Theorem addcanprg 7927

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 7925	. . . . 5 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) -> ( 1st ` B ) C_ ( 1st ` C ) )
2		3ancomb 1013	. . . . . . 7 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) <-> ( A e. P. /\ C e. P. /\ B e. P. ) )
3		eqcom 2234	. . . . . . 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 7925	. . . . . 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 3254	. . . 4 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) -> ( 1st ` B ) = ( 1st ` C ) )
8		addcanprlemu 7926	. . . . 5 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) -> ( 2nd ` B ) C_ ( 2nd ` C ) )
9		addcanprlemu 7926	. . . . . 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 3254	. . . 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 7783	. . . . 5 |- ( ( B e. P. /\ C e. P. ) -> ( B = C <-> ( ( 1st ` B ) = ( 1st ` C ) /\ ( 2nd ` B ) = ( 2nd ` C ) ) ) )
14	13	3adant1 1042	. . . 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 1005   = wceq 1398   e. wcel 2203   C_ wss 3210   ` cfv 5351  (class class class)co 6049   1stc1st 6331   2ndc2nd 6332   P.cnp 7602   +P. cpp 7604

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-eprel 4409  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-irdg 6600  df-1o 6646  df-2o 6647  df-oadd 6650  df-omul 6651  df-er 6766  df-ec 6768  df-qs 6772  df-ni 7615  df-pli 7616  df-mi 7617  df-lti 7618  df-plpq 7655  df-mpq 7656  df-enq 7658  df-nqqs 7659  df-plqqs 7660  df-mqqs 7661  df-1nqqs 7662  df-rq 7663  df-ltnqqs 7664  df-enq0 7735  df-nq0 7736  df-0nq0 7737  df-plq0 7738  df-mq0 7739  df-inp 7777  df-iplp 7779

This theorem is referenced by:  lteupri  7928  ltaprg  7930  enrer  8046  mulcmpblnr  8052  mulgt0sr  8089  srpospr  8094