Theorem prsrlt 8007

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 7774	. . . . 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 7799	. . . 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 1273	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( ( 1P +P. B ) +P. 1P ) = ( 1P +P. ( B +P. 1P ) ) )
6	5	breq2d 4100	. 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 7839	. . . 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 1273	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( A <P B <-> ( 1P +P. A ) <P ( 1P +P. B ) ) )
10		addcomprg 7798	. . . . 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 4098	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( ( A +P. 1P ) <P ( 1P +P. B ) <-> ( 1P +P. A ) <P ( 1P +P. B ) ) )
13		ltaprg 7839	. . . . 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 7757	. . . . 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 7757	. . . . 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 7798	. . . . 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 6192	. . 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 7757	. . . 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 7967	. . 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 1274	. 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 1004   = wceq 1397   e. wcel 2202   <.cop 3672   class class class wbr 4088  (class class class)co 6018   [cec 6700   P.cnp 7511   1Pc1p 7512   +P. cpp 7513   <P cltp 7515   ~R cer 7516   <R cltr 7523

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-eprel 4386  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-1o 6582  df-2o 6583  df-oadd 6586  df-omul 6587  df-er 6702  df-ec 6704  df-qs 6708  df-ni 7524  df-pli 7525  df-mi 7526  df-lti 7527  df-plpq 7564  df-mpq 7565  df-enq 7567  df-nqqs 7568  df-plqqs 7569  df-mqqs 7570  df-1nqqs 7571  df-rq 7572  df-ltnqqs 7573  df-enq0 7644  df-nq0 7645  df-0nq0 7646  df-plq0 7647  df-mq0 7648  df-inp 7686  df-i1p 7687  df-iplp 7688  df-iltp 7690  df-enr 7946  df-nr 7947  df-ltr 7950

This theorem is referenced by:  caucvgsrlemcau  8013  caucvgsrlembound  8014  caucvgsrlemgt1  8015  ltrennb  8074