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Theorem addcanprg 7122
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 7120 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  ->  ( 1st `  B
)  C_  ( 1st `  C ) )
2 3ancomb 930 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  <->  ( A  e.  P.  /\  C  e. 
P.  /\  B  e.  P. ) )
3 eqcom 2087 . . . . . . 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 7120 . . . . . 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 3031 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  ->  ( 1st `  B
)  =  ( 1st `  C ) )
8 addcanprlemu 7121 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A  +P.  B
)  =  ( A  +P.  C ) )  ->  ( 2nd `  B
)  C_  ( 2nd `  C ) )
9 addcanprlemu 7121 . . . . . 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 3031 . . . 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 6978 . . . . 5  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( B  =  C  <-> 
( ( 1st `  B
)  =  ( 1st `  C )  /\  ( 2nd `  B )  =  ( 2nd `  C
) ) ) )
14133adant1 959 . . . 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 922    = wceq 1287    e. wcel 1436    C_ wss 2988   ` cfv 4983  (class class class)co 5615   1stc1st 5868   2ndc2nd 5869   P.cnp 6797    +P. cpp 6799
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3931  ax-sep 3934  ax-nul 3942  ax-pow 3986  ax-pr 4012  ax-un 4236  ax-setind 4328  ax-iinf 4378
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-int 3674  df-iun 3717  df-br 3823  df-opab 3877  df-mpt 3878  df-tr 3914  df-eprel 4092  df-id 4096  df-po 4099  df-iso 4100  df-iord 4169  df-on 4171  df-suc 4174  df-iom 4381  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-dm 4423  df-rn 4424  df-res 4425  df-ima 4426  df-iota 4948  df-fun 4985  df-fn 4986  df-f 4987  df-f1 4988  df-fo 4989  df-f1o 4990  df-fv 4991  df-ov 5618  df-oprab 5619  df-mpt2 5620  df-1st 5870  df-2nd 5871  df-recs 6026  df-irdg 6091  df-1o 6137  df-2o 6138  df-oadd 6141  df-omul 6142  df-er 6246  df-ec 6248  df-qs 6252  df-ni 6810  df-pli 6811  df-mi 6812  df-lti 6813  df-plpq 6850  df-mpq 6851  df-enq 6853  df-nqqs 6854  df-plqqs 6855  df-mqqs 6856  df-1nqqs 6857  df-rq 6858  df-ltnqqs 6859  df-enq0 6930  df-nq0 6931  df-0nq0 6932  df-plq0 6933  df-mq0 6934  df-inp 6972  df-iplp 6974
This theorem is referenced by:  lteupri  7123  ltaprg  7125  enrer  7228  mulcmpblnr  7234  mulgt0sr  7270  srpospr  7275
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