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Theorem mappsrpr 10119
Description: Mapping from positive signed reals to positive reals. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)
Hypothesis
Ref Expression
mappsrpr.2 𝐶R
Assertion
Ref Expression
mappsrpr ((𝐶 +R -1R) <R (𝐶 +R [⟨𝐴, 1P⟩] ~R ) ↔ 𝐴P)

Proof of Theorem mappsrpr
StepHypRef Expression
1 df-m1r 10074 . . . 4 -1R = [⟨1P, (1P +P 1P)⟩] ~R
21breq1i 4809 . . 3 (-1R <R [⟨𝐴, 1P⟩] ~R ↔ [⟨1P, (1P +P 1P)⟩] ~R <R [⟨𝐴, 1P⟩] ~R )
3 ltsrpr 10088 . . 3 ([⟨1P, (1P +P 1P)⟩] ~R <R [⟨𝐴, 1P⟩] ~R ↔ (1P +P 1P)<P ((1P +P 1P) +P 𝐴))
42, 3bitri 264 . 2 (-1R <R [⟨𝐴, 1P⟩] ~R ↔ (1P +P 1P)<P ((1P +P 1P) +P 𝐴))
5 mappsrpr.2 . . 3 𝐶R
6 ltasr 10111 . . 3 (𝐶R → (-1R <R [⟨𝐴, 1P⟩] ~R ↔ (𝐶 +R -1R) <R (𝐶 +R [⟨𝐴, 1P⟩] ~R )))
75, 6ax-mp 5 . 2 (-1R <R [⟨𝐴, 1P⟩] ~R ↔ (𝐶 +R -1R) <R (𝐶 +R [⟨𝐴, 1P⟩] ~R ))
8 ltrelpr 10010 . . . . . 6 <P ⊆ (P × P)
98brel 5323 . . . . 5 ((1P +P 1P)<P ((1P +P 1P) +P 𝐴) → ((1P +P 1P) ∈ P ∧ ((1P +P 1P) +P 𝐴) ∈ P))
109simprd 482 . . . 4 ((1P +P 1P)<P ((1P +P 1P) +P 𝐴) → ((1P +P 1P) +P 𝐴) ∈ P)
11 dmplp 10024 . . . . . 6 dom +P = (P × P)
12 0npr 10004 . . . . . 6 ¬ ∅ ∈ P
1311, 12ndmovrcl 6983 . . . . 5 (((1P +P 1P) +P 𝐴) ∈ P → ((1P +P 1P) ∈ P𝐴P))
1413simprd 482 . . . 4 (((1P +P 1P) +P 𝐴) ∈ P𝐴P)
1510, 14syl 17 . . 3 ((1P +P 1P)<P ((1P +P 1P) +P 𝐴) → 𝐴P)
16 1pr 10027 . . . . 5 1PP
17 addclpr 10030 . . . . 5 ((1PP ∧ 1PP) → (1P +P 1P) ∈ P)
1816, 16, 17mp2an 710 . . . 4 (1P +P 1P) ∈ P
19 ltaddpr 10046 . . . 4 (((1P +P 1P) ∈ P𝐴P) → (1P +P 1P)<P ((1P +P 1P) +P 𝐴))
2018, 19mpan 708 . . 3 (𝐴P → (1P +P 1P)<P ((1P +P 1P) +P 𝐴))
2115, 20impbii 199 . 2 ((1P +P 1P)<P ((1P +P 1P) +P 𝐴) ↔ 𝐴P)
224, 7, 213bitr3i 290 1 ((𝐶 +R -1R) <R (𝐶 +R [⟨𝐴, 1P⟩] ~R ) ↔ 𝐴P)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wcel 2137  cop 4325   class class class wbr 4802  (class class class)co 6811  [cec 7907  Pcnp 9871  1Pc1p 9872   +P cpp 9873  <P cltp 9875   ~R cer 9876  Rcnr 9877  -1Rcm1r 9880   +R cplr 9881   <R cltr 9883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112  ax-inf2 8709
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-reu 3055  df-rmo 3056  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-uni 4587  df-int 4626  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-tr 4903  df-id 5172  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-we 5225  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-pred 5839  df-ord 5885  df-on 5886  df-lim 5887  df-suc 5888  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-om 7229  df-1st 7331  df-2nd 7332  df-wrecs 7574  df-recs 7635  df-rdg 7673  df-1o 7727  df-oadd 7731  df-omul 7732  df-er 7909  df-ec 7911  df-qs 7915  df-ni 9884  df-pli 9885  df-mi 9886  df-lti 9887  df-plpq 9920  df-mpq 9921  df-ltpq 9922  df-enq 9923  df-nq 9924  df-erq 9925  df-plq 9926  df-mq 9927  df-1nq 9928  df-rq 9929  df-ltnq 9930  df-np 9993  df-1p 9994  df-plp 9995  df-ltp 9997  df-enr 10067  df-nr 10068  df-plr 10069  df-ltr 10071  df-m1r 10074
This theorem is referenced by:  map2psrpr  10121  supsrlem  10122
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