Theorem mappsrpr 6370

Description: Mapping from positive signed reals to positive reals.

Hypothesis

Ref	Expression
mappsrpr.1	|- A e. _V

Assertion

Ref	Expression
mappsrpr	|- (0R <R [<.(A +P. 1P), 1P>.] ~R <-> A e. P.)

Proof of Theorem mappsrpr

Step	Hyp	Ref	Expression
1		df-0r 6323	. . 3 |- 0R = [<.1P, 1P>.] ~R
2	1	breq1i 3345	. 2 |- (0R <R [<.(A +P. 1P), 1P>.] ~R <-> [<.1P, 1P>.] ~R <R [<.(A +P. 1P), 1P>.] ~R )
3		1pr 6269	. . . 4 |- 1P e. P.
4	3	elisseti 2301	. . 3 |- 1P e. _V
5		oprex 4907	. . 3 |- (A +P. 1P) e. _V
6	4, 4, 5, 4	ltsrpr 6338	. 2 |- ([<.1P, 1P>.] ~R <R [<.(A +P. 1P), 1P>.] ~R <-> (1P +P. 1P) <P (1P +P. (A +P. 1P)))
7		mappsrpr.1	. . . . . . 7 |- A e. _V
8	7, 4	addcompr 6275	. . . . . 6 |- (A +P. 1P) = (1P +P. A)
9	8	opreq2i 4893	. . . . 5 |- (1P +P. (A +P. 1P)) = (1P +P. (1P +P. A))
10	4, 7	addasspr 6276	. . . . 5 |- ((1P +P. 1P) +P. A) = (1P +P. (1P +P. A))
11	9, 10	eqtr4i 1911	. . . 4 |- (1P +P. (A +P. 1P)) = ((1P +P. 1P) +P. A)
12	11	breq2i 3346	. . 3 |- ((1P +P. 1P) <P (1P +P. (A +P. 1P)) <-> (1P +P. 1P) <P ((1P +P. 1P) +P. A))
13		oprex 4907	. . . . . . 7 |- ((1P +P. 1P) +P. A) e. _V
14		ltrelpr 6253	. . . . . . 7 |- <P C_ (P. X. P.)
15	13, 14	brel 4048	. . . . . 6 |- ((1P +P. 1P) <P ((1P +P. 1P) +P. A) -> ((1P +P. 1P) e. P. /\ ((1P +P. 1P) +P. A) e. P.))
16	15	simprd 352	. . . . 5 |- ((1P +P. 1P) <P ((1P +P. 1P) +P. A) -> ((1P +P. 1P) +P. A) e. P.)
17		dmplp 6267	. . . . . . 7 |- dom +P. = (P. X. P.)
18		0npr 6248	. . . . . . 7 |- -. (/) e. P.
19	7, 17, 18	ndmoprrcl 4979	. . . . . 6 |- (((1P +P. 1P) +P. A) e. P. -> ((1P +P. 1P) e. P. /\ A e. P.))
20	19	simprd 352	. . . . 5 |- (((1P +P. 1P) +P. A) e. P. -> A e. P.)
21	16, 20	syl 12	. . . 4 |- ((1P +P. 1P) <P ((1P +P. 1P) +P. A) -> A e. P.)
22		addclpr 6272	. . . . . 6 |- ((1P e. P. /\ 1P e. P.) -> (1P +P. 1P) e. P.)
23	3, 3, 22	mp2an 761	. . . . 5 |- (1P +P. 1P) e. P.
24		ltaddpr 6292	. . . . 5 |- (((1P +P. 1P) e. P. /\ A e. P.) -> (1P +P. 1P) <P ((1P +P. 1P) +P. A))
25	23, 24	mpan 759	. . . 4 |- (A e. P. -> (1P +P. 1P) <P ((1P +P. 1P) +P. A))
26	21, 25	impbii 174	. . 3 |- ((1P +P. 1P) <P ((1P +P. 1P) +P. A) <-> A e. P.)
27	12, 26	bitri 190	. 2 |- ((1P +P. 1P) <P (1P +P. (A +P. 1P)) <-> A e. P.)
28	2, 6, 27	3bitri 194	1 |- (0R <R [<.(A +P. 1P), 1P>.] ~R <-> A e. P.)

Syntax hints:   <-> wb 163   e. wcel 1300  _Vcvv 2292  <.cop 3046   class class class wbr 3338  (class class class)co 4884  [cec 5316  P.cnp 6137  1Pc1p 6138   +P. cpp 6139   <P cltp 6141   ~R cer 6144  0Rc0r 6146   <R cltr 6151

This theorem is referenced by:  map2psrpr 6372  suppsrlem 6373  suppsr 6374

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-opr 4886  df-oprab 4887  df-1st 5020  df-2nd 5021  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-ltp 6242  df-enr 6318  df-nr 6319  df-ltr 6322  df-0r 6323

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