Theorem addcanprg 7629

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 7627	. . . . 5 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) -> ( 1st ` B ) C_ ( 1st ` C ) )
2		3ancomb 987	. . . . . . 7 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) <-> ( A e. P. /\ C e. P. /\ B e. P. ) )
3		eqcom 2189	. . . . . . 7 |- ( ( A +P. B ) = ( A +P. C ) <-> ( A +P. C ) = ( A +P. B ) )
4	2, 3	anbi12i 460	. . . . . 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 7627	. . . . . 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 3184	. . . 4 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) -> ( 1st ` B ) = ( 1st ` C ) )
8		addcanprlemu 7628	. . . . 5 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) -> ( 2nd ` B ) C_ ( 2nd ` C ) )
9		addcanprlemu 7628	. . . . . 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 3184	. . . 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 7485	. . . . 5 |- ( ( B e. P. /\ C e. P. ) -> ( B = C <-> ( ( 1st ` B ) = ( 1st ` C ) /\ ( 2nd ` B ) = ( 2nd ` C ) ) ) )
14	13	3adant1 1016	. . . 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 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 979   = wceq 1363   e. wcel 2158   C_ wss 3141   ` cfv 5228  (class class class)co 5888   1stc1st 6153   2ndc2nd 6154   P.cnp 7304   +P. cpp 7306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-eprel 4301  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-recs 6320  df-irdg 6385  df-1o 6431  df-2o 6432  df-oadd 6435  df-omul 6436  df-er 6549  df-ec 6551  df-qs 6555  df-ni 7317  df-pli 7318  df-mi 7319  df-lti 7320  df-plpq 7357  df-mpq 7358  df-enq 7360  df-nqqs 7361  df-plqqs 7362  df-mqqs 7363  df-1nqqs 7364  df-rq 7365  df-ltnqqs 7366  df-enq0 7437  df-nq0 7438  df-0nq0 7439  df-plq0 7440  df-mq0 7441  df-inp 7479  df-iplp 7481

This theorem is referenced by:  lteupri  7630  ltaprg  7632  enrer  7748  mulcmpblnr  7754  mulgt0sr  7791  srpospr  7796