B )  +o  C
)  =  ( A  +o  ( B  +o  C ) ) ) )
4140com12 29 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( C  e.  om  ->  ( ( A  +o  B )  +o  C
)  =  ( A  +o  ( B  +o  C ) ) ) )
42413impia 1150 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721   (/)c0 3596   suc csuc 4551   omcom 4812  (class class class)co 6048    +o coa 6688
This theorem is referenced by:  nndi  6833  nnmsucr  6835  omopthlem1  6865  omopthlem2  6866  addasspi  8736
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-recs 6600  df-rdg 6635  df-oadd 6695
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