Theorem addcanprg 7676

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 7674	. . . . 5 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) -> ( 1st ` B ) C_ ( 1st ` C ) )
2		3ancomb 988	. . . . . . 7 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) <-> ( A e. P. /\ C e. P. /\ B e. P. ) )
3		eqcom 2195	. . . . . . 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 7674	. . . . . 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 3196	. . . 4 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) -> ( 1st ` B ) = ( 1st ` C ) )
8		addcanprlemu 7675	. . . . 5 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) -> ( 2nd ` B ) C_ ( 2nd ` C ) )
9		addcanprlemu 7675	. . . . . 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 3196	. . . 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 7532	. . . . 5 |- ( ( B e. P. /\ C e. P. ) -> ( B = C <-> ( ( 1st ` B ) = ( 1st ` C ) /\ ( 2nd ` B ) = ( 2nd ` C ) ) ) )
14	13	3adant1 1017	. . . 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 980   = wceq 1364   e. wcel 2164   C_ wss 3153   ` cfv 5254  (class class class)co 5918   1stc1st 6191   2ndc2nd 6192   P.cnp 7351   +P. cpp 7353

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-eprel 4320  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-1o 6469  df-2o 6470  df-oadd 6473  df-omul 6474  df-er 6587  df-ec 6589  df-qs 6593  df-ni 7364  df-pli 7365  df-mi 7366  df-lti 7367  df-plpq 7404  df-mpq 7405  df-enq 7407  df-nqqs 7408  df-plqqs 7409  df-mqqs 7410  df-1nqqs 7411  df-rq 7412  df-ltnqqs 7413  df-enq0 7484  df-nq0 7485  df-0nq0 7486  df-plq0 7487  df-mq0 7488  df-inp 7526  df-iplp 7528

This theorem is referenced by:  lteupri  7677  ltaprg  7679  enrer  7795  mulcmpblnr  7801  mulgt0sr  7838  srpospr  7843