Theorem srpospr 7705

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 7649	. . 3 |- R. = ( ( P. X. P. ) /. ~R )
2		breq2 3971	. . . 4 |- ( [ <. a , b >. ] ~R = A -> ( 0R <R [ <. a , b >. ] ~R <-> 0R <R A ) )
3		eqeq2 2167	. . . . 5 |- ( [ <. a , b >. ] ~R = A -> ( [ <. ( x +P. 1P ) , 1P >. ] ~R = [ <. a , b >. ] ~R <-> [ <. ( x +P. 1P ) , 1P >. ] ~R = A ) )
4	3	reubidv 2640	. . . 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 233	. . 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 7670	. . . . . . . 8 |- ( 0R <R [ <. a , b >. ] ~R <-> b <P a )
7	6	biimpi 119	. . . . . . 7 |- ( 0R <R [ <. a , b >. ] ~R -> b <P a )
8	7	adantl 275	. . . . . 6 |- ( ( ( a e. P. /\ b e. P. ) /\ 0R <R [ <. a , b >. ] ~R ) -> b <P a )
9		lteupri 7539	. . . . . 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 109	. . . . . . . . 9 |- ( ( ( ( a e. P. /\ b e. P. ) /\ 0R <R [ <. a , b >. ] ~R ) /\ x e. P. ) -> x e. P. )
12		1pr 7476	. . . . . . . . . 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 7459	. . . . . . . . 9 |- ( ( x e. P. /\ 1P e. P. ) -> ( x +P. 1P ) e. P. )
15	11, 13, 14	syl2anc 409	. . . . . . . 8 |- ( ( ( ( a e. P. /\ b e. P. ) /\ 0R <R [ <. a , b >. ] ~R ) /\ x e. P. ) -> ( x +P. 1P ) e. P. )
16		simplll 523	. . . . . . . 8 |- ( ( ( ( a e. P. /\ b e. P. ) /\ 0R <R [ <. a , b >. ] ~R ) /\ x e. P. ) -> a e. P. )
17		simpllr 524	. . . . . . . 8 |- ( ( ( ( a e. P. /\ b e. P. ) /\ 0R <R [ <. a , b >. ] ~R ) /\ x e. P. ) -> b e. P. )
18		enreceq 7658	. . . . . . . 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 1221	. . . . . . 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 7500	. . . . . . . . . . . 12 |- ( ( x e. P. /\ 1P e. P. ) -> ( x +P. 1P ) = ( 1P +P. x ) )
21	11, 13, 20	syl2anc 409	. . . . . . . . . . 11 |- ( ( ( ( a e. P. /\ b e. P. ) /\ 0R <R [ <. a , b >. ] ~R ) /\ x e. P. ) -> ( x +P. 1P ) = ( 1P +P. x ) )
22	21	oveq1d 5841	. . . . . . . . . 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 7501	. . . . . . . . . . 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 1220	. . . . . . . . . 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 2190	. . . . . . . . 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 2166	. . . . . . . 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 7459	. . . . . . . . . . 11 |- ( ( x e. P. /\ b e. P. ) -> ( x +P. b ) e. P. )
28	11, 17, 27	syl2anc 409	. . . . . . . . . 10 |- ( ( ( ( a e. P. /\ b e. P. ) /\ 0R <R [ <. a , b >. ] ~R ) /\ x e. P. ) -> ( x +P. b ) e. P. )
29		addcanprg 7538	. . . . . . . . . 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 1220	. . . . . . . . 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 5834	. . . . . . . . 9 |- ( ( x +P. b ) = a -> ( 1P +P. ( x +P. b ) ) = ( 1P +P. a ) )
32	30, 31	impbid1 141	. . . . . . . 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 187	. . . . . . 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 7500	. . . . . . . . 9 |- ( ( x e. P. /\ b e. P. ) -> ( x +P. b ) = ( b +P. x ) )
35	11, 17, 34	syl2anc 409	. . . . . . . 8 |- ( ( ( ( a e. P. /\ b e. P. ) /\ 0R <R [ <. a , b >. ] ~R ) /\ x e. P. ) -> ( x +P. b ) = ( b +P. x ) )
36	35	eqeq1d 2166	. . . . . . 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 214	. . . . . 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 2639	. . . . 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 146	. . . 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 114	. . 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 6569	. 2 |- ( A e. R. -> ( 0R <R A -> E! x e. P. [ <. ( x +P. 1P ) , 1P >. ] ~R = A ) )
42	41	imp 123	1 |- ( ( A e. R. /\ 0R <R A ) -> E! x e. P. [ <. ( x +P. 1P ) , 1P >. ] ~R = A )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1335   e. wcel 2128   E!wreu 2437   <.cop 3564   class class class wbr 3967  (class class class)co 5826   [cec 6480   P.cnp 7213   1Pc1p 7214   +P. cpp 7215   <P cltp 7217   ~R cer 7218   R.cnr 7219   0Rc0r 7220   <R cltr 7225

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4081  ax-sep 4084  ax-nul 4092  ax-pow 4137  ax-pr 4171  ax-un 4395  ax-setind 4498  ax-iinf 4549

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4028  df-mpt 4029  df-tr 4065  df-eprel 4251  df-id 4255  df-po 4258  df-iso 4259  df-iord 4328  df-on 4330  df-suc 4333  df-iom 4552  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-iota 5137  df-fun 5174  df-fn 5175  df-f 5176  df-f1 5177  df-fo 5178  df-f1o 5179  df-fv 5180  df-ov 5829  df-oprab 5830  df-mpo 5831  df-1st 6090  df-2nd 6091  df-recs 6254  df-irdg 6319  df-1o 6365  df-2o 6366  df-oadd 6369  df-omul 6370  df-er 6482  df-ec 6484  df-qs 6488  df-ni 7226  df-pli 7227  df-mi 7228  df-lti 7229  df-plpq 7266  df-mpq 7267  df-enq 7269  df-nqqs 7270  df-plqqs 7271  df-mqqs 7272  df-1nqqs 7273  df-rq 7274  df-ltnqqs 7275  df-enq0 7346  df-nq0 7347  df-0nq0 7348  df-plq0 7349  df-mq0 7350  df-inp 7388  df-i1p 7389  df-iplp 7390  df-iltp 7392  df-enr 7648  df-nr 7649  df-ltr 7652  df-0r 7653

This theorem is referenced by:  prsrriota  7710  caucvgsrlemcl  7711  caucvgsrlemgt1  7717