MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mappsrpr Structured version   Visualization version   GIF version

Theorem mappsrpr 9785
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 9740 . . . 4 -1R = [⟨1P, (1P +P 1P)⟩] ~R
21breq1i 4584 . . 3 (-1R <R [⟨𝐴, 1P⟩] ~R ↔ [⟨1P, (1P +P 1P)⟩] ~R <R [⟨𝐴, 1P⟩] ~R )
3 ltsrpr 9754 . . 3 ([⟨1P, (1P +P 1P)⟩] ~R <R [⟨𝐴, 1P⟩] ~R ↔ (1P +P 1P)<P ((1P +P 1P) +P 𝐴))
42, 3bitri 262 . 2 (-1R <R [⟨𝐴, 1P⟩] ~R ↔ (1P +P 1P)<P ((1P +P 1P) +P 𝐴))
5 mappsrpr.2 . . 3 𝐶R
6 ltasr 9777 . . 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 9676 . . . . . 6 <P ⊆ (P × P)
98brel 5079 . . . . 5 ((1P +P 1P)<P ((1P +P 1P) +P 𝐴) → ((1P +P 1P) ∈ P ∧ ((1P +P 1P) +P 𝐴) ∈ P))
109simprd 477 . . . 4 ((1P +P 1P)<P ((1P +P 1P) +P 𝐴) → ((1P +P 1P) +P 𝐴) ∈ P)
11 dmplp 9690 . . . . . 6 dom +P = (P × P)
12 0npr 9670 . . . . . 6 ¬ ∅ ∈ P
1311, 12ndmovrcl 6695 . . . . 5 (((1P +P 1P) +P 𝐴) ∈ P → ((1P +P 1P) ∈ P𝐴P))
1413simprd 477 . . . 4 (((1P +P 1P) +P 𝐴) ∈ P𝐴P)
1510, 14syl 17 . . 3 ((1P +P 1P)<P ((1P +P 1P) +P 𝐴) → 𝐴P)
16 1pr 9693 . . . . 5 1PP
17 addclpr 9696 . . . . 5 ((1PP ∧ 1PP) → (1P +P 1P) ∈ P)
1816, 16, 17mp2an 703 . . . 4 (1P +P 1P) ∈ P
19 ltaddpr 9712 . . . 4 (((1P +P 1P) ∈ P𝐴P) → (1P +P 1P)<P ((1P +P 1P) +P 𝐴))
2018, 19mpan 701 . . 3 (𝐴P → (1P +P 1P)<P ((1P +P 1P) +P 𝐴))
2115, 20impbii 197 . 2 ((1P +P 1P)<P ((1P +P 1P) +P 𝐴) ↔ 𝐴P)
224, 7, 213bitr3i 288 1 ((𝐶 +R -1R) <R (𝐶 +R [⟨𝐴, 1P⟩] ~R ) ↔ 𝐴P)
Colors of variables: wff setvar class
Syntax hints:  wb 194  wcel 1976  cop 4130   class class class wbr 4577  (class class class)co 6526  [cec 7604  Pcnp 9537  1Pc1p 9538   +P cpp 9539  <P cltp 9541   ~R cer 9542  Rcnr 9543  -1Rcm1r 9546   +R cplr 9547   <R cltr 9549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824  ax-inf2 8398
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4938  df-id 4942  df-po 4948  df-so 4949  df-fr 4986  df-we 4988  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-pred 5582  df-ord 5628  df-on 5629  df-lim 5630  df-suc 5631  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-omul 7429  df-er 7606  df-ec 7608  df-qs 7612  df-ni 9550  df-pli 9551  df-mi 9552  df-lti 9553  df-plpq 9586  df-mpq 9587  df-ltpq 9588  df-enq 9589  df-nq 9590  df-erq 9591  df-plq 9592  df-mq 9593  df-1nq 9594  df-rq 9595  df-ltnq 9596  df-np 9659  df-1p 9660  df-plp 9661  df-ltp 9663  df-enr 9733  df-nr 9734  df-plr 9735  df-ltr 9737  df-m1r 9740
This theorem is referenced by:  map2psrpr  9787  supsrlem  9788
  Copyright terms: Public domain W3C validator