Theorem srpospr 8140

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 8084	. . 3 |- R. = ( ( P. X. P. ) /. ~R )
2		breq2 4129	. . . 4 |- ( [ <. a , b >. ] ~R = A -> ( 0R <R [ <. a , b >. ] ~R <-> 0R <R A ) )
3		eqeq2 2248	. . . . 5 |- ( [ <. a , b >. ] ~R = A -> ( [ <. ( x +P. 1P ) , 1P >. ] ~R = [ <. a , b >. ] ~R <-> [ <. ( x +P. 1P ) , 1P >. ] ~R = A ) )
4	3	reubidv 2737	. . . 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 8105	. . . . . . . 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 7974	. . . . . 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 7911	. . . . . . . . . 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 7894	. . . . . . . . 9 |- ( ( x e. P. /\ 1P e. P. ) -> ( x +P. 1P ) e. P. )
15	11, 13, 14	syl2anc 415	. . . . . . . 8 |- ( ( ( ( a e. P. /\ b e. P. ) /\ 0R <R [ <. a , b >. ] ~R ) /\ x e. P. ) -> ( x +P. 1P ) e. P. )
16		simplll 539	. . . . . . . 8 |- ( ( ( ( a e. P. /\ b e. P. ) /\ 0R <R [ <. a , b >. ] ~R ) /\ x e. P. ) -> a e. P. )
17		simpllr 540	. . . . . . . 8 |- ( ( ( ( a e. P. /\ b e. P. ) /\ 0R <R [ <. a , b >. ] ~R ) /\ x e. P. ) -> b e. P. )
18		enreceq 8093	. . . . . . . 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 1279	. . . . . . 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 7935	. . . . . . . . . . . 12 |- ( ( x e. P. /\ 1P e. P. ) -> ( x +P. 1P ) = ( 1P +P. x ) )
21	11, 13, 20	syl2anc 415	. . . . . . . . . . 11 |- ( ( ( ( a e. P. /\ b e. P. ) /\ 0R <R [ <. a , b >. ] ~R ) /\ x e. P. ) -> ( x +P. 1P ) = ( 1P +P. x ) )
22	21	oveq1d 6090	. . . . . . . . . 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 7936	. . . . . . . . . . 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 1278	. . . . . . . . . 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 2271	. . . . . . . . 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 2247	. . . . . . . 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 7894	. . . . . . . . . . 11 |- ( ( x e. P. /\ b e. P. ) -> ( x +P. b ) e. P. )
28	11, 17, 27	syl2anc 415	. . . . . . . . . 10 |- ( ( ( ( a e. P. /\ b e. P. ) /\ 0R <R [ <. a , b >. ] ~R ) /\ x e. P. ) -> ( x +P. b ) e. P. )
29		addcanprg 7973	. . . . . . . . . 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 1278	. . . . . . . . 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 6083	. . . . . . . . 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 7935	. . . . . . . . 9 |- ( ( x e. P. /\ b e. P. ) -> ( x +P. b ) = ( b +P. x ) )
35	11, 17, 34	syl2anc 415	. . . . . . . 8 |- ( ( ( ( a e. P. /\ b e. P. ) /\ 0R <R [ <. a , b >. ] ~R ) /\ x e. P. ) -> ( x +P. b ) = ( b +P. x ) )
36	35	eqeq1d 2247	. . . . . . 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 2736	. . . . 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 6886	. 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 1402   e. wcel 2209   E!wreu 2530   <.cop 3708   class class class wbr 4125  (class class class)co 6075   [cec 6795   P.cnp 7648   1Pc1p 7649   +P. cpp 7650   <P cltp 7652   ~R cer 7653   R.cnr 7654   0Rc0r 7655   <R cltr 7660

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 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 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem 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-rmo 2536  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 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-eprel 4429  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-1o 6677  df-2o 6678  df-oadd 6681  df-omul 6682  df-er 6797  df-ec 6799  df-qs 6803  df-ni 7661  df-pli 7662  df-mi 7663  df-lti 7664  df-plpq 7701  df-mpq 7702  df-enq 7704  df-nqqs 7705  df-plqqs 7706  df-mqqs 7707  df-1nqqs 7708  df-rq 7709  df-ltnqqs 7710  df-enq0 7781  df-nq0 7782  df-0nq0 7783  df-plq0 7784  df-mq0 7785  df-inp 7823  df-i1p 7824  df-iplp 7825  df-iltp 7827  df-enr 8083  df-nr 8084  df-ltr 8087  df-0r 8088

This theorem is referenced by:  prsrriota  8145  caucvgsrlemcl  8146  caucvgsrlemgt1  8152