Theorem map2psrpr 6372

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 6332	. . . . 5 |- <R C_ (R. X. R.)
3	1, 2	brel 4048	. . . 4 |- (0R <R A -> (0R e. R. /\ A e. R.))
4	3	simprd 352	. . 3 |- (0R <R A -> A e. R.)
5		df-nr 6319	. . . 4 |- R. = ((P. X. P.)/. ~R )
6		breq2 3342	. . . . 5 |- ([<.y, z>.] ~R = A -> (0R <R [<.y, z>.] ~R <-> 0R <R A))
7		eqeq2 1893	. . . . . . 7 |- ([<.y, z>.] ~R = A -> ([<.(x +P. 1P), 1P>.] ~R = [<.y, z>.] ~R <-> [<.(x +P. 1P), 1P>.] ~R = A))
8	7	anbi2d 678	. . . . . 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 1657	. . . . 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 688	. . . 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 6329	. . . . . . . . . 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 6269	. . . . . . . . . . . 12 |- 1P e. P.
13		addclpr 6272	. . . . . . . . . . . 12 |- ((x e. P. /\ 1P e. P.) -> (x +P. 1P) e. P.)
14	12, 13	mpan2 760	. . . . . . . . . . 11 |- (x e. P. -> (x +P. 1P) e. P.)
15	14, 12	jctir 317	. . . . . . . . . 10 |- (x e. P. -> ((x +P. 1P) e. P. /\ 1P e. P.))
16	11, 15	sylan 497	. . . . . . . . 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 2301	. . . . . . . . . . . 12 |- 1P e. _V
18		visset 2295	. . . . . . . . . . . 12 |- z e. _V
19	17, 18	addasspr 6276	. . . . . . . . . . 11 |- ((x +P. 1P) +P. z) = (x +P. (1P +P. z))
20		visset 2295	. . . . . . . . . . . 12 |- x e. _V
21		oprex 4907	. . . . . . . . . . . 12 |- (1P +P. z) e. _V
22	20, 21	addcompr 6275	. . . . . . . . . . 11 |- (x +P. (1P +P. z)) = ((1P +P. z) +P. x)
23	19, 22	eqtri 1908	. . . . . . . . . 10 |- ((x +P. 1P) +P. z) = ((1P +P. z) +P. x)
24	23	eqeq1i 1891	. . . . . . . . 9 |- (((x +P. 1P) +P. z) = (1P +P. y) <-> ((1P +P. z) +P. x) = (1P +P. y))
25	16, 24	syl6bb 595	. . . . . . . 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 403	. . . . . . 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 709	. . . . . 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 1657	. . . . 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 6323	. . . . . . . 8 |- 0R = [<.1P, 1P>.] ~R
30	29	breq1i 3345	. . . . . . 7 |- (0R <R [<.y, z>.] ~R <-> [<.1P, 1P>.] ~R <R [<.y, z>.] ~R )
31		visset 2295	. . . . . . . 8 |- y e. _V
32	17, 17, 31, 18	ltsrpr 6338	. . . . . . 7 |- ([<.1P, 1P>.] ~R <R [<.y, z>.] ~R <-> (1P +P. z) <P (1P +P. y))
33	30, 32	bitri 190	. . . . . 6 |- (0R <R [<.y, z>.] ~R <-> (1P +P. z) <P (1P +P. y))
34		oprex 4907	. . . . . . 7 |- (1P +P. y) e. _V
35	34	ltexpri 6301	. . . . . 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 216	. . . . 5 |- (0R <R [<.y, z>.] ~R -> E.x(x e. P. /\ ((1P +P. z) +P. x) = (1P +P. y)))
37	28, 36	syl5bir 227	. . . 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 5362	. . 3 |- (A e. R. -> (0R <R A -> E.x(x e. P. /\ [<.(x +P. 1P), 1P>.] ~R = A)))
39	4, 38	mpcom 60	. 2 |- (0R <R A -> E.x(x e. P. /\ [<.(x +P. 1P), 1P>.] ~R = A))
40		breq2 3342	. . . . 5 |- ([<.(x +P. 1P), 1P>.] ~R = A -> (0R <R [<.(x +P. 1P), 1P>.] ~R <-> 0R <R A))
41	20	mappsrpr 6370	. . . . 5 |- (0R <R [<.(x +P. 1P), 1P>.] ~R <-> x e. P.)
42	40, 41	syl5bbr 593	. . . 4 |- ([<.(x +P. 1P), 1P>.] ~R = A -> (x e. P. <-> 0R <R A))
43	42	biimpac 462	. . 3 |- ((x e. P. /\ [<.(x +P. 1P), 1P>.] ~R = A) -> 0R <R A)
44	43	19.23aiv 1674	. 2 |- (E.x(x e. P. /\ [<.(x +P. 1P), 1P>.] ~R = A) -> 0R <R A)
45	39, 44	impbii 174	1 |- (0R <R A <-> E.x(x e. P. /\ [<.(x +P. 1P), 1P>.] ~R = A))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  _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  R.cnr 6145  0Rc0r 6146   <R cltr 6151

This theorem is referenced by:  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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