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Theorem addcanpi 8768
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 8761 . . . . . . . . . 10  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( A  +N  B
)  e.  N. )
2 eleq1 2495 . . . . . . . . . 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 8759 . . . . . . . . 9  |-  dom  +N  =  ( N.  X.  N. )
6 0npi 8751 . . . . . . . . 9  |-  -.  (/)  e.  N.
75, 6ndmovrcl 6225 . . . . . . . 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 8753 . . . . . . . . . 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 8753 . . . . . . . . . 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 2449 . . . . . . . 8  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  C  e.  N. )  ->  ( ( A  +N  B )  =  ( A  +N  C
)  <->  ( A  +o  B )  =  ( A  +o  C ) ) )
15 pinn 8747 . . . . . . . . . 10  |-  ( A  e.  N.  ->  A  e.  om )
16 pinn 8747 . . . . . . . . . 10  |-  ( B  e.  N.  ->  B  e.  om )
17 pinn 8747 . . . . . . . . . 10  |-  ( C  e.  N.  ->  C  e.  om )
18 nnacan 6863 . . . . . . . . . . 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 6081 . 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 1652    e. wcel 1725   omcom 4837  (class class class)co 6073    +o coa 6713   N.cnpi 8711    +N cpli 8712
This theorem is referenced by:  adderpqlem  8823
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-1st 6341  df-2nd 6342  df-recs 6625  df-rdg 6660  df-oadd 6720  df-ni 8741  df-pli 8742
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