Theorem addcanprg 7578

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 7576	. . . . 5 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) -> ( 1st ` B ) C_ ( 1st ` C ) )
2		3ancomb 981	. . . . . . 7 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) <-> ( A e. P. /\ C e. P. /\ B e. P. ) )
3		eqcom 2172	. . . . . . 7 |- ( ( A +P. B ) = ( A +P. C ) <-> ( A +P. C ) = ( A +P. B ) )
4	2, 3	anbi12i 457	. . . . . 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 7576	. . . . . 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 120	. . . . 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 3164	. . . 4 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) -> ( 1st ` B ) = ( 1st ` C ) )
8		addcanprlemu 7577	. . . . 5 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) -> ( 2nd ` B ) C_ ( 2nd ` C ) )
9		addcanprlemu 7577	. . . . . 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 120	. . . . 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 3164	. . . 4 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) -> ( 2nd ` B ) = ( 2nd ` C ) )
12	7, 11	jca 304	. . 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 7434	. . . . 5 |- ( ( B e. P. /\ C e. P. ) -> ( B = C <-> ( ( 1st ` B ) = ( 1st ` C ) /\ ( 2nd ` B ) = ( 2nd ` C ) ) ) )
14	13	3adant1 1010	. . . 4 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( B = C <-> ( ( 1st ` B ) = ( 1st ` C ) /\ ( 2nd ` B ) = ( 2nd ` C ) ) ) )
15	14	adantr 274	. . 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 166	. 2 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) -> B = C )
17	16	ex 114	1 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( ( A +P. B ) = ( A +P. C ) -> B = C ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 973   = wceq 1348   e. wcel 2141   C_ wss 3121   ` cfv 5198  (class class class)co 5853   1stc1st 6117   2ndc2nd 6118   P.cnp 7253   +P. cpp 7255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-iplp 7430

This theorem is referenced by:  lteupri  7579  ltaprg  7581  enrer  7697  mulcmpblnr  7703  mulgt0sr  7740  srpospr  7745