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Theorem addcanpi 8539
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 8532 . . . . . . . . . 10  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( A  +N  B
)  e.  N. )
2 eleq1 2356 . . . . . . . . . 10  |-  ( ( A  +N  B )  =  ( A  +N  C )  ->  (
( A  +N  B
)  e.  N.  <->  ( A  +N  C )  e.  N. ) )
31, 2syl5ib 210 . . . . . . . . 9  |-  ( ( A  +N  B )  =  ( A  +N  C )  ->  (
( A  e.  N.  /\  B  e.  N. )  ->  ( A  +N  C
)  e.  N. )
)
43imp 418 . . . . . . . 8  |-  ( ( ( A  +N  B
)  =  ( A  +N  C )  /\  ( A  e.  N.  /\  B  e.  N. )
)  ->  ( A  +N  C )  e.  N. )
5 dmaddpi 8530 . . . . . . . . 9  |-  dom  +N  =  ( N.  X.  N. )
6 0npi 8522 . . . . . . . . 9  |-  -.  (/)  e.  N.
75, 6ndmovrcl 6022 . . . . . . . 8  |-  ( ( A  +N  C )  e.  N.  ->  ( A  e.  N.  /\  C  e.  N. ) )
8 simpr 447 . . . . . . . 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 8524 . . . . . . . . . 10  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( A  +N  B
)  =  ( A  +o  B ) )
1110adantr 451 . . . . . . . . 9  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  C  e.  N. )  ->  ( A  +N  B )  =  ( A  +o  B ) )
12 addpiord 8524 . . . . . . . . . 10  |-  ( ( A  e.  N.  /\  C  e.  N. )  ->  ( A  +N  C
)  =  ( A  +o  C ) )
1312adantlr 695 . . . . . . . . 9  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  C  e.  N. )  ->  ( A  +N  C )  =  ( A  +o  C ) )
1411, 13eqeq12d 2310 . . . . . . . 8  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  C  e.  N. )  ->  ( ( A  +N  B )  =  ( A  +N  C
)  <->  ( A  +o  B )  =  ( A  +o  C ) ) )
15 pinn 8518 . . . . . . . . . 10  |-  ( A  e.  N.  ->  A  e.  om )
16 pinn 8518 . . . . . . . . . 10  |-  ( B  e.  N.  ->  B  e.  om )
17 pinn 8518 . . . . . . . . . 10  |-  ( C  e.  N.  ->  C  e.  om )
18 nnacan 6642 . . . . . . . . . . 11  |-  ( ( A  e.  om  /\  B  e.  om  /\  C  e.  om )  ->  (
( A  +o  B
)  =  ( A  +o  C )  <->  B  =  C ) )
1918biimpd 198 . . . . . . . . . 10  |-  ( ( A  e.  om  /\  B  e.  om  /\  C  e.  om )  ->  (
( A  +o  B
)  =  ( A  +o  C )  ->  B  =  C )
)
2015, 16, 17, 19syl3an 1224 . . . . . . . . 9  |-  ( ( A  e.  N.  /\  B  e.  N.  /\  C  e.  N. )  ->  (
( A  +o  B
)  =  ( A  +o  C )  ->  B  =  C )
)
21203expa 1151 . . . . . . . 8  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  C  e.  N. )  ->  ( ( A  +o  B )  =  ( A  +o  C
)  ->  B  =  C ) )
2214, 21sylbid 206 . . . . . . 7  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  C  e.  N. )  ->  ( ( A  +N  B )  =  ( A  +N  C
)  ->  B  =  C ) )
239, 22sylan2 460 . . . . . 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 588 . . . . 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 573 . . . 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 43 . . 3  |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( A  +N  B
)  =  ( A  +N  C ) )  ->  B  =  C )
2726ex 423 . 2  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( ( A  +N  B )  =  ( A  +N  C )  ->  B  =  C ) )
28 oveq2 5882 . 2  |-  ( B  =  C  ->  ( A  +N  B )  =  ( A  +N  C
) )
2927, 28impbid1 194 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   omcom 4672  (class class class)co 5874    +o coa 6492   N.cnpi 8482    +N cpli 8483
This theorem is referenced by:  adderpqlem  8594
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-recs 6404  df-rdg 6439  df-oadd 6499  df-ni 8512  df-pli 8513
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