Theorem prsrlt 7061

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 6842	. . . . 5 |- 1P e. P.
2	1	a1i 9	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> 1P e. P. )
3		simpr 108	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> B e. P. )
4		addassprg 6867	. . . 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 1170	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( ( 1P +P. B ) +P. 1P ) = ( 1P +P. ( B +P. 1P ) ) )
6	5	breq2d 3818	. 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 107	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> A e. P. )
8		ltaprg 6907	. . . 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 1170	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( A <P B <-> ( 1P +P. A ) <P ( 1P +P. B ) ) )
10		addcomprg 6866	. . . . 5 |- ( ( A e. P. /\ 1P e. P. ) -> ( A +P. 1P ) = ( 1P +P. A ) )
11	7, 2, 10	syl2anc 403	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> ( A +P. 1P ) = ( 1P +P. A ) )
12	11	breq1d 3816	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( ( A +P. 1P ) <P ( 1P +P. B ) <-> ( 1P +P. A ) <P ( 1P +P. B ) ) )
13		ltaprg 6907	. . . . 5 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> ( f <P g <-> ( h +P. f ) <P ( h +P. g ) ) )
14	13	adantl 271	. . . 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 6825	. . . . 5 |- ( ( A e. P. /\ 1P e. P. ) -> ( A +P. 1P ) e. P. )
16	7, 2, 15	syl2anc 403	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> ( A +P. 1P ) e. P. )
17		addclpr 6825	. . . . 5 |- ( ( 1P e. P. /\ B e. P. ) -> ( 1P +P. B ) e. P. )
18	2, 3, 17	syl2anc 403	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> ( 1P +P. B ) e. P. )
19		addcomprg 6866	. . . . 5 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) = ( g +P. f ) )
20	19	adantl 271	. . . 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 5722	. . 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 214	. 2 |- ( ( A e. P. /\ B e. P. ) -> ( A <P B <-> ( ( A +P. 1P ) +P. 1P ) <P ( ( 1P +P. B ) +P. 1P ) ) )
23		addclpr 6825	. . . 4 |- ( ( B e. P. /\ 1P e. P. ) -> ( B +P. 1P ) e. P. )
24	3, 2, 23	syl2anc 403	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( B +P. 1P ) e. P. )
25		ltsrprg 7022	. . 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 1171	. 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 218	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 102   <-> wb 103   /\ w3a 920   = wceq 1285   e. wcel 1434   <.cop 3420   class class class wbr 3806  (class class class)co 5564   [cec 6192   P.cnp 6579   1Pc1p 6580   +P. cpp 6581   <P cltp 6583   ~R cer 6584   <R cltr 6591

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3914  ax-sep 3917  ax-nul 3925  ax-pow 3969  ax-pr 3993  ax-un 4217  ax-setind 4309  ax-iinf 4358

This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2826  df-csb 2919  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-nul 3269  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-int 3658  df-iun 3701  df-br 3807  df-opab 3861  df-mpt 3862  df-tr 3897  df-eprel 4073  df-id 4077  df-po 4080  df-iso 4081  df-iord 4150  df-on 4152  df-suc 4155  df-iom 4361  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-rn 4403  df-res 4404  df-ima 4405  df-iota 4918  df-fun 4955  df-fn 4956  df-f 4957  df-f1 4958  df-fo 4959  df-f1o 4960  df-fv 4961  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-1st 5819  df-2nd 5820  df-recs 5975  df-irdg 6040  df-1o 6086  df-2o 6087  df-oadd 6090  df-omul 6091  df-er 6194  df-ec 6196  df-qs 6200  df-ni 6592  df-pli 6593  df-mi 6594  df-lti 6595  df-plpq 6632  df-mpq 6633  df-enq 6635  df-nqqs 6636  df-plqqs 6637  df-mqqs 6638  df-1nqqs 6639  df-rq 6640  df-ltnqqs 6641  df-enq0 6712  df-nq0 6713  df-0nq0 6714  df-plq0 6715  df-mq0 6716  df-inp 6754  df-i1p 6755  df-iplp 6756  df-iltp 6758  df-enr 7001  df-nr 7002  df-ltr 7005

This theorem is referenced by:  caucvgsrlemcau  7067  caucvgsrlembound  7068  caucvgsrlemgt1  7069  ltrennb  7120