Theorem ltpsrpr 5219

Description: Mapping of order from positive signed reals to positive reals.

Hypotheses

Ref	Expression
ltpsrpr.1	|- A e. V
ltpsrpr.2	|- B e. V

Assertion

Ref	Expression
ltpsrpr	|- ([<.(A +P. 1P), 1P>.] ~R <R [<.(B +P. 1P), 1P>.] ~R <-> A <P B)

Proof of Theorem ltpsrpr

Step	Hyp	Ref	Expression
1		ltpsrpr.1	. . . 4 |- A e. V
2		1pr 5117	. . . . 5 |- 1P e. P.
3	2	elisseti 1818	. . . 4 |- 1P e. V
4	1, 3	addcompr 5123	. . 3 |- (A +P. 1P) = (1P +P. A)
5		ltpsrpr.2	. . . 4 |- B e. V
6	5, 3	addcompr 5123	. . 3 |- (B +P. 1P) = (1P +P. B)
7	4, 6	breq12i 2628	. 2 |- ((A +P. 1P) <P (B +P. 1P) <-> (1P +P. A) <P (1P +P. B))
8		oprex 3983	. . . . 5 |- (A +P. 1P) e. V
9	8, 3	addcompr 5123	. . . 4 |- ((A +P. 1P) +P. 1P) = (1P +P. (A +P. 1P))
10	9	breq1i 2626	. . 3 |- (((A +P. 1P) +P. 1P) <P (1P +P. (B +P. 1P)) <-> (1P +P. (A +P. 1P)) <P (1P +P. (B +P. 1P)))
11		oprex 3983	. . . 4 |- (B +P. 1P) e. V
12	8, 3, 11, 3	ltsrpr 5186	. . 3 |- ([<.(A +P. 1P), 1P>.] ~R <R [<.(B +P. 1P), 1P>.] ~R <-> ((A +P. 1P) +P. 1P) <P (1P +P. (B +P. 1P)))
13	8, 11	ltapr 5151	. . . 4 |- (1P e. P. -> ((A +P. 1P) <P (B +P. 1P) <-> (1P +P. (A +P. 1P)) <P (1P +P. (B +P. 1P))))
14	2, 13	ax-mp 7	. . 3 |- ((A +P. 1P) <P (B +P. 1P) <-> (1P +P. (A +P. 1P)) <P (1P +P. (B +P. 1P)))
15	10, 12, 14	3bitr4 183	. 2 |- ([<.(A +P. 1P), 1P>.] ~R <R [<.(B +P. 1P), 1P>.] ~R <-> (A +P. 1P) <P (B +P. 1P))
16	1, 5	ltapr 5151	. . 3 |- (1P e. P. -> (A <P B <-> (1P +P. A) <P (1P +P. B)))
17	2, 16	ax-mp 7	. 2 |- (A <P B <-> (1P +P. A) <P (1P +P. B))
18	7, 15, 17	3bitr4 183	1 |- ([<.(A +P. 1P), 1P>.] ~R <R [<.(B +P. 1P), 1P>.] ~R <-> A <P B)

Syntax hints:   <-> wb 146   e. wcel 958  Vcvv 1811  <.cop 2411   class class class wbr 2619  (class class class)co 3963  [cec 4259  P.cnp 4985  1Pc1p 4986   +P. cpp 4987   <P cltp 4989   ~R cer 4992   <R cltr 4999

This theorem is referenced by:  suppsr 5222

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-oadd 4135  df-omul 4136  df-er 4261  df-ec 4263  df-qs 4266  df-ni 5000  df-pli 5001  df-mi 5002  df-lti 5003  df-plpq 5035  df-mpq 5036  df-enq 5037  df-nq 5038  df-plq 5039  df-mq 5040  df-rq 5041  df-ltq 5042  df-1q 5043  df-np 5086  df-1p 5087  df-plp 5088  df-ltp 5090  df-enr 5166  df-nr 5167  df-ltr 5170

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