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Theorem addcanpi 9721
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 9714 . . . . . . . . . 10 |- ( ( A e. N. /\ B e. N. ) -> ( A +N B ) e. N. )
2 eleq1 2689 . . . . . . . . . 10 |- ( ( A +N B ) = ( A +N C ) -> ( ( A +N B ) e. N. <-> ( A +N C ) e. N. ) )
31, 2syl5ib 234 . . . . . . . . 9 |- ( ( A +N B ) = ( A +N C ) -> ( ( A e. N. /\ B e. N. ) -> ( A +N C ) e. N. ) )
43imp 445 . . . . . . . 8 |- ( ( ( A +N B ) = ( A +N C ) /\ ( A e. N. /\ B e. N. ) ) -> ( A +N C ) e. N. )
5 dmaddpi 9712 . . . . . . . . 9 |- dom +N = ( N. X. N. )
6 0npi 9704 . . . . . . . . 9 |- -. (/) e. N.
75, 6ndmovrcl 6820 . . . . . . . 8 |- ( ( A +N C ) e. N. -> ( A e. N. /\ C e. N. ) )
8 simpr 477 . . . . . . . 8 |- ( ( A e. N. /\ C e. N. ) -> C e. N. )
94, 7, 83syl 18 . . . . . . 7 |- ( ( ( A +N B ) = ( A +N C ) /\ ( A e. N. /\ B e. N. ) ) -> C e. N. )
10 addpiord 9706 . . . . . . . . . 10 |- ( ( A e. N. /\ B e. N. ) -> ( A +N B ) = ( A +o B ) )
1110adantr 481 . . . . . . . . 9 |- ( ( ( A e. N. /\ B e. N. ) /\ C e. N. ) -> ( A +N B ) = ( A +o B ) )
12 addpiord 9706 . . . . . . . . . 10 |- ( ( A e. N. /\ C e. N. ) -> ( A +N C ) = ( A +o C ) )
1312adantlr 751 . . . . . . . . 9 |- ( ( ( A e. N. /\ B e. N. ) /\ C e. N. ) -> ( A +N C ) = ( A +o C ) )
1411, 13eqeq12d 2637 . . . . . . . 8 |- ( ( ( A e. N. /\ B e. N. ) /\ C e. N. ) -> ( ( A +N B ) = ( A +N C ) <-> ( A +o B ) = ( A +o C ) ) )
15 pinn 9700 . . . . . . . . . 10 |- ( A e. N. -> A e. om )
16 pinn 9700 . . . . . . . . . 10 |- ( B e. N. -> B e. om )
17 pinn 9700 . . . . . . . . . 10 |- ( C e. N. -> C e. om )
18 nnacan 7708 . . . . . . . . . . 11 |- ( ( A e. om /\ B e. om /\ C e. om ) -> ( ( A +o B ) = ( A +o C ) <-> B = C ) )
1918biimpd 219 . . . . . . . . . 10 |- ( ( A e. om /\ B e. om /\ C e. om ) -> ( ( A +o B ) = ( A +o C ) -> B = C ) )
2015, 16, 17, 19syl3an 1368 . . . . . . . . 9 |- ( ( A e. N. /\ B e. N. /\ C e. N. ) -> ( ( A +o B ) = ( A +o C ) -> B = C ) )
21203expa 1265 . . . . . . . 8 |- ( ( ( A e. N. /\ B e. N. ) /\ C e. N. ) -> ( ( A +o B ) = ( A +o C ) -> B = C ) )
2214, 21sylbid 230 . . . . . . 7 |- ( ( ( A e. N. /\ B e. N. ) /\ C e. N. ) -> ( ( A +N B ) = ( A +N C ) -> B = C ) )
239, 22sylan2 491 . . . . . 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 631 . . . . 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 613 . . . 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 52 . . 3 |- ( ( ( A e. N. /\ B e. N. ) /\ ( A +N B ) = ( A +N C ) ) -> B = C )
2726ex 450 . 2 |- ( ( A e. N. /\ B e. N. ) -> ( ( A +N B ) = ( A +N C ) -> B = C ) )
28 oveq2 6658 . 2 |- ( B = C -> ( A +N B ) = ( A +N C ) )
2927, 28impbid1 215 1 |- ( ( A e. N. /\ B e. N. ) -> ( ( A +N B ) = ( A +N C ) <-> B = C ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990  (class class class)co 6650   omcom 7065   +o coa 7557   N.cnpi 9666   +N cpli 9667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564  df-ni 9694  df-pli 9695
This theorem is referenced by:  adderpqlem  9776
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