Theorem mappsrpr 5198

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 5151	. . 3 |- 0R = [<.1P, 1P>.] ~R
2	1	breq1i 2621	. 2 |- (0R <R [<.(A +P. 1P), 1P>.] ~R <-> [<.1P, 1P>.] ~R <R [<.(A +P. 1P), 1P>.] ~R )
3		1pr 5097	. . . 4 |- 1P e. P.
4	3	elisseti 1814	. . 3 |- 1P e. V
5		oprex 3974	. . 3 |- (A +P. 1P) e. V
6	4, 4, 5, 4	ltsrpr 5166	. 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 5103	. . . . . 6 |- (A +P. 1P) = (1P +P. A)
9	8	opreq2i 3963	. . . . 5 |- (1P +P. (A +P. 1P)) = (1P +P. (1P +P. A))
10	4, 7	addasspr 5104	. . . . 5 |- ((1P +P. 1P) +P. A) = (1P +P. (1P +P. A))
11	9, 10	eqtr4 1495	. . . 4 |- (1P +P. (A +P. 1P)) = ((1P +P. 1P) +P. A)
12	11	breq2i 2622	. . 3 |- ((1P +P. 1P) <P (1P +P. (A +P. 1P)) <-> (1P +P. 1P) <P ((1P +P. 1P) +P. A))
13		oprex 3974	. . . . . . 7 |- ((1P +P. 1P) +P. A) e. V
14		ltrelpr 5081	. . . . . . 7 |- <P (_ (P. X. P.)
15	13, 14	brel 3218	. . . . . 6 |- ((1P +P. 1P) <P ((1P +P. 1P) +P. A) -> ((1P +P. 1P) e. P. /\ ((1P +P. 1P) +P. A) e. P.))
16	15	pm3.27d 325	. . . . 5 |- ((1P +P. 1P) <P ((1P +P. 1P) +P. A) -> ((1P +P. 1P) +P. A) e. P.)
17		dmplp 5095	. . . . . . 7 |- dom +P. = (P. X. P.)
18		0npr 5076	. . . . . . 7 |- -. (/) e. P.
19	7, 17, 18	ndmoprrcl 4038	. . . . . 6 |- (((1P +P. 1P) +P. A) e. P. -> ((1P +P. 1P) e. P. /\ A e. P.))
20	19	pm3.27d 325	. . . . 5 |- (((1P +P. 1P) +P. A) e. P. -> A e. P.)
21	16, 20	syl 10	. . . 4 |- ((1P +P. 1P) <P ((1P +P. 1P) +P. A) -> A e. P.)
22		addclpr 5100	. . . . . 6 |- ((1P e. P. /\ 1P e. P.) -> (1P +P. 1P) e. P.)
23	3, 3, 22	mp2an 696	. . . . 5 |- (1P +P. 1P) e. P.
24		ltaddpr 5120	. . . . 5 |- (((1P +P. 1P) e. P. /\ A e. P.) -> (1P +P. 1P) <P ((1P +P. 1P) +P. A))
25	23, 24	mpan 694	. . . 4 |- (A e. P. -> (1P +P. 1P) <P ((1P +P. 1P) +P. A))
26	21, 25	impbi 157	. . 3 |- ((1P +P. 1P) <P ((1P +P. 1P) +P. A) <-> A e. P.)
27	12, 26	bitr 173	. 2 |- ((1P +P. 1P) <P (1P +P. (A +P. 1P)) <-> A e. P.)
28	2, 6, 27	3bitr 177	1 |- (0R <R [<.(A +P. 1P), 1P>.] ~R <-> A e. P.)

Syntax hints:   <-> wb 146   e. wcel 956  Vcvv 1807  <.cop 2407   class class class wbr 2614  (class class class)co 3954  [cec 4249  P.cnp 4965  1Pc1p 4966   +P. cpp 4967   <P cltp 4969   ~R cer 4972  0Rc0r 4974   <R cltr 4979

This theorem is referenced by:  map2psrpr 5200  suppsrlem 5201  suppsr 5202

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-inf2 4605

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-fv 3193  df-rdg 3923  df-opr 3956  df-oprab 3957  df-1st 4069  df-2nd 4070  df-1o 4123  df-oadd 4125  df-omul 4126  df-er 4251  df-ec 4253  df-qs 4256  df-ni 4980  df-pli 4981  df-mi 4982  df-lti 4983  df-plpq 5015  df-mpq 5016  df-enq 5017  df-nq 5018  df-plq 5019  df-mq 5020  df-rq 5021  df-ltq 5022  df-1q 5023  df-np 5066  df-1p 5067  df-plp 5068  df-ltp 5070  df-enr 5146  df-nr 5147  df-ltr 5150  df-0r 5151

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