Theorem prsrlt 8154

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 7921	. . . . 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 7946	. . . 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 1278	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( ( 1P +P. B ) +P. 1P ) = ( 1P +P. ( B +P. 1P ) ) )
6	5	breq2d 4142	. 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 7986	. . . 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 1278	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( A <P B <-> ( 1P +P. A ) <P ( 1P +P. B ) ) )
10		addcomprg 7945	. . . . 5 |- ( ( A e. P. /\ 1P e. P. ) -> ( A +P. 1P ) = ( 1P +P. A ) )
11	7, 2, 10	syl2anc 415	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> ( A +P. 1P ) = ( 1P +P. A ) )
12	11	breq1d 4140	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( ( A +P. 1P ) <P ( 1P +P. B ) <-> ( 1P +P. A ) <P ( 1P +P. B ) ) )
13		ltaprg 7986	. . . . 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 7904	. . . . 5 |- ( ( A e. P. /\ 1P e. P. ) -> ( A +P. 1P ) e. P. )
16	7, 2, 15	syl2anc 415	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> ( A +P. 1P ) e. P. )
17		addclpr 7904	. . . . 5 |- ( ( 1P e. P. /\ B e. P. ) -> ( 1P +P. B ) e. P. )
18	2, 3, 17	syl2anc 415	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> ( 1P +P. B ) e. P. )
19		addcomprg 7945	. . . . 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 6259	. . 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 7904	. . . 4 |- ( ( B e. P. /\ 1P e. P. ) -> ( B +P. 1P ) e. P. )
24	3, 2, 23	syl2anc 415	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( B +P. 1P ) e. P. )
25		ltsrprg 8114	. . 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 1279	. 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 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   <.cop 3712   class class class wbr 4130  (class class class)co 6085   [cec 6805   P.cnp 7658   1Pc1p 7659   +P. cpp 7660   <P cltp 7662   ~R cer 7663   <R cltr 7670

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-eprel 4434  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-1o 6687  df-2o 6688  df-oadd 6691  df-omul 6692  df-er 6807  df-ec 6809  df-qs 6813  df-ni 7671  df-pli 7672  df-mi 7673  df-lti 7674  df-plpq 7711  df-mpq 7712  df-enq 7714  df-nqqs 7715  df-plqqs 7716  df-mqqs 7717  df-1nqqs 7718  df-rq 7719  df-ltnqqs 7720  df-enq0 7791  df-nq0 7792  df-0nq0 7793  df-plq0 7794  df-mq0 7795  df-inp 7833  df-i1p 7834  df-iplp 7835  df-iltp 7837  df-enr 8093  df-nr 8094  df-ltr 8097

This theorem is used by:  caucvgsrlemcau  8160  caucvgsrlembound  8161  caucvgsrlemgt1  8162  ltrennb  8221