Theorem prsrlt 7728

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 7495	. . . . 5 |- 1P e. P.
2	1	a1i 9	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> 1P e. P. )
3		simpr 109	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> B e. P. )
4		addassprg 7520	. . . 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 1228	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( ( 1P +P. B ) +P. 1P ) = ( 1P +P. ( B +P. 1P ) ) )
6	5	breq2d 3994	. 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 108	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> A e. P. )
8		ltaprg 7560	. . . 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 1228	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( A <P B <-> ( 1P +P. A ) <P ( 1P +P. B ) ) )
10		addcomprg 7519	. . . . 5 |- ( ( A e. P. /\ 1P e. P. ) -> ( A +P. 1P ) = ( 1P +P. A ) )
11	7, 2, 10	syl2anc 409	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> ( A +P. 1P ) = ( 1P +P. A ) )
12	11	breq1d 3992	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( ( A +P. 1P ) <P ( 1P +P. B ) <-> ( 1P +P. A ) <P ( 1P +P. B ) ) )
13		ltaprg 7560	. . . . 5 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> ( f <P g <-> ( h +P. f ) <P ( h +P. g ) ) )
14	13	adantl 275	. . . 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 7478	. . . . 5 |- ( ( A e. P. /\ 1P e. P. ) -> ( A +P. 1P ) e. P. )
16	7, 2, 15	syl2anc 409	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> ( A +P. 1P ) e. P. )
17		addclpr 7478	. . . . 5 |- ( ( 1P e. P. /\ B e. P. ) -> ( 1P +P. B ) e. P. )
18	2, 3, 17	syl2anc 409	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> ( 1P +P. B ) e. P. )
19		addcomprg 7519	. . . . 5 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) = ( g +P. f ) )
20	19	adantl 275	. . . 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 6011	. . 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 215	. 2 |- ( ( A e. P. /\ B e. P. ) -> ( A <P B <-> ( ( A +P. 1P ) +P. 1P ) <P ( ( 1P +P. B ) +P. 1P ) ) )
23		addclpr 7478	. . . 4 |- ( ( B e. P. /\ 1P e. P. ) -> ( B +P. 1P ) e. P. )
24	3, 2, 23	syl2anc 409	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( B +P. 1P ) e. P. )
25		ltsrprg 7688	. . 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 1229	. 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 219	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 103   <-> wb 104   /\ w3a 968   = wceq 1343   e. wcel 2136   <.cop 3579   class class class wbr 3982  (class class class)co 5842   [cec 6499   P.cnp 7232   1Pc1p 7233   +P. cpp 7234   <P cltp 7236   ~R cer 7237   <R cltr 7244

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-2o 6385  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-pli 7246  df-mi 7247  df-lti 7248  df-plpq 7285  df-mpq 7286  df-enq 7288  df-nqqs 7289  df-plqqs 7290  df-mqqs 7291  df-1nqqs 7292  df-rq 7293  df-ltnqqs 7294  df-enq0 7365  df-nq0 7366  df-0nq0 7367  df-plq0 7368  df-mq0 7369  df-inp 7407  df-i1p 7408  df-iplp 7409  df-iltp 7411  df-enr 7667  df-nr 7668  df-ltr 7671

This theorem is referenced by:  caucvgsrlemcau  7734  caucvgsrlembound  7735  caucvgsrlemgt1  7736  ltrennb  7795