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Theorem addcanprg 6945
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 6943 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  ->  ( 1st `  B
)  C_  ( 1st `  C ) )
2 3ancomb 928 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  <->  ( A  e.  P.  /\  C  e. 
P.  /\  B  e.  P. ) )
3 eqcom 2085 . . . . . . 7  |-  ( ( A  +P.  B )  =  ( A  +P.  C )  <->  ( A  +P.  C )  =  ( A  +P.  B ) )
42, 3anbi12i 448 . . . . . 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 6943 . . . . . 6  |-  ( ( ( A  e.  P.  /\  C  e.  P.  /\  B  e.  P. )  /\  ( A  +P.  C
)  =  ( A  +P.  B ) )  ->  ( 1st `  C
)  C_  ( 1st `  B ) )
64, 5sylbi 119 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  ->  ( 1st `  C
)  C_  ( 1st `  B ) )
71, 6eqssd 3026 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  ->  ( 1st `  B
)  =  ( 1st `  C ) )
8 addcanprlemu 6944 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  ->  ( 2nd `  B
)  C_  ( 2nd `  C ) )
9 addcanprlemu 6944 . . . . . 6  |-  ( ( ( A  e.  P.  /\  C  e.  P.  /\  B  e.  P. )  /\  ( A  +P.  C
)  =  ( A  +P.  B ) )  ->  ( 2nd `  C
)  C_  ( 2nd `  B ) )
104, 9sylbi 119 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  ->  ( 2nd `  C
)  C_  ( 2nd `  B ) )
118, 10eqssd 3026 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  ->  ( 2nd `  B
)  =  ( 2nd `  C ) )
127, 11jca 300 . . 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 6801 . . . . 5  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( B  =  C  <-> 
( ( 1st `  B
)  =  ( 1st `  C )  /\  ( 2nd `  B )  =  ( 2nd `  C
) ) ) )
14133adant1 957 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( B  =  C  <->  ( ( 1st `  B )  =  ( 1st `  C
)  /\  ( 2nd `  B )  =  ( 2nd `  C ) ) ) )
1514adantr 270 . . 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 165 . 2  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  ->  B  =  C )
1716ex 113 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 102    <-> wb 103    /\ w3a 920    = wceq 1285    e. wcel 1434    C_ wss 2983   ` cfv 4953  (class class class)co 5565   1stc1st 5818   2ndc2nd 5819   P.cnp 6620    +P. cpp 6622
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3914  ax-sep 3917  ax-nul 3925  ax-pow 3969  ax-pr 3993  ax-un 4217  ax-setind 4309  ax-iinf 4358
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2613  df-sbc 2826  df-csb 2919  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-nul 3269  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-int 3658  df-iun 3701  df-br 3807  df-opab 3861  df-mpt 3862  df-tr 3897  df-eprel 4073  df-id 4077  df-po 4080  df-iso 4081  df-iord 4150  df-on 4152  df-suc 4155  df-iom 4361  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-rn 4403  df-res 4404  df-ima 4405  df-iota 4918  df-fun 4955  df-fn 4956  df-f 4957  df-f1 4958  df-fo 4959  df-f1o 4960  df-fv 4961  df-ov 5568  df-oprab 5569  df-mpt2 5570  df-1st 5820  df-2nd 5821  df-recs 5976  df-irdg 6041  df-1o 6087  df-2o 6088  df-oadd 6091  df-omul 6092  df-er 6195  df-ec 6197  df-qs 6201  df-ni 6633  df-pli 6634  df-mi 6635  df-lti 6636  df-plpq 6673  df-mpq 6674  df-enq 6676  df-nqqs 6677  df-plqqs 6678  df-mqqs 6679  df-1nqqs 6680  df-rq 6681  df-ltnqqs 6682  df-enq0 6753  df-nq0 6754  df-0nq0 6755  df-plq0 6756  df-mq0 6757  df-inp 6795  df-iplp 6797
This theorem is referenced by:  lteupri  6946  ltaprg  6948  enrer  7051  mulcmpblnr  7057  mulgt0sr  7093  srpospr  7098
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