Theorem addcanprg 7989

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

Step	Hyp	Ref	Expression
1		addcanprleml 7987	. . . . 5 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) -> ( 1st ` B ) C_ ( 1st ` C ) )
2		3ancomb 1017	. . . . . . 7 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) <-> ( A e. P. /\ C e. P. /\ B e. P. ) )
3		eqcom 2240	. . . . . . 7 |- ( ( A +P. B ) = ( A +P. C ) <-> ( A +P. C ) = ( A +P. B ) )
4	2, 3	anbi12i 464	. . . . . 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 7987	. . . . . 6 |- ( ( ( A e. P. /\ C e. P. /\ B e. P. ) /\ ( A +P. C ) = ( A +P. B ) ) -> ( 1st ` C ) C_ ( 1st ` B ) )
6	4, 5	sylbi 121	. . . . 5 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) -> ( 1st ` C ) C_ ( 1st ` B ) )
7	1, 6	eqssd 3265	. . . 4 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) -> ( 1st ` B ) = ( 1st ` C ) )
8		addcanprlemu 7988	. . . . 5 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) -> ( 2nd ` B ) C_ ( 2nd ` C ) )
9		addcanprlemu 7988	. . . . . 6 |- ( ( ( A e. P. /\ C e. P. /\ B e. P. ) /\ ( A +P. C ) = ( A +P. B ) ) -> ( 2nd ` C ) C_ ( 2nd ` B ) )
10	4, 9	sylbi 121	. . . . 5 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) -> ( 2nd ` C ) C_ ( 2nd ` B ) )
11	8, 10	eqssd 3265	. . . 4 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) -> ( 2nd ` B ) = ( 2nd ` C ) )
12	7, 11	jca 306	. . 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 7845	. . . . 5 |- ( ( B e. P. /\ C e. P. ) -> ( B = C <-> ( ( 1st ` B ) = ( 1st ` C ) /\ ( 2nd ` B ) = ( 2nd ` C ) ) ) )
14	13	3adant1 1046	. . . 4 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( B = C <-> ( ( 1st ` B ) = ( 1st ` C ) /\ ( 2nd ` B ) = ( 2nd ` C ) ) ) )
15	14	adantr 276	. . 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 ) ) ) )
16	12, 15	mpbird 167	. 2 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) -> B = C )
17	16	ex 115	1 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( ( A +P. B ) = ( A +P. C ) -> B = C ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   C_ wss 3220   ` cfv 5380  (class class class)co 6088   1stc1st 6376   2ndc2nd 6377   P.cnp 7664   +P. cpp 7666

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-eprel 4436  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-1o 6691  df-2o 6692  df-oadd 6695  df-omul 6696  df-er 6811  df-ec 6813  df-qs 6817  df-ni 7677  df-pli 7678  df-mi 7679  df-lti 7680  df-plpq 7717  df-mpq 7718  df-enq 7720  df-nqqs 7721  df-plqqs 7722  df-mqqs 7723  df-1nqqs 7724  df-rq 7725  df-ltnqqs 7726  df-enq0 7797  df-nq0 7798  df-0nq0 7799  df-plq0 7800  df-mq0 7801  df-inp 7839  df-iplp 7841

This theorem is used by:  lteupri  7990  ltaprg  7992  enrer  8108  mulcmpblnr  8114  mulgt0sr  8151  srpospr  8156