Theorem srpospr 8063

Description: Mapping from a signed real greater than zero to a positive real. (Contributed by Jim Kingdon, 25-Jun-2021.)

Assertion

Ref	Expression
srpospr	|- ( ( A e. R. /\ 0R <R A ) -> E! x e. P. [ <. ( x +P. 1P ) , 1P >. ] ~R = A )

Distinct variable group:   x, A

Proof of Theorem srpospr

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nr 8007	. . 3 |- R. = ( ( P. X. P. ) /. ~R )
2		breq2 4097	. . . 4 |- ( [ <. a , b >. ] ~R = A -> ( 0R <R [ <. a , b >. ] ~R <-> 0R <R A ) )
3		eqeq2 2241	. . . . 5 |- ( [ <. a , b >. ] ~R = A -> ( [ <. ( x +P. 1P ) , 1P >. ] ~R = [ <. a , b >. ] ~R <-> [ <. ( x +P. 1P ) , 1P >. ] ~R = A ) )
4	3	reubidv 2719	. . . 4 |- ( [ <. a , b >. ] ~R = A -> ( E! x e. P. [ <. ( x +P. 1P ) , 1P >. ] ~R = [ <. a , b >. ] ~R <-> E! x e. P. [ <. ( x +P. 1P ) , 1P >. ] ~R = A ) )
5	2, 4	imbi12d 234	. . 3 |- ( [ <. a , b >. ] ~R = A -> ( ( 0R <R [ <. a , b >. ] ~R -> E! x e. P. [ <. ( x +P. 1P ) , 1P >. ] ~R = [ <. a , b >. ] ~R ) <-> ( 0R <R A -> E! x e. P. [ <. ( x +P. 1P ) , 1P >. ] ~R = A ) ) )
6		gt0srpr 8028	. . . . . . . 8 |- ( 0R <R [ <. a , b >. ] ~R <-> b <P a )
7	6	biimpi 120	. . . . . . 7 |- ( 0R <R [ <. a , b >. ] ~R -> b <P a )
8	7	adantl 277	. . . . . 6 |- ( ( ( a e. P. /\ b e. P. ) /\ 0R <R [ <. a , b >. ] ~R ) -> b <P a )
9		lteupri 7897	. . . . . 6 |- ( b <P a -> E! x e. P. ( b +P. x ) = a )
10	8, 9	syl 14	. . . . 5 |- ( ( ( a e. P. /\ b e. P. ) /\ 0R <R [ <. a , b >. ] ~R ) -> E! x e. P. ( b +P. x ) = a )
11		simpr 110	. . . . . . . . 9 |- ( ( ( ( a e. P. /\ b e. P. ) /\ 0R <R [ <. a , b >. ] ~R ) /\ x e. P. ) -> x e. P. )
12		1pr 7834	. . . . . . . . . 10 |- 1P e. P.
13	12	a1i 9	. . . . . . . . 9 |- ( ( ( ( a e. P. /\ b e. P. ) /\ 0R <R [ <. a , b >. ] ~R ) /\ x e. P. ) -> 1P e. P. )
14		addclpr 7817	. . . . . . . . 9 |- ( ( x e. P. /\ 1P e. P. ) -> ( x +P. 1P ) e. P. )
15	11, 13, 14	syl2anc 411	. . . . . . . 8 |- ( ( ( ( a e. P. /\ b e. P. ) /\ 0R <R [ <. a , b >. ] ~R ) /\ x e. P. ) -> ( x +P. 1P ) e. P. )
16		simplll 535	. . . . . . . 8 |- ( ( ( ( a e. P. /\ b e. P. ) /\ 0R <R [ <. a , b >. ] ~R ) /\ x e. P. ) -> a e. P. )
17		simpllr 536	. . . . . . . 8 |- ( ( ( ( a e. P. /\ b e. P. ) /\ 0R <R [ <. a , b >. ] ~R ) /\ x e. P. ) -> b e. P. )
18		enreceq 8016	. . . . . . . 8 |- ( ( ( ( x +P. 1P ) e. P. /\ 1P e. P. ) /\ ( a e. P. /\ b e. P. ) ) -> ( [ <. ( x +P. 1P ) , 1P >. ] ~R = [ <. a , b >. ] ~R <-> ( ( x +P. 1P ) +P. b ) = ( 1P +P. a ) ) )
19	15, 13, 16, 17, 18	syl22anc 1275	. . . . . . 7 |- ( ( ( ( a e. P. /\ b e. P. ) /\ 0R <R [ <. a , b >. ] ~R ) /\ x e. P. ) -> ( [ <. ( x +P. 1P ) , 1P >. ] ~R = [ <. a , b >. ] ~R <-> ( ( x +P. 1P ) +P. b ) = ( 1P +P. a ) ) )
20		addcomprg 7858	. . . . . . . . . . . 12 |- ( ( x e. P. /\ 1P e. P. ) -> ( x +P. 1P ) = ( 1P +P. x ) )
21	11, 13, 20	syl2anc 411	. . . . . . . . . . 11 |- ( ( ( ( a e. P. /\ b e. P. ) /\ 0R <R [ <. a , b >. ] ~R ) /\ x e. P. ) -> ( x +P. 1P ) = ( 1P +P. x ) )
22	21	oveq1d 6043	. . . . . . . . . 10 |- ( ( ( ( a e. P. /\ b e. P. ) /\ 0R <R [ <. a , b >. ] ~R ) /\ x e. P. ) -> ( ( x +P. 1P ) +P. b ) = ( ( 1P +P. x ) +P. b ) )
23		addassprg 7859	. . . . . . . . . . 11 |- ( ( 1P e. P. /\ x e. P. /\ b e. P. ) -> ( ( 1P +P. x ) +P. b ) = ( 1P +P. ( x +P. b ) ) )
24	13, 11, 17, 23	syl3anc 1274	. . . . . . . . . 10 |- ( ( ( ( a e. P. /\ b e. P. ) /\ 0R <R [ <. a , b >. ] ~R ) /\ x e. P. ) -> ( ( 1P +P. x ) +P. b ) = ( 1P +P. ( x +P. b ) ) )
25	22, 24	eqtrd 2264	. . . . . . . . 9 |- ( ( ( ( a e. P. /\ b e. P. ) /\ 0R <R [ <. a , b >. ] ~R ) /\ x e. P. ) -> ( ( x +P. 1P ) +P. b ) = ( 1P +P. ( x +P. b ) ) )
26	25	eqeq1d 2240	. . . . . . . 8 |- ( ( ( ( a e. P. /\ b e. P. ) /\ 0R <R [ <. a , b >. ] ~R ) /\ x e. P. ) -> ( ( ( x +P. 1P ) +P. b ) = ( 1P +P. a ) <-> ( 1P +P. ( x +P. b ) ) = ( 1P +P. a ) ) )
27		addclpr 7817	. . . . . . . . . . 11 |- ( ( x e. P. /\ b e. P. ) -> ( x +P. b ) e. P. )
28	11, 17, 27	syl2anc 411	. . . . . . . . . 10 |- ( ( ( ( a e. P. /\ b e. P. ) /\ 0R <R [ <. a , b >. ] ~R ) /\ x e. P. ) -> ( x +P. b ) e. P. )
29		addcanprg 7896	. . . . . . . . . 10 |- ( ( 1P e. P. /\ ( x +P. b ) e. P. /\ a e. P. ) -> ( ( 1P +P. ( x +P. b ) ) = ( 1P +P. a ) -> ( x +P. b ) = a ) )
30	13, 28, 16, 29	syl3anc 1274	. . . . . . . . 9 |- ( ( ( ( a e. P. /\ b e. P. ) /\ 0R <R [ <. a , b >. ] ~R ) /\ x e. P. ) -> ( ( 1P +P. ( x +P. b ) ) = ( 1P +P. a ) -> ( x +P. b ) = a ) )
31		oveq2 6036	. . . . . . . . 9 |- ( ( x +P. b ) = a -> ( 1P +P. ( x +P. b ) ) = ( 1P +P. a ) )
32	30, 31	impbid1 142	. . . . . . . 8 |- ( ( ( ( a e. P. /\ b e. P. ) /\ 0R <R [ <. a , b >. ] ~R ) /\ x e. P. ) -> ( ( 1P +P. ( x +P. b ) ) = ( 1P +P. a ) <-> ( x +P. b ) = a ) )
33	26, 32	bitrd 188	. . . . . . 7 |- ( ( ( ( a e. P. /\ b e. P. ) /\ 0R <R [ <. a , b >. ] ~R ) /\ x e. P. ) -> ( ( ( x +P. 1P ) +P. b ) = ( 1P +P. a ) <-> ( x +P. b ) = a ) )
34		addcomprg 7858	. . . . . . . . 9 |- ( ( x e. P. /\ b e. P. ) -> ( x +P. b ) = ( b +P. x ) )
35	11, 17, 34	syl2anc 411	. . . . . . . 8 |- ( ( ( ( a e. P. /\ b e. P. ) /\ 0R <R [ <. a , b >. ] ~R ) /\ x e. P. ) -> ( x +P. b ) = ( b +P. x ) )
36	35	eqeq1d 2240	. . . . . . 7 |- ( ( ( ( a e. P. /\ b e. P. ) /\ 0R <R [ <. a , b >. ] ~R ) /\ x e. P. ) -> ( ( x +P. b ) = a <-> ( b +P. x ) = a ) )
37	19, 33, 36	3bitrrd 215	. . . . . 6 |- ( ( ( ( a e. P. /\ b e. P. ) /\ 0R <R [ <. a , b >. ] ~R ) /\ x e. P. ) -> ( ( b +P. x ) = a <-> [ <. ( x +P. 1P ) , 1P >. ] ~R = [ <. a , b >. ] ~R ) )
38	37	reubidva 2718	. . . . 5 |- ( ( ( a e. P. /\ b e. P. ) /\ 0R <R [ <. a , b >. ] ~R ) -> ( E! x e. P. ( b +P. x ) = a <-> E! x e. P. [ <. ( x +P. 1P ) , 1P >. ] ~R = [ <. a , b >. ] ~R ) )
39	10, 38	mpbid 147	. . . 4 |- ( ( ( a e. P. /\ b e. P. ) /\ 0R <R [ <. a , b >. ] ~R ) -> E! x e. P. [ <. ( x +P. 1P ) , 1P >. ] ~R = [ <. a , b >. ] ~R )
40	39	ex 115	. . 3 |- ( ( a e. P. /\ b e. P. ) -> ( 0R <R [ <. a , b >. ] ~R -> E! x e. P. [ <. ( x +P. 1P ) , 1P >. ] ~R = [ <. a , b >. ] ~R ) )
41	1, 5, 40	ecoptocl 6834	. 2 |- ( A e. R. -> ( 0R <R A -> E! x e. P. [ <. ( x +P. 1P ) , 1P >. ] ~R = A ) )
42	41	imp 124	1 |- ( ( A e. R. /\ 0R <R A ) -> E! x e. P. [ <. ( x +P. 1P ) , 1P >. ] ~R = A )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2202   E!wreu 2513   <.cop 3676   class class class wbr 4093  (class class class)co 6028   [cec 6743   P.cnp 7571   1Pc1p 7572   +P. cpp 7573   <P cltp 7575   ~R cer 7576   R.cnr 7577   0Rc0r 7578   <R cltr 7583

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-eprel 4392  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-1o 6625  df-2o 6626  df-oadd 6629  df-omul 6630  df-er 6745  df-ec 6747  df-qs 6751  df-ni 7584  df-pli 7585  df-mi 7586  df-lti 7587  df-plpq 7624  df-mpq 7625  df-enq 7627  df-nqqs 7628  df-plqqs 7629  df-mqqs 7630  df-1nqqs 7631  df-rq 7632  df-ltnqqs 7633  df-enq0 7704  df-nq0 7705  df-0nq0 7706  df-plq0 7707  df-mq0 7708  df-inp 7746  df-i1p 7747  df-iplp 7748  df-iltp 7750  df-enr 8006  df-nr 8007  df-ltr 8010  df-0r 8011

This theorem is referenced by:  prsrriota  8068  caucvgsrlemcl  8069  caucvgsrlemgt1  8075