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Theorem map2psrpr 10123
Description: Equivalence for positive signed real. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)
Hypothesis
Ref Expression
map2psrpr.2 𝐶R
Assertion
Ref Expression
map2psrpr ((𝐶 +R -1R) <R 𝐴 ↔ ∃𝑥P (𝐶 +R [⟨𝑥, 1P⟩] ~R ) = 𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶

Proof of Theorem map2psrpr
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelsr 10081 . . . . 5 <R ⊆ (R × R)
21brel 5325 . . . 4 ((𝐶 +R -1R) <R 𝐴 → ((𝐶 +R -1R) ∈ R𝐴R))
32simprd 482 . . 3 ((𝐶 +R -1R) <R 𝐴𝐴R)
4 map2psrpr.2 . . . . . 6 𝐶R
5 ltasr 10113 . . . . . 6 (𝐶R → (-1R <R ((𝐶 ·R -1R) +R 𝐴) ↔ (𝐶 +R -1R) <R (𝐶 +R ((𝐶 ·R -1R) +R 𝐴))))
64, 5ax-mp 5 . . . . 5 (-1R <R ((𝐶 ·R -1R) +R 𝐴) ↔ (𝐶 +R -1R) <R (𝐶 +R ((𝐶 ·R -1R) +R 𝐴)))
7 pn0sr 10114 . . . . . . . . . 10 (𝐶R → (𝐶 +R (𝐶 ·R -1R)) = 0R)
84, 7ax-mp 5 . . . . . . . . 9 (𝐶 +R (𝐶 ·R -1R)) = 0R
98oveq1i 6823 . . . . . . . 8 ((𝐶 +R (𝐶 ·R -1R)) +R 𝐴) = (0R +R 𝐴)
10 addasssr 10101 . . . . . . . 8 ((𝐶 +R (𝐶 ·R -1R)) +R 𝐴) = (𝐶 +R ((𝐶 ·R -1R) +R 𝐴))
11 addcomsr 10100 . . . . . . . 8 (0R +R 𝐴) = (𝐴 +R 0R)
129, 10, 113eqtr3i 2790 . . . . . . 7 (𝐶 +R ((𝐶 ·R -1R) +R 𝐴)) = (𝐴 +R 0R)
13 0idsr 10110 . . . . . . 7 (𝐴R → (𝐴 +R 0R) = 𝐴)
1412, 13syl5eq 2806 . . . . . 6 (𝐴R → (𝐶 +R ((𝐶 ·R -1R) +R 𝐴)) = 𝐴)
1514breq2d 4816 . . . . 5 (𝐴R → ((𝐶 +R -1R) <R (𝐶 +R ((𝐶 ·R -1R) +R 𝐴)) ↔ (𝐶 +R -1R) <R 𝐴))
166, 15syl5bb 272 . . . 4 (𝐴R → (-1R <R ((𝐶 ·R -1R) +R 𝐴) ↔ (𝐶 +R -1R) <R 𝐴))
17 m1r 10095 . . . . . . . 8 -1RR
18 mulclsr 10097 . . . . . . . 8 ((𝐶R ∧ -1RR) → (𝐶 ·R -1R) ∈ R)
194, 17, 18mp2an 710 . . . . . . 7 (𝐶 ·R -1R) ∈ R
20 addclsr 10096 . . . . . . 7 (((𝐶 ·R -1R) ∈ R𝐴R) → ((𝐶 ·R -1R) +R 𝐴) ∈ R)
2119, 20mpan 708 . . . . . 6 (𝐴R → ((𝐶 ·R -1R) +R 𝐴) ∈ R)
22 df-nr 10070 . . . . . . 7 R = ((P × P) / ~R )
23 breq2 4808 . . . . . . . 8 ([⟨𝑦, 𝑧⟩] ~R = ((𝐶 ·R -1R) +R 𝐴) → (-1R <R [⟨𝑦, 𝑧⟩] ~R ↔ -1R <R ((𝐶 ·R -1R) +R 𝐴)))
24 eqeq2 2771 . . . . . . . . 9 ([⟨𝑦, 𝑧⟩] ~R = ((𝐶 ·R -1R) +R 𝐴) → ([⟨𝑥, 1P⟩] ~R = [⟨𝑦, 𝑧⟩] ~R ↔ [⟨𝑥, 1P⟩] ~R = ((𝐶 ·R -1R) +R 𝐴)))
2524rexbidv 3190 . . . . . . . 8 ([⟨𝑦, 𝑧⟩] ~R = ((𝐶 ·R -1R) +R 𝐴) → (∃𝑥P [⟨𝑥, 1P⟩] ~R = [⟨𝑦, 𝑧⟩] ~R ↔ ∃𝑥P [⟨𝑥, 1P⟩] ~R = ((𝐶 ·R -1R) +R 𝐴)))
2623, 25imbi12d 333 . . . . . . 7 ([⟨𝑦, 𝑧⟩] ~R = ((𝐶 ·R -1R) +R 𝐴) → ((-1R <R [⟨𝑦, 𝑧⟩] ~R → ∃𝑥P [⟨𝑥, 1P⟩] ~R = [⟨𝑦, 𝑧⟩] ~R ) ↔ (-1R <R ((𝐶 ·R -1R) +R 𝐴) → ∃𝑥P [⟨𝑥, 1P⟩] ~R = ((𝐶 ·R -1R) +R 𝐴))))
27 df-m1r 10076 . . . . . . . . . . 11 -1R = [⟨1P, (1P +P 1P)⟩] ~R
2827breq1i 4811 . . . . . . . . . 10 (-1R <R [⟨𝑦, 𝑧⟩] ~R ↔ [⟨1P, (1P +P 1P)⟩] ~R <R [⟨𝑦, 𝑧⟩] ~R )
29 addasspr 10036 . . . . . . . . . . . 12 ((1P +P 1P) +P 𝑦) = (1P +P (1P +P 𝑦))
3029breq2i 4812 . . . . . . . . . . 11 ((1P +P 𝑧)<P ((1P +P 1P) +P 𝑦) ↔ (1P +P 𝑧)<P (1P +P (1P +P 𝑦)))
31 ltsrpr 10090 . . . . . . . . . . 11 ([⟨1P, (1P +P 1P)⟩] ~R <R [⟨𝑦, 𝑧⟩] ~R ↔ (1P +P 𝑧)<P ((1P +P 1P) +P 𝑦))
32 1pr 10029 . . . . . . . . . . . 12 1PP
33 ltapr 10059 . . . . . . . . . . . 12 (1PP → (𝑧<P (1P +P 𝑦) ↔ (1P +P 𝑧)<P (1P +P (1P +P 𝑦))))
3432, 33ax-mp 5 . . . . . . . . . . 11 (𝑧<P (1P +P 𝑦) ↔ (1P +P 𝑧)<P (1P +P (1P +P 𝑦)))
3530, 31, 343bitr4i 292 . . . . . . . . . 10 ([⟨1P, (1P +P 1P)⟩] ~R <R [⟨𝑦, 𝑧⟩] ~R𝑧<P (1P +P 𝑦))
3628, 35bitri 264 . . . . . . . . 9 (-1R <R [⟨𝑦, 𝑧⟩] ~R𝑧<P (1P +P 𝑦))
37 ltexpri 10057 . . . . . . . . 9 (𝑧<P (1P +P 𝑦) → ∃𝑥P (𝑧 +P 𝑥) = (1P +P 𝑦))
3836, 37sylbi 207 . . . . . . . 8 (-1R <R [⟨𝑦, 𝑧⟩] ~R → ∃𝑥P (𝑧 +P 𝑥) = (1P +P 𝑦))
39 enreceq 10079 . . . . . . . . . . . 12 (((𝑥P ∧ 1PP) ∧ (𝑦P𝑧P)) → ([⟨𝑥, 1P⟩] ~R = [⟨𝑦, 𝑧⟩] ~R ↔ (𝑥 +P 𝑧) = (1P +P 𝑦)))
4032, 39mpanl2 719 . . . . . . . . . . 11 ((𝑥P ∧ (𝑦P𝑧P)) → ([⟨𝑥, 1P⟩] ~R = [⟨𝑦, 𝑧⟩] ~R ↔ (𝑥 +P 𝑧) = (1P +P 𝑦)))
41 addcompr 10035 . . . . . . . . . . . 12 (𝑧 +P 𝑥) = (𝑥 +P 𝑧)
4241eqeq1i 2765 . . . . . . . . . . 11 ((𝑧 +P 𝑥) = (1P +P 𝑦) ↔ (𝑥 +P 𝑧) = (1P +P 𝑦))
4340, 42syl6bbr 278 . . . . . . . . . 10 ((𝑥P ∧ (𝑦P𝑧P)) → ([⟨𝑥, 1P⟩] ~R = [⟨𝑦, 𝑧⟩] ~R ↔ (𝑧 +P 𝑥) = (1P +P 𝑦)))
4443ancoms 468 . . . . . . . . 9 (((𝑦P𝑧P) ∧ 𝑥P) → ([⟨𝑥, 1P⟩] ~R = [⟨𝑦, 𝑧⟩] ~R ↔ (𝑧 +P 𝑥) = (1P +P 𝑦)))
4544rexbidva 3187 . . . . . . . 8 ((𝑦P𝑧P) → (∃𝑥P [⟨𝑥, 1P⟩] ~R = [⟨𝑦, 𝑧⟩] ~R ↔ ∃𝑥P (𝑧 +P 𝑥) = (1P +P 𝑦)))
4638, 45syl5ibr 236 . . . . . . 7 ((𝑦P𝑧P) → (-1R <R [⟨𝑦, 𝑧⟩] ~R → ∃𝑥P [⟨𝑥, 1P⟩] ~R = [⟨𝑦, 𝑧⟩] ~R ))
4722, 26, 46ecoptocl 8004 . . . . . 6 (((𝐶 ·R -1R) +R 𝐴) ∈ R → (-1R <R ((𝐶 ·R -1R) +R 𝐴) → ∃𝑥P [⟨𝑥, 1P⟩] ~R = ((𝐶 ·R -1R) +R 𝐴)))
4821, 47syl 17 . . . . 5 (𝐴R → (-1R <R ((𝐶 ·R -1R) +R 𝐴) → ∃𝑥P [⟨𝑥, 1P⟩] ~R = ((𝐶 ·R -1R) +R 𝐴)))
49 oveq2 6821 . . . . . . . 8 ([⟨𝑥, 1P⟩] ~R = ((𝐶 ·R -1R) +R 𝐴) → (𝐶 +R [⟨𝑥, 1P⟩] ~R ) = (𝐶 +R ((𝐶 ·R -1R) +R 𝐴)))
5049, 14sylan9eqr 2816 . . . . . . 7 ((𝐴R ∧ [⟨𝑥, 1P⟩] ~R = ((𝐶 ·R -1R) +R 𝐴)) → (𝐶 +R [⟨𝑥, 1P⟩] ~R ) = 𝐴)
5150ex 449 . . . . . 6 (𝐴R → ([⟨𝑥, 1P⟩] ~R = ((𝐶 ·R -1R) +R 𝐴) → (𝐶 +R [⟨𝑥, 1P⟩] ~R ) = 𝐴))
5251reximdv 3154 . . . . 5 (𝐴R → (∃𝑥P [⟨𝑥, 1P⟩] ~R = ((𝐶 ·R -1R) +R 𝐴) → ∃𝑥P (𝐶 +R [⟨𝑥, 1P⟩] ~R ) = 𝐴))
5348, 52syld 47 . . . 4 (𝐴R → (-1R <R ((𝐶 ·R -1R) +R 𝐴) → ∃𝑥P (𝐶 +R [⟨𝑥, 1P⟩] ~R ) = 𝐴))
5416, 53sylbird 250 . . 3 (𝐴R → ((𝐶 +R -1R) <R 𝐴 → ∃𝑥P (𝐶 +R [⟨𝑥, 1P⟩] ~R ) = 𝐴))
553, 54mpcom 38 . 2 ((𝐶 +R -1R) <R 𝐴 → ∃𝑥P (𝐶 +R [⟨𝑥, 1P⟩] ~R ) = 𝐴)
564mappsrpr 10121 . . . . 5 ((𝐶 +R -1R) <R (𝐶 +R [⟨𝑥, 1P⟩] ~R ) ↔ 𝑥P)
57 breq2 4808 . . . . 5 ((𝐶 +R [⟨𝑥, 1P⟩] ~R ) = 𝐴 → ((𝐶 +R -1R) <R (𝐶 +R [⟨𝑥, 1P⟩] ~R ) ↔ (𝐶 +R -1R) <R 𝐴))
5856, 57syl5bbr 274 . . . 4 ((𝐶 +R [⟨𝑥, 1P⟩] ~R ) = 𝐴 → (𝑥P ↔ (𝐶 +R -1R) <R 𝐴))
5958biimpac 504 . . 3 ((𝑥P ∧ (𝐶 +R [⟨𝑥, 1P⟩] ~R ) = 𝐴) → (𝐶 +R -1R) <R 𝐴)
6059rexlimiva 3166 . 2 (∃𝑥P (𝐶 +R [⟨𝑥, 1P⟩] ~R ) = 𝐴 → (𝐶 +R -1R) <R 𝐴)
6155, 60impbii 199 1 ((𝐶 +R -1R) <R 𝐴 ↔ ∃𝑥P (𝐶 +R [⟨𝑥, 1P⟩] ~R ) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wrex 3051  cop 4327   class class class wbr 4804  (class class class)co 6813  [cec 7909  Pcnp 9873  1Pc1p 9874   +P cpp 9875  <P cltp 9877   ~R cer 9878  Rcnr 9879  0Rc0r 9880  -1Rcm1r 9882   +R cplr 9883   ·R cmr 9884   <R cltr 9885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-omul 7734  df-er 7911  df-ec 7913  df-qs 7917  df-ni 9886  df-pli 9887  df-mi 9888  df-lti 9889  df-plpq 9922  df-mpq 9923  df-ltpq 9924  df-enq 9925  df-nq 9926  df-erq 9927  df-plq 9928  df-mq 9929  df-1nq 9930  df-rq 9931  df-ltnq 9932  df-np 9995  df-1p 9996  df-plp 9997  df-mp 9998  df-ltp 9999  df-enr 10069  df-nr 10070  df-plr 10071  df-mr 10072  df-ltr 10073  df-0r 10074  df-1r 10075  df-m1r 10076
This theorem is referenced by:  supsrlem  10124
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