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

Theorem ltresr 9921
Description: Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
Assertion
Ref Expression
ltresr (⟨𝐴, 0R⟩ <𝐵, 0R⟩ ↔ 𝐴 <R 𝐵)

Proof of Theorem ltresr
Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelre 9915 . . . 4 < ⊆ (ℝ × ℝ)
21brel 5138 . . 3 (⟨𝐴, 0R⟩ <𝐵, 0R⟩ → (⟨𝐴, 0R⟩ ∈ ℝ ∧ ⟨𝐵, 0R⟩ ∈ ℝ))
3 opelreal 9911 . . . 4 (⟨𝐴, 0R⟩ ∈ ℝ ↔ 𝐴R)
4 opelreal 9911 . . . 4 (⟨𝐵, 0R⟩ ∈ ℝ ↔ 𝐵R)
53, 4anbi12i 732 . . 3 ((⟨𝐴, 0R⟩ ∈ ℝ ∧ ⟨𝐵, 0R⟩ ∈ ℝ) ↔ (𝐴R𝐵R))
62, 5sylib 208 . 2 (⟨𝐴, 0R⟩ <𝐵, 0R⟩ → (𝐴R𝐵R))
7 ltrelsr 9849 . . 3 <R ⊆ (R × R)
87brel 5138 . 2 (𝐴 <R 𝐵 → (𝐴R𝐵R))
9 opex 4903 . . . . . . 7 𝐴, 0R⟩ ∈ V
10 opex 4903 . . . . . . 7 𝐵, 0R⟩ ∈ V
11 eleq1 2686 . . . . . . . . 9 (𝑥 = ⟨𝐴, 0R⟩ → (𝑥 ∈ ℝ ↔ ⟨𝐴, 0R⟩ ∈ ℝ))
1211anbi1d 740 . . . . . . . 8 (𝑥 = ⟨𝐴, 0R⟩ → ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ↔ (⟨𝐴, 0R⟩ ∈ ℝ ∧ 𝑦 ∈ ℝ)))
13 eqeq1 2625 . . . . . . . . . . 11 (𝑥 = ⟨𝐴, 0R⟩ → (𝑥 = ⟨𝑧, 0R⟩ ↔ ⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩))
1413anbi1d 740 . . . . . . . . . 10 (𝑥 = ⟨𝐴, 0R⟩ → ((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ↔ (⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩)))
1514anbi1d 740 . . . . . . . . 9 (𝑥 = ⟨𝐴, 0R⟩ → (((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤) ↔ ((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤)))
16152exbidv 1849 . . . . . . . 8 (𝑥 = ⟨𝐴, 0R⟩ → (∃𝑧𝑤((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤) ↔ ∃𝑧𝑤((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤)))
1712, 16anbi12d 746 . . . . . . 7 (𝑥 = ⟨𝐴, 0R⟩ → (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∃𝑧𝑤((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤)) ↔ ((⟨𝐴, 0R⟩ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∃𝑧𝑤((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤))))
18 eleq1 2686 . . . . . . . . 9 (𝑦 = ⟨𝐵, 0R⟩ → (𝑦 ∈ ℝ ↔ ⟨𝐵, 0R⟩ ∈ ℝ))
1918anbi2d 739 . . . . . . . 8 (𝑦 = ⟨𝐵, 0R⟩ → ((⟨𝐴, 0R⟩ ∈ ℝ ∧ 𝑦 ∈ ℝ) ↔ (⟨𝐴, 0R⟩ ∈ ℝ ∧ ⟨𝐵, 0R⟩ ∈ ℝ)))
20 eqeq1 2625 . . . . . . . . . . 11 (𝑦 = ⟨𝐵, 0R⟩ → (𝑦 = ⟨𝑤, 0R⟩ ↔ ⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩))
2120anbi2d 739 . . . . . . . . . 10 (𝑦 = ⟨𝐵, 0R⟩ → ((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ↔ (⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ ⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩)))
2221anbi1d 740 . . . . . . . . 9 (𝑦 = ⟨𝐵, 0R⟩ → (((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤) ↔ ((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ ⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤)))
23222exbidv 1849 . . . . . . . 8 (𝑦 = ⟨𝐵, 0R⟩ → (∃𝑧𝑤((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤) ↔ ∃𝑧𝑤((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ ⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤)))
2419, 23anbi12d 746 . . . . . . 7 (𝑦 = ⟨𝐵, 0R⟩ → (((⟨𝐴, 0R⟩ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∃𝑧𝑤((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤)) ↔ ((⟨𝐴, 0R⟩ ∈ ℝ ∧ ⟨𝐵, 0R⟩ ∈ ℝ) ∧ ∃𝑧𝑤((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ ⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤))))
25 df-lt 9909 . . . . . . 7 < = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∃𝑧𝑤((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤))}
269, 10, 17, 24, 25brab 4968 . . . . . 6 (⟨𝐴, 0R⟩ <𝐵, 0R⟩ ↔ ((⟨𝐴, 0R⟩ ∈ ℝ ∧ ⟨𝐵, 0R⟩ ∈ ℝ) ∧ ∃𝑧𝑤((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ ⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤)))
2726baib 943 . . . . 5 ((⟨𝐴, 0R⟩ ∈ ℝ ∧ ⟨𝐵, 0R⟩ ∈ ℝ) → (⟨𝐴, 0R⟩ <𝐵, 0R⟩ ↔ ∃𝑧𝑤((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ ⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤)))
28 vex 3193 . . . . . . . . . . 11 𝑧 ∈ V
2928eqresr 9918 . . . . . . . . . 10 (⟨𝑧, 0R⟩ = ⟨𝐴, 0R⟩ ↔ 𝑧 = 𝐴)
30 eqcom 2628 . . . . . . . . . 10 (⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ↔ ⟨𝑧, 0R⟩ = ⟨𝐴, 0R⟩)
31 eqcom 2628 . . . . . . . . . 10 (𝐴 = 𝑧𝑧 = 𝐴)
3229, 30, 313bitr4i 292 . . . . . . . . 9 (⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ↔ 𝐴 = 𝑧)
33 vex 3193 . . . . . . . . . . 11 𝑤 ∈ V
3433eqresr 9918 . . . . . . . . . 10 (⟨𝑤, 0R⟩ = ⟨𝐵, 0R⟩ ↔ 𝑤 = 𝐵)
35 eqcom 2628 . . . . . . . . . 10 (⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩ ↔ ⟨𝑤, 0R⟩ = ⟨𝐵, 0R⟩)
36 eqcom 2628 . . . . . . . . . 10 (𝐵 = 𝑤𝑤 = 𝐵)
3734, 35, 363bitr4i 292 . . . . . . . . 9 (⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩ ↔ 𝐵 = 𝑤)
3832, 37anbi12i 732 . . . . . . . 8 ((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ ⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩) ↔ (𝐴 = 𝑧𝐵 = 𝑤))
3928, 33opth2 4919 . . . . . . . 8 (⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ↔ (𝐴 = 𝑧𝐵 = 𝑤))
4038, 39bitr4i 267 . . . . . . 7 ((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ ⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩) ↔ ⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩)
4140anbi1i 730 . . . . . 6 (((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ ⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ 𝑧 <R 𝑤))
42412exbii 1772 . . . . 5 (∃𝑧𝑤((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ ⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤) ↔ ∃𝑧𝑤(⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ 𝑧 <R 𝑤))
4327, 42syl6bb 276 . . . 4 ((⟨𝐴, 0R⟩ ∈ ℝ ∧ ⟨𝐵, 0R⟩ ∈ ℝ) → (⟨𝐴, 0R⟩ <𝐵, 0R⟩ ↔ ∃𝑧𝑤(⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ 𝑧 <R 𝑤)))
443, 4, 43syl2anbr 497 . . 3 ((𝐴R𝐵R) → (⟨𝐴, 0R⟩ <𝐵, 0R⟩ ↔ ∃𝑧𝑤(⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ 𝑧 <R 𝑤)))
45 breq12 4628 . . . 4 ((𝑧 = 𝐴𝑤 = 𝐵) → (𝑧 <R 𝑤𝐴 <R 𝐵))
4645copsex2g 4928 . . 3 ((𝐴R𝐵R) → (∃𝑧𝑤(⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ 𝑧 <R 𝑤) ↔ 𝐴 <R 𝐵))
4744, 46bitrd 268 . 2 ((𝐴R𝐵R) → (⟨𝐴, 0R⟩ <𝐵, 0R⟩ ↔ 𝐴 <R 𝐵))
486, 8, 47pm5.21nii 368 1 (⟨𝐴, 0R⟩ <𝐵, 0R⟩ ↔ 𝐴 <R 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1987  cop 4161   class class class wbr 4623  Rcnr 9647  0Rc0r 9648   <R cltr 9653  cr 9895   < cltrr 9900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-inf2 8498
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-omul 7525  df-er 7702  df-ec 7704  df-qs 7708  df-ni 9654  df-pli 9655  df-mi 9656  df-lti 9657  df-plpq 9690  df-mpq 9691  df-ltpq 9692  df-enq 9693  df-nq 9694  df-erq 9695  df-plq 9696  df-mq 9697  df-1nq 9698  df-rq 9699  df-ltnq 9700  df-np 9763  df-1p 9764  df-enr 9837  df-nr 9838  df-ltr 9841  df-0r 9842  df-r 9906  df-lt 9909
This theorem is referenced by:  ltresr2  9922  axpre-lttri  9946  axpre-lttrn  9947  axpre-ltadd  9948  axpre-mulgt0  9949  axpre-sup  9950
  Copyright terms: Public domain W3C validator