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Theorem addcanprg 6712
Description: Addition cancellation law for positive reals. Proposition 9-3.5(vi) of [Gleason] p. 123. (Contributed by Jim Kingdon, 24-Dec-2019.)
Assertion
Ref Expression
addcanprg  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( A  +P.  B
)  =  ( A  +P.  C )  ->  B  =  C )
)

Proof of Theorem addcanprg
StepHypRef Expression
1 addcanprleml 6710 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  ->  ( 1st `  B
)  C_  ( 1st `  C ) )
2 3ancomb 893 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  <->  ( A  e.  P.  /\  C  e. 
P.  /\  B  e.  P. ) )
3 eqcom 2042 . . . . . . 7  |-  ( ( A  +P.  B )  =  ( A  +P.  C )  <->  ( A  +P.  C )  =  ( A  +P.  B ) )
42, 3anbi12i 433 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  <-> 
( ( A  e. 
P.  /\  C  e.  P.  /\  B  e.  P. )  /\  ( A  +P.  C )  =  ( A  +P.  B ) ) )
5 addcanprleml 6710 . . . . . 6  |-  ( ( ( A  e.  P.  /\  C  e.  P.  /\  B  e.  P. )  /\  ( A  +P.  C
)  =  ( A  +P.  B ) )  ->  ( 1st `  C
)  C_  ( 1st `  B ) )
64, 5sylbi 114 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  ->  ( 1st `  C
)  C_  ( 1st `  B ) )
71, 6eqssd 2962 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  ->  ( 1st `  B
)  =  ( 1st `  C ) )
8 addcanprlemu 6711 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  ->  ( 2nd `  B
)  C_  ( 2nd `  C ) )
9 addcanprlemu 6711 . . . . . 6  |-  ( ( ( A  e.  P.  /\  C  e.  P.  /\  B  e.  P. )  /\  ( A  +P.  C
)  =  ( A  +P.  B ) )  ->  ( 2nd `  C
)  C_  ( 2nd `  B ) )
104, 9sylbi 114 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  ->  ( 2nd `  C
)  C_  ( 2nd `  B ) )
118, 10eqssd 2962 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  ->  ( 2nd `  B
)  =  ( 2nd `  C ) )
127, 11jca 290 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  ->  ( ( 1st `  B )  =  ( 1st `  C )  /\  ( 2nd `  B
)  =  ( 2nd `  C ) ) )
13 preqlu 6568 . . . . 5  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( B  =  C  <-> 
( ( 1st `  B
)  =  ( 1st `  C )  /\  ( 2nd `  B )  =  ( 2nd `  C
) ) ) )
14133adant1 922 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( B  =  C  <->  ( ( 1st `  B )  =  ( 1st `  C
)  /\  ( 2nd `  B )  =  ( 2nd `  C ) ) ) )
1514adantr 261 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  ->  ( B  =  C  <->  ( ( 1st `  B )  =  ( 1st `  C )  /\  ( 2nd `  B
)  =  ( 2nd `  C ) ) ) )
1612, 15mpbird 156 . 2  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  ->  B  =  C )
1716ex 108 1  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( A  +P.  B
)  =  ( A  +P.  C )  ->  B  =  C )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98    /\ w3a 885    = wceq 1243    e. wcel 1393    C_ wss 2917   ` cfv 4902  (class class class)co 5512   1stc1st 5765   2ndc2nd 5766   P.cnp 6387    +P. cpp 6389
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6400  df-pli 6401  df-mi 6402  df-lti 6403  df-plpq 6440  df-mpq 6441  df-enq 6443  df-nqqs 6444  df-plqqs 6445  df-mqqs 6446  df-1nqqs 6447  df-rq 6448  df-ltnqqs 6449  df-enq0 6520  df-nq0 6521  df-0nq0 6522  df-plq0 6523  df-mq0 6524  df-inp 6562  df-iplp 6564
This theorem is referenced by:  lteupri  6713  ltaprg  6715  enrer  6818  mulcmpblnr  6824  mulgt0sr  6860  srpospr  6865
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