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Theorem nnaass 6857
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 6081 . . . . . 6  |-  ( x  =  C  ->  (
( A  +o  B
)  +o  x )  =  ( ( A  +o  B )  +o  C ) )
2 oveq2 6081 . . . . . . 7  |-  ( x  =  C  ->  ( B  +o  x )  =  ( B  +o  C
) )
32oveq2d 6089 . . . . . 6  |-  ( x  =  C  ->  ( A  +o  ( B  +o  x ) )  =  ( A  +o  ( B  +o  C ) ) )
41, 3eqeq12d 2449 . . . . 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 6081 . . . . . 6  |-  ( x  =  (/)  ->  ( ( A  +o  B )  +o  x )  =  ( ( A  +o  B )  +o  (/) ) )
7 oveq2 6081 . . . . . . 7  |-  ( x  =  (/)  ->  ( B  +o  x )  =  ( B  +o  (/) ) )
87oveq2d 6089 . . . . . 6  |-  ( x  =  (/)  ->  ( A  +o  ( B  +o  x ) )  =  ( A  +o  ( B  +o  (/) ) ) )
96, 8eqeq12d 2449 . . . . 5  |-  ( x  =  (/)  ->  ( ( ( A  +o  B
)  +o  x )  =  ( A  +o  ( B  +o  x
) )  <->  ( ( A  +o  B )  +o  (/) )  =  ( A  +o  ( B  +o  (/) ) ) ) )
10 oveq2 6081 . . . . . 6  |-  ( x  =  y  ->  (
( A  +o  B
)  +o  x )  =  ( ( A  +o  B )  +o  y ) )
11 oveq2 6081 . . . . . . 7  |-  ( x  =  y  ->  ( B  +o  x )  =  ( B  +o  y
) )
1211oveq2d 6089 . . . . . 6  |-  ( x  =  y  ->  ( A  +o  ( B  +o  x ) )  =  ( A  +o  ( B  +o  y ) ) )
1310, 12eqeq12d 2449 . . . . 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 6081 . . . . . 6  |-  ( x  =  suc  y  -> 
( ( A  +o  B )  +o  x
)  =  ( ( A  +o  B )  +o  suc  y ) )
15 oveq2 6081 . . . . . . 7  |-  ( x  =  suc  y  -> 
( B  +o  x
)  =  ( B  +o  suc  y ) )
1615oveq2d 6089 . . . . . 6  |-  ( x  =  suc  y  -> 
( A  +o  ( B  +o  x ) )  =  ( A  +o  ( B  +o  suc  y
) ) )
1714, 16eqeq12d 2449 . . . . 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 6846 . . . . . . 7  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  +o  B
)  e.  om )
19 nna0 6839 . . . . . . 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 6839 . . . . . . . 8  |-  ( B  e.  om  ->  ( B  +o  (/) )  =  B )
2221oveq2d 6089 . . . . . . 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 2470 . . . . 5  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( ( A  +o  B )  +o  (/) )  =  ( A  +o  ( B  +o  (/) ) ) )
25 suceq 4638 . . . . . . 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 6841 . . . . . . . . 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 6841 . . . . . . . . . . . 12  |-  ( ( B  e.  om  /\  y  e.  om )  ->  ( B  +o  suc  y )  =  suc  ( B  +o  y
) )
2928oveq2d 6089 . . . . . . . . . . 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 6846 . . . . . . . . . . 11  |-  ( ( B  e.  om  /\  y  e.  om )  ->  ( B  +o  y
)  e.  om )
32 nnasuc 6841 . . . . . . . . . . 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 2467 . . . . . . . . 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 2449 . . . . . . 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 4865 . . . 4  |-  ( x  e.  om  ->  (
( A  e.  om  /\  B  e.  om )  ->  ( ( A  +o  B )  +o  x
)  =  ( A  +o  ( B  +o  x ) ) ) )
405, 39vtoclga 3009 . . 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 1652    e. wcel 1725   (/)c0 3620   suc csuc 4575   omcom 4837  (class class class)co 6073    +o coa 6713
This theorem is referenced by:  nndi  6858  nnmsucr  6860  omopthlem1  6890  omopthlem2  6891  addasspi  8764
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-recs 6625  df-rdg 6660  df-oadd 6720
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