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Theorem ltresr 10163
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 10157 . . . 4 < ⊆ (ℝ × ℝ)
21brel 5308 . . 3 (⟨𝐴, 0R⟩ <𝐵, 0R⟩ → (⟨𝐴, 0R⟩ ∈ ℝ ∧ ⟨𝐵, 0R⟩ ∈ ℝ))
3 opelreal 10153 . . . 4 (⟨𝐴, 0R⟩ ∈ ℝ ↔ 𝐴R)
4 opelreal 10153 . . . 4 (⟨𝐵, 0R⟩ ∈ ℝ ↔ 𝐵R)
53, 4anbi12i 604 . . 3 ((⟨𝐴, 0R⟩ ∈ ℝ ∧ ⟨𝐵, 0R⟩ ∈ ℝ) ↔ (𝐴R𝐵R))
62, 5sylib 208 . 2 (⟨𝐴, 0R⟩ <𝐵, 0R⟩ → (𝐴R𝐵R))
7 ltrelsr 10091 . . 3 <R ⊆ (R × R)
87brel 5308 . 2 (𝐴 <R 𝐵 → (𝐴R𝐵R))
9 opex 5060 . . . . . . 7 𝐴, 0R⟩ ∈ V
10 opex 5060 . . . . . . 7 𝐵, 0R⟩ ∈ V
11 eleq1 2838 . . . . . . . . 9 (𝑥 = ⟨𝐴, 0R⟩ → (𝑥 ∈ ℝ ↔ ⟨𝐴, 0R⟩ ∈ ℝ))
1211anbi1d 607 . . . . . . . 8 (𝑥 = ⟨𝐴, 0R⟩ → ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ↔ (⟨𝐴, 0R⟩ ∈ ℝ ∧ 𝑦 ∈ ℝ)))
13 eqeq1 2775 . . . . . . . . . . 11 (𝑥 = ⟨𝐴, 0R⟩ → (𝑥 = ⟨𝑧, 0R⟩ ↔ ⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩))
1413anbi1d 607 . . . . . . . . . 10 (𝑥 = ⟨𝐴, 0R⟩ → ((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ↔ (⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩)))
1514anbi1d 607 . . . . . . . . 9 (𝑥 = ⟨𝐴, 0R⟩ → (((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤) ↔ ((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤)))
16152exbidv 2004 . . . . . . . 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 608 . . . . . . 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 2838 . . . . . . . . 9 (𝑦 = ⟨𝐵, 0R⟩ → (𝑦 ∈ ℝ ↔ ⟨𝐵, 0R⟩ ∈ ℝ))
1918anbi2d 606 . . . . . . . 8 (𝑦 = ⟨𝐵, 0R⟩ → ((⟨𝐴, 0R⟩ ∈ ℝ ∧ 𝑦 ∈ ℝ) ↔ (⟨𝐴, 0R⟩ ∈ ℝ ∧ ⟨𝐵, 0R⟩ ∈ ℝ)))
20 eqeq1 2775 . . . . . . . . . . 11 (𝑦 = ⟨𝐵, 0R⟩ → (𝑦 = ⟨𝑤, 0R⟩ ↔ ⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩))
2120anbi2d 606 . . . . . . . . . 10 (𝑦 = ⟨𝐵, 0R⟩ → ((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ↔ (⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ ⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩)))
2221anbi1d 607 . . . . . . . . 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 2004 . . . . . . . 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 608 . . . . . . 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 10151 . . . . . . 7 < = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∃𝑧𝑤((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤))}
269, 10, 17, 24, 25brab 5131 . . . . . 6 (⟨𝐴, 0R⟩ <𝐵, 0R⟩ ↔ ((⟨𝐴, 0R⟩ ∈ ℝ ∧ ⟨𝐵, 0R⟩ ∈ ℝ) ∧ ∃𝑧𝑤((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ ⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤)))
2726baib 517 . . . . 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 3354 . . . . . . . . . . 11 𝑧 ∈ V
2928eqresr 10160 . . . . . . . . . 10 (⟨𝑧, 0R⟩ = ⟨𝐴, 0R⟩ ↔ 𝑧 = 𝐴)
30 eqcom 2778 . . . . . . . . . 10 (⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ↔ ⟨𝑧, 0R⟩ = ⟨𝐴, 0R⟩)
31 eqcom 2778 . . . . . . . . . 10 (𝐴 = 𝑧𝑧 = 𝐴)
3229, 30, 313bitr4i 292 . . . . . . . . 9 (⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ↔ 𝐴 = 𝑧)
33 vex 3354 . . . . . . . . . . 11 𝑤 ∈ V
3433eqresr 10160 . . . . . . . . . 10 (⟨𝑤, 0R⟩ = ⟨𝐵, 0R⟩ ↔ 𝑤 = 𝐵)
35 eqcom 2778 . . . . . . . . . 10 (⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩ ↔ ⟨𝑤, 0R⟩ = ⟨𝐵, 0R⟩)
36 eqcom 2778 . . . . . . . . . 10 (𝐵 = 𝑤𝑤 = 𝐵)
3734, 35, 363bitr4i 292 . . . . . . . . 9 (⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩ ↔ 𝐵 = 𝑤)
3832, 37anbi12i 604 . . . . . . . 8 ((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ ⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩) ↔ (𝐴 = 𝑧𝐵 = 𝑤))
3928, 33opth2 5076 . . . . . . . 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 602 . . . . . 6 (((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ ⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ 𝑧 <R 𝑤))
42412exbii 1925 . . . . 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 578 . . 3 ((𝐴R𝐵R) → (⟨𝐴, 0R⟩ <𝐵, 0R⟩ ↔ ∃𝑧𝑤(⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ 𝑧 <R 𝑤)))
45 breq12 4791 . . . 4 ((𝑧 = 𝐴𝑤 = 𝐵) → (𝑧 <R 𝑤𝐴 <R 𝐵))
4645copsex2g 5085 . . 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 367 1 (⟨𝐴, 0R⟩ <𝐵, 0R⟩ ↔ 𝐴 <R 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382   = wceq 1631  wex 1852  wcel 2145  cop 4322   class class class wbr 4786  Rcnr 9889  0Rc0r 9890   <R cltr 9895  cr 10137   < cltrr 10142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-omul 7718  df-er 7896  df-ec 7898  df-qs 7902  df-ni 9896  df-pli 9897  df-mi 9898  df-lti 9899  df-plpq 9932  df-mpq 9933  df-ltpq 9934  df-enq 9935  df-nq 9936  df-erq 9937  df-plq 9938  df-mq 9939  df-1nq 9940  df-rq 9941  df-ltnq 9942  df-np 10005  df-1p 10006  df-enr 10079  df-nr 10080  df-ltr 10083  df-0r 10084  df-r 10148  df-lt 10151
This theorem is referenced by:  ltresr2  10164  axpre-lttri  10188  axpre-lttrn  10189  axpre-ltadd  10190  axpre-mulgt0  10191  axpre-sup  10192
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