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Theorem addcanpi 8709
Description: Addition cancellation law for positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
addcanpi  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( ( A  +N  B )  =  ( A  +N  C )  <-> 
B  =  C ) )

Proof of Theorem addcanpi
StepHypRef Expression
1 addclpi 8702 . . . . . . . . . 10  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( A  +N  B
)  e.  N. )
2 eleq1 2447 . . . . . . . . . 10  |-  ( ( A  +N  B )  =  ( A  +N  C )  ->  (
( A  +N  B
)  e.  N.  <->  ( A  +N  C )  e.  N. ) )
31, 2syl5ib 211 . . . . . . . . 9  |-  ( ( A  +N  B )  =  ( A  +N  C )  ->  (
( A  e.  N.  /\  B  e.  N. )  ->  ( A  +N  C
)  e.  N. )
)
43imp 419 . . . . . . . 8  |-  ( ( ( A  +N  B
)  =  ( A  +N  C )  /\  ( A  e.  N.  /\  B  e.  N. )
)  ->  ( A  +N  C )  e.  N. )
5 dmaddpi 8700 . . . . . . . . 9  |-  dom  +N  =  ( N.  X.  N. )
6 0npi 8692 . . . . . . . . 9  |-  -.  (/)  e.  N.
75, 6ndmovrcl 6172 . . . . . . . 8  |-  ( ( A  +N  C )  e.  N.  ->  ( A  e.  N.  /\  C  e.  N. ) )
8 simpr 448 . . . . . . . 8  |-  ( ( A  e.  N.  /\  C  e.  N. )  ->  C  e.  N. )
94, 7, 83syl 19 . . . . . . 7  |-  ( ( ( A  +N  B
)  =  ( A  +N  C )  /\  ( A  e.  N.  /\  B  e.  N. )
)  ->  C  e.  N. )
10 addpiord 8694 . . . . . . . . . 10  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( A  +N  B
)  =  ( A  +o  B ) )
1110adantr 452 . . . . . . . . 9  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  C  e.  N. )  ->  ( A  +N  B )  =  ( A  +o  B ) )
12 addpiord 8694 . . . . . . . . . 10  |-  ( ( A  e.  N.  /\  C  e.  N. )  ->  ( A  +N  C
)  =  ( A  +o  C ) )
1312adantlr 696 . . . . . . . . 9  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  C  e.  N. )  ->  ( A  +N  C )  =  ( A  +o  C ) )
1411, 13eqeq12d 2401 . . . . . . . 8  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  C  e.  N. )  ->  ( ( A  +N  B )  =  ( A  +N  C
)  <->  ( A  +o  B )  =  ( A  +o  C ) ) )
15 pinn 8688 . . . . . . . . . 10  |-  ( A  e.  N.  ->  A  e.  om )
16 pinn 8688 . . . . . . . . . 10  |-  ( B  e.  N.  ->  B  e.  om )
17 pinn 8688 . . . . . . . . . 10  |-  ( C  e.  N.  ->  C  e.  om )
18 nnacan 6807 . . . . . . . . . . 11  |-  ( ( A  e.  om  /\  B  e.  om  /\  C  e.  om )  ->  (
( A  +o  B
)  =  ( A  +o  C )  <->  B  =  C ) )
1918biimpd 199 . . . . . . . . . 10  |-  ( ( A  e.  om  /\  B  e.  om  /\  C  e.  om )  ->  (
( A  +o  B
)  =  ( A  +o  C )  ->  B  =  C )
)
2015, 16, 17, 19syl3an 1226 . . . . . . . . 9  |-  ( ( A  e.  N.  /\  B  e.  N.  /\  C  e.  N. )  ->  (
( A  +o  B
)  =  ( A  +o  C )  ->  B  =  C )
)
21203expa 1153 . . . . . . . 8  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  C  e.  N. )  ->  ( ( A  +o  B )  =  ( A  +o  C
)  ->  B  =  C ) )
2214, 21sylbid 207 . . . . . . 7  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  C  e.  N. )  ->  ( ( A  +N  B )  =  ( A  +N  C
)  ->  B  =  C ) )
239, 22sylan2 461 . . . . . 6  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( ( A  +N  B )  =  ( A  +N  C )  /\  ( A  e. 
N.  /\  B  e.  N. ) ) )  -> 
( ( A  +N  B )  =  ( A  +N  C )  ->  B  =  C ) )
2423exp32 589 . . . . 5  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( ( A  +N  B )  =  ( A  +N  C )  ->  ( ( A  e.  N.  /\  B  e.  N. )  ->  (
( A  +N  B
)  =  ( A  +N  C )  ->  B  =  C )
) ) )
2524imp4b 574 . . . 4  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( A  +N  B
)  =  ( A  +N  C ) )  ->  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( A  +N  B
)  =  ( A  +N  C ) )  ->  B  =  C ) )
2625pm2.43i 45 . . 3  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( A  +N  B
)  =  ( A  +N  C ) )  ->  B  =  C )
2726ex 424 . 2  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( ( A  +N  B )  =  ( A  +N  C )  ->  B  =  C ) )
28 oveq2 6028 . 2  |-  ( B  =  C  ->  ( A  +N  B )  =  ( A  +N  C
) )
2927, 28impbid1 195 1  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( ( A  +N  B )  =  ( A  +N  C )  <-> 
B  =  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   omcom 4785  (class class class)co 6020    +o coa 6657   N.cnpi 8652    +N cpli 8653
This theorem is referenced by:  adderpqlem  8764
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-recs 6569  df-rdg 6604  df-oadd 6664  df-ni 8682  df-pli 8683
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