Theorem srpospr 7591

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 7535	. . 3 |- R. = ( ( P. X. P. ) /. ~R )
2		breq2 3933	. . . 4 |- ( [ <. a , b >. ] ~R = A -> ( 0R <R [ <. a , b >. ] ~R <-> 0R <R A ) )
3		eqeq2 2149	. . . . 5 |- ( [ <. a , b >. ] ~R = A -> ( [ <. ( x +P. 1P ) , 1P >. ] ~R = [ <. a , b >. ] ~R <-> [ <. ( x +P. 1P ) , 1P >. ] ~R = A ) )
4	3	reubidv 2614	. . . 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 7556	. . . . . . . 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 7425	. . . . . 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 7362	. . . . . . . . . 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 7345	. . . . . . . . 9 |- ( ( x e. P. /\ 1P e. P. ) -> ( x +P. 1P ) e. P. )
15	11, 13, 14	syl2anc 408	. . . . . . . 8 |- ( ( ( ( a e. P. /\ b e. P. ) /\ 0R <R [ <. a , b >. ] ~R ) /\ x e. P. ) -> ( x +P. 1P ) e. P. )
16		simplll 522	. . . . . . . 8 |- ( ( ( ( a e. P. /\ b e. P. ) /\ 0R <R [ <. a , b >. ] ~R ) /\ x e. P. ) -> a e. P. )
17		simpllr 523	. . . . . . . 8 |- ( ( ( ( a e. P. /\ b e. P. ) /\ 0R <R [ <. a , b >. ] ~R ) /\ x e. P. ) -> b e. P. )
18		enreceq 7544	. . . . . . . 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 1217	. . . . . . 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 7386	. . . . . . . . . . . 12 |- ( ( x e. P. /\ 1P e. P. ) -> ( x +P. 1P ) = ( 1P +P. x ) )
21	11, 13, 20	syl2anc 408	. . . . . . . . . . 11 |- ( ( ( ( a e. P. /\ b e. P. ) /\ 0R <R [ <. a , b >. ] ~R ) /\ x e. P. ) -> ( x +P. 1P ) = ( 1P +P. x ) )
22	21	oveq1d 5789	. . . . . . . . . 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 7387	. . . . . . . . . . 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 1216	. . . . . . . . . 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 2172	. . . . . . . . 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 2148	. . . . . . . 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 7345	. . . . . . . . . . 11 |- ( ( x e. P. /\ b e. P. ) -> ( x +P. b ) e. P. )
28	11, 17, 27	syl2anc 408	. . . . . . . . . 10 |- ( ( ( ( a e. P. /\ b e. P. ) /\ 0R <R [ <. a , b >. ] ~R ) /\ x e. P. ) -> ( x +P. b ) e. P. )
29		addcanprg 7424	. . . . . . . . . 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 1216	. . . . . . . . 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 5782	. . . . . . . . 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 7386	. . . . . . . . 9 |- ( ( x e. P. /\ b e. P. ) -> ( x +P. b ) = ( b +P. x ) )
35	11, 17, 34	syl2anc 408	. . . . . . . 8 |- ( ( ( ( a e. P. /\ b e. P. ) /\ 0R <R [ <. a , b >. ] ~R ) /\ x e. P. ) -> ( x +P. b ) = ( b +P. x ) )
36	35	eqeq1d 2148	. . . . . . 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 2613	. . . . 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 6516	. 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 1331   e. wcel 1480   E!wreu 2418   <.cop 3530   class class class wbr 3929  (class class class)co 5774   [cec 6427   P.cnp 7099   1Pc1p 7100   +P. cpp 7101   <P cltp 7103   ~R cer 7104   R.cnr 7105   0Rc0r 7106   <R cltr 7111

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7112  df-pli 7113  df-mi 7114  df-lti 7115  df-plpq 7152  df-mpq 7153  df-enq 7155  df-nqqs 7156  df-plqqs 7157  df-mqqs 7158  df-1nqqs 7159  df-rq 7160  df-ltnqqs 7161  df-enq0 7232  df-nq0 7233  df-0nq0 7234  df-plq0 7235  df-mq0 7236  df-inp 7274  df-i1p 7275  df-iplp 7276  df-iltp 7278  df-enr 7534  df-nr 7535  df-ltr 7538  df-0r 7539

This theorem is referenced by:  prsrriota  7596  caucvgsrlemcl  7597  caucvgsrlemgt1  7603