Theorem map2psrpr 5232

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 5192	. . . . 5 |- <R (_ (R. X. R.)
3	1, 2	brel 3229	. . . 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 5179	. . . 4 |- R. = ((P. X. P.)/. ~R )
6		breq2 2628	. . . . 5 |- ([<.y, z>.] ~R = A -> (0R <R [<.y, z>.] ~R <-> 0R <R A))
7		eqeq2 1487	. . . . . . 7 |- ([<.y, z>.] ~R = A -> ([<.(x +P. 1P), 1P>.] ~R = [<.y, z>.] ~R <-> [<.(x +P. 1P), 1P>.] ~R = A))
8	7	anbi2d 618	. . . . . 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 1281	. . . . 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 628	. . . 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 5189	. . . . . . . . . 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 5129	. . . . . . . . . . . 12 |- 1P e. P.
13		addclpr 5132	. . . . . . . . . . . 12 |- ((x e. P. /\ 1P e. P.) -> (x +P. 1P) e. P.)
14	12, 13	mpan2 698	. . . . . . . . . . 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 450	. . . . . . . . 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 1821	. . . . . . . . . . . 12 |- 1P e. V
18		visset 1816	. . . . . . . . . . . 12 |- z e. V
19	17, 18	addasspr 5136	. . . . . . . . . . 11 |- ((x +P. 1P) +P. z) = (x +P. (1P +P. z))
20		visset 1816	. . . . . . . . . . . 12 |- x e. V
21		oprex 3989	. . . . . . . . . . . 12 |- (1P +P. z) e. V
22	20, 21	addcompr 5135	. . . . . . . . . . 11 |- (x +P. (1P +P. z)) = ((1P +P. z) +P. x)
23	19, 22	eqtr 1498	. . . . . . . . . 10 |- ((x +P. 1P) +P. z) = ((1P +P. z) +P. x)
24	23	eqeq1i 1485	. . . . . . . . 9 |- (((x +P. 1P) +P. z) = (1P +P. y) <-> ((1P +P. z) +P. x) = (1P +P. y))
25	16, 24	syl6bb 538	. . . . . . . 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 649	. . . . . 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 1281	. . . . 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 5183	. . . . . . . 8 |- 0R = [<.1P, 1P>.] ~R
30	29	breq1i 2631	. . . . . . 7 |- (0R <R [<.y, z>.] ~R <-> [<.1P, 1P>.] ~R <R [<.y, z>.] ~R )
31		visset 1816	. . . . . . . 8 |- y e. V
32	17, 17, 31, 18	ltsrpr 5198	. . . . . . 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 3989	. . . . . . 7 |- (1P +P. y) e. V
35	34	ltexpri 5161	. . . . . 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 4309	. . 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 2628	. . . . 5 |- ([<.(x +P. 1P), 1P>.] ~R = A -> (0R <R [<.(x +P. 1P), 1P>.] ~R <-> 0R <R A))
41	20	mappsrpr 5230	. . . . 5 |- (0R <R [<.(x +P. 1P), 1P>.] ~R <-> x e. P.)
42	40, 41	syl5bbr 536	. . . 4 |- ([<.(x +P. 1P), 1P>.] ~R = A -> (x e. P. <-> 0R <R A))
43	42	biimpac 420	. . 3 |- ((x e. P. /\ [<.(x +P. 1P), 1P>.] ~R = A) -> 0R <R A)
44	43	19.23aiv 1297	. 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 958   e. wcel 960  E.wex 982  Vcvv 1814  <.cop 2415   class class class wbr 2624  (class class class)co 3969  [cec 4265  P.cnp 4997  1Pc1p 4998   +P. cpp 4999   <P cltp 5001   ~R cer 5004  R.cnr 5005  0Rc0r 5006   <R cltr 5011

This theorem is referenced by:  suppsrlem 5233  suppsr 5234

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-ltp 5102  df-enr 5178  df-nr 5179  df-ltr 5182  df-0r 5183

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