Theorem map2psrpr 5192

Description: Equivalence for positive signed real.

Hypothesis

Ref	Expression
map2psrpr.1	|- A e. V

Assertion

Ref	Expression
map2psrpr	|- (0R <R A <-> E.x(x e. P. /\ [<.(x +P. 1P), 1P>.] ~R = A))

Distinct variable group:   x,A

Proof of Theorem map2psrpr

Step	Hyp	Ref	Expression
1		map2psrpr.1	. . . . 5 |- A e. V
2		ltrelsr 5152	. . . . 5 |- <R (_ (R. X. R.)
3	1, 2	brel 3213	. . . 4 |- (0R <R A -> (0R e. R. /\ A e. R.))
4	3	pm3.27d 325	. . 3 |- (0R <R A -> A e. R.)
5		df-nr 5139	. . . 4 |- R. = ((P. X. P.)/. ~R )
6		breq2 2613	. . . . 5 |- ([<.y, z>.] ~R = A -> (0R <R [<.y, z>.] ~R <-> 0R <R A))
7		eqeq2 1476	. . . . . . 7 |- ([<.y, z>.] ~R = A -> ([<.(x +P. 1P), 1P>.] ~R = [<.y, z>.] ~R <-> [<.(x +P. 1P), 1P>.] ~R = A))
8	7	anbi2d 614	. . . . . 6 |- ([<.y, z>.] ~R = A -> ((x e. P. /\ [<.(x +P. 1P), 1P>.] ~R = [<.y, z>.] ~R ) <-> (x e. P. /\ [<.(x +P. 1P), 1P>.] ~R = A)))
9	8	exbidv 1274	. . . . 5 |- ([<.y, z>.] ~R = A -> (E.x(x e. P. /\ [<.(x +P. 1P), 1P>.] ~R = [<.y, z>.] ~R ) <-> E.x(x e. P. /\ [<.(x +P. 1P), 1P>.] ~R = A)))
10	6, 9	imbi12d 624	. . . 4 |- ([<.y, z>.] ~R = A -> ((0R <R [<.y, z>.] ~R -> E.x(x e. P. /\ [<.(x +P. 1P), 1P>.] ~R = [<.y, z>.] ~R )) <-> (0R <R A -> E.x(x e. P. /\ [<.(x +P. 1P), 1P>.] ~R = A))))
11		enreceq 5149	. . . . . . . . . 10 |- ((((x +P. 1P) e. P. /\ 1P e. P.) /\ (y e. P. /\ z e. P.)) -> ([<.(x +P. 1P), 1P>.] ~R = [<.y, z>.] ~R <-> ((x +P. 1P) +P. z) = (1P +P. y)))
12		1pr 5089	. . . . . . . . . . . 12 |- 1P e. P.
13		addclpr 5092	. . . . . . . . . . . 12 |- ((x e. P. /\ 1P e. P.) -> (x +P. 1P) e. P.)
14	12, 13	mpan2 694	. . . . . . . . . . 11 |- (x e. P. -> (x +P. 1P) e. P.)
15	14, 12	jctir 293	. . . . . . . . . 10 |- (x e. P. -> ((x +P. 1P) e. P. /\ 1P e. P.))
16	11, 15	sylan 448	. . . . . . . . 9 |- ((x e. P. /\ (y e. P. /\ z e. P.)) -> ([<.(x +P. 1P), 1P>.] ~R = [<.y, z>.] ~R <-> ((x +P. 1P) +P. z) = (1P +P. y)))
17	12	elisseti 1809	. . . . . . . . . . . 12 |- 1P e. V
18		visset 1804	. . . . . . . . . . . 12 |- z e. V
19	17, 18	addasspr 5096	. . . . . . . . . . 11 |- ((x +P. 1P) +P. z) = (x +P. (1P +P. z))
20		visset 1804	. . . . . . . . . . . 12 |- x e. V
21		oprex 3968	. . . . . . . . . . . 12 |- (1P +P. z) e. V
22	20, 21	addcompr 5095	. . . . . . . . . . 11 |- (x +P. (1P +P. z)) = ((1P +P. z) +P. x)
23	19, 22	eqtr 1487	. . . . . . . . . 10 |- ((x +P. 1P) +P. z) = ((1P +P. z) +P. x)
24	23	eqeq1i 1474	. . . . . . . . 9 |- (((x +P. 1P) +P. z) = (1P +P. y) <-> ((1P +P. z) +P. x) = (1P +P. y))
25	16, 24	syl6bb 534	. . . . . . . 8 |- ((x e. P. /\ (y e. P. /\ z e. P.)) -> ([<.(x +P. 1P), 1P>.] ~R = [<.y, z>.] ~R <-> ((1P +P. z) +P. x) = (1P +P. y)))
26	25	expcom 374	. . . . . . 7 |- ((y e. P. /\ z e. P.) -> (x e. P. -> ([<.(x +P. 1P), 1P>.] ~R = [<.y, z>.] ~R <-> ((1P +P. z) +P. x) = (1P +P. y))))
27	26	pm5.32d 645	. . . . . 6 |- ((y e. P. /\ z e. P.) -> ((x e. P. /\ [<.(x +P. 1P), 1P>.] ~R = [<.y, z>.] ~R ) <-> (x e. P. /\ ((1P +P. z) +P. x) = (1P +P. y))))
28	27	exbidv 1274	. . . . 5 |- ((y e. P. /\ z e. P.) -> (E.x(x e. P. /\ [<.(x +P. 1P), 1P>.] ~R = [<.y, z>.] ~R ) <-> E.x(x e. P. /\ ((1P +P. z) +P. x) = (1P +P. y))))
29		df-0r 5143	. . . . . . . 8 |- 0R = [<.1P, 1P>.] ~R
30	29	breq1i 2616	. . . . . . 7 |- (0R <R [<.y, z>.] ~R <-> [<.1P, 1P>.] ~R <R [<.y, z>.] ~R )
31		visset 1804	. . . . . . . 8 |- y e. V
32	17, 17, 31, 18	ltsrpr 5158	. . . . . . 7 |- ([<.1P, 1P>.] ~R <R [<.y, z>.] ~R <-> (1P +P. z) <P (1P +P. y))
33	30, 32	bitr 173	. . . . . 6 |- (0R <R [<.y, z>.] ~R <-> (1P +P. z) <P (1P +P. y))
34		oprex 3968	. . . . . . 7 |- (1P +P. y) e. V
35	34	ltexpri 5121	. . . . . 6 |- ((1P +P. z) <P (1P +P. y) -> E.x(x e. P. /\ ((1P +P. z) +P. x) = (1P +P. y)))
36	33, 35	sylbi 199	. . . . 5 |- (0R <R [<.y, z>.] ~R -> E.x(x e. P. /\ ((1P +P. z) +P. x) = (1P +P. y)))
37	28, 36	syl5bir 210	. . . 4 |- ((y e. P. /\ z e. P.) -> (0R <R [<.y, z>.] ~R -> E.x(x e. P. /\ [<.(x +P. 1P), 1P>.] ~R = [<.y, z>.] ~R )))
38	5, 10, 37	ecoptocl 4287	. . 3 |- (A e. R. -> (0R <R A -> E.x(x e. P. /\ [<.(x +P. 1P), 1P>.] ~R = A)))
39	4, 38	mpcom 49	. 2 |- (0R <R A -> E.x(x e. P. /\ [<.(x +P. 1P), 1P>.] ~R = A))
40		breq2 2613	. . . . 5 |- ([<.(x +P. 1P), 1P>.] ~R = A -> (0R <R [<.(x +P. 1P), 1P>.] ~R <-> 0R <R A))
41	20	mappsrpr 5190	. . . . 5 |- (0R <R [<.(x +P. 1P), 1P>.] ~R <-> x e. P.)
42	40, 41	syl5bbr 532	. . . 4 |- ([<.(x +P. 1P), 1P>.] ~R = A -> (x e. P. <-> 0R <R A))
43	42	biimpac 418	. . 3 |- ((x e. P. /\ [<.(x +P. 1P), 1P>.] ~R = A) -> 0R <R A)
44	43	19.23aiv 1290	. 2 |- (E.x(x e. P. /\ [<.(x +P. 1P), 1P>.] ~R = A) -> 0R <R A)
45	39, 44	impbi 157	1 |- (0R <R A <-> E.x(x e. P. /\ [<.(x +P. 1P), 1P>.] ~R = A))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  E.wex 977  Vcvv 1802  <.cop 2401   class class class wbr 2609  (class class class)co 3948  [cec 4243  P.cnp 4957  1Pc1p 4958   +P. cpp 4959   <P cltp 4961   ~R cer 4964  R.cnr 4965  0Rc0r 4966   <R cltr 4971

This theorem is referenced by:  suppsrlem 5193  suppsr 5194

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-1p 5059  df-plp 5060  df-ltp 5062  df-enr 5138  df-nr 5139  df-ltr 5142  df-0r 5143

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