Theorem prsrlt 7786

Description: Mapping from positive real ordering to signed real ordering. (Contributed by Jim Kingdon, 29-Jun-2021.)

Assertion

Ref	Expression
prsrlt	|- ( ( A e. P. /\ B e. P. ) -> ( A <P B <-> [ <. ( A +P. 1P ) , 1P >. ] ~R <R [ <. ( B +P. 1P ) , 1P >. ] ~R ) )

Proof of Theorem prsrlt

Dummy variables f g h are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1pr 7553	. . . . 5 |- 1P e. P.
2	1	a1i 9	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> 1P e. P. )
3		simpr 110	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> B e. P. )
4		addassprg 7578	. . . 4 |- ( ( 1P e. P. /\ B e. P. /\ 1P e. P. ) -> ( ( 1P +P. B ) +P. 1P ) = ( 1P +P. ( B +P. 1P ) ) )
5	2, 3, 2, 4	syl3anc 1238	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( ( 1P +P. B ) +P. 1P ) = ( 1P +P. ( B +P. 1P ) ) )
6	5	breq2d 4016	. 2 |- ( ( A e. P. /\ B e. P. ) -> ( ( ( A +P. 1P ) +P. 1P ) <P ( ( 1P +P. B ) +P. 1P ) <-> ( ( A +P. 1P ) +P. 1P ) <P ( 1P +P. ( B +P. 1P ) ) ) )
7		simpl 109	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> A e. P. )
8		ltaprg 7618	. . . 4 |- ( ( A e. P. /\ B e. P. /\ 1P e. P. ) -> ( A <P B <-> ( 1P +P. A ) <P ( 1P +P. B ) ) )
9	7, 3, 2, 8	syl3anc 1238	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( A <P B <-> ( 1P +P. A ) <P ( 1P +P. B ) ) )
10		addcomprg 7577	. . . . 5 |- ( ( A e. P. /\ 1P e. P. ) -> ( A +P. 1P ) = ( 1P +P. A ) )
11	7, 2, 10	syl2anc 411	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> ( A +P. 1P ) = ( 1P +P. A ) )
12	11	breq1d 4014	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( ( A +P. 1P ) <P ( 1P +P. B ) <-> ( 1P +P. A ) <P ( 1P +P. B ) ) )
13		ltaprg 7618	. . . . 5 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> ( f <P g <-> ( h +P. f ) <P ( h +P. g ) ) )
14	13	adantl 277	. . . 4 |- ( ( ( A e. P. /\ B e. P. ) /\ ( f e. P. /\ g e. P. /\ h e. P. ) ) -> ( f <P g <-> ( h +P. f ) <P ( h +P. g ) ) )
15		addclpr 7536	. . . . 5 |- ( ( A e. P. /\ 1P e. P. ) -> ( A +P. 1P ) e. P. )
16	7, 2, 15	syl2anc 411	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> ( A +P. 1P ) e. P. )
17		addclpr 7536	. . . . 5 |- ( ( 1P e. P. /\ B e. P. ) -> ( 1P +P. B ) e. P. )
18	2, 3, 17	syl2anc 411	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> ( 1P +P. B ) e. P. )
19		addcomprg 7577	. . . . 5 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) = ( g +P. f ) )
20	19	adantl 277	. . . 4 |- ( ( ( A e. P. /\ B e. P. ) /\ ( f e. P. /\ g e. P. ) ) -> ( f +P. g ) = ( g +P. f ) )
21	14, 16, 18, 2, 20	caovord2d 6044	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( ( A +P. 1P ) <P ( 1P +P. B ) <-> ( ( A +P. 1P ) +P. 1P ) <P ( ( 1P +P. B ) +P. 1P ) ) )
22	9, 12, 21	3bitr2d 216	. 2 |- ( ( A e. P. /\ B e. P. ) -> ( A <P B <-> ( ( A +P. 1P ) +P. 1P ) <P ( ( 1P +P. B ) +P. 1P ) ) )
23		addclpr 7536	. . . 4 |- ( ( B e. P. /\ 1P e. P. ) -> ( B +P. 1P ) e. P. )
24	3, 2, 23	syl2anc 411	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( B +P. 1P ) e. P. )
25		ltsrprg 7746	. . 3 |- ( ( ( ( A +P. 1P ) e. P. /\ 1P e. P. ) /\ ( ( B +P. 1P ) e. P. /\ 1P e. P. ) ) -> ( [ <. ( A +P. 1P ) , 1P >. ] ~R <R [ <. ( B +P. 1P ) , 1P >. ] ~R <-> ( ( A +P. 1P ) +P. 1P ) <P ( 1P +P. ( B +P. 1P ) ) ) )
26	16, 2, 24, 2, 25	syl22anc 1239	. 2 |- ( ( A e. P. /\ B e. P. ) -> ( [ <. ( A +P. 1P ) , 1P >. ] ~R <R [ <. ( B +P. 1P ) , 1P >. ] ~R <-> ( ( A +P. 1P ) +P. 1P ) <P ( 1P +P. ( B +P. 1P ) ) ) )
27	6, 22, 26	3bitr4d 220	1 |- ( ( A e. P. /\ B e. P. ) -> ( A <P B <-> [ <. ( A +P. 1P ) , 1P >. ] ~R <R [ <. ( B +P. 1P ) , 1P >. ] ~R ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 978   = wceq 1353   e. wcel 2148   <.cop 3596   class class class wbr 4004  (class class class)co 5875   [cec 6533   P.cnp 7290   1Pc1p 7291   +P. cpp 7292   <P cltp 7294   ~R cer 7295   <R cltr 7302

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-eprel 4290  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-irdg 6371  df-1o 6417  df-2o 6418  df-oadd 6421  df-omul 6422  df-er 6535  df-ec 6537  df-qs 6541  df-ni 7303  df-pli 7304  df-mi 7305  df-lti 7306  df-plpq 7343  df-mpq 7344  df-enq 7346  df-nqqs 7347  df-plqqs 7348  df-mqqs 7349  df-1nqqs 7350  df-rq 7351  df-ltnqqs 7352  df-enq0 7423  df-nq0 7424  df-0nq0 7425  df-plq0 7426  df-mq0 7427  df-inp 7465  df-i1p 7466  df-iplp 7467  df-iltp 7469  df-enr 7725  df-nr 7726  df-ltr 7729

This theorem is referenced by:  caucvgsrlemcau  7792  caucvgsrlembound  7793  caucvgsrlemgt1  7794  ltrennb  7853