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Theorem ltresr 11083
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 11077 . . . 4 < ⊆ (ℝ × ℝ)
21brel 5702 . . 3 (⟨𝐴, 0R⟩ <𝐵, 0R⟩ → (⟨𝐴, 0R⟩ ∈ ℝ ∧ ⟨𝐵, 0R⟩ ∈ ℝ))
3 opelreal 11073 . . . 4 (⟨𝐴, 0R⟩ ∈ ℝ ↔ 𝐴R)
4 opelreal 11073 . . . 4 (⟨𝐵, 0R⟩ ∈ ℝ ↔ 𝐵R)
53, 4anbi12i 628 . . 3 ((⟨𝐴, 0R⟩ ∈ ℝ ∧ ⟨𝐵, 0R⟩ ∈ ℝ) ↔ (𝐴R𝐵R))
62, 5sylib 217 . 2 (⟨𝐴, 0R⟩ <𝐵, 0R⟩ → (𝐴R𝐵R))
7 ltrelsr 11011 . . 3 <R ⊆ (R × R)
87brel 5702 . 2 (𝐴 <R 𝐵 → (𝐴R𝐵R))
9 opex 5426 . . . . . . 7 𝐴, 0R⟩ ∈ V
10 opex 5426 . . . . . . 7 𝐵, 0R⟩ ∈ V
11 eleq1 2826 . . . . . . . . 9 (𝑥 = ⟨𝐴, 0R⟩ → (𝑥 ∈ ℝ ↔ ⟨𝐴, 0R⟩ ∈ ℝ))
1211anbi1d 631 . . . . . . . 8 (𝑥 = ⟨𝐴, 0R⟩ → ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ↔ (⟨𝐴, 0R⟩ ∈ ℝ ∧ 𝑦 ∈ ℝ)))
13 eqeq1 2741 . . . . . . . . . . 11 (𝑥 = ⟨𝐴, 0R⟩ → (𝑥 = ⟨𝑧, 0R⟩ ↔ ⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩))
1413anbi1d 631 . . . . . . . . . 10 (𝑥 = ⟨𝐴, 0R⟩ → ((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ↔ (⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩)))
1514anbi1d 631 . . . . . . . . 9 (𝑥 = ⟨𝐴, 0R⟩ → (((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤) ↔ ((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤)))
16152exbidv 1928 . . . . . . . 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 632 . . . . . . 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 2826 . . . . . . . . 9 (𝑦 = ⟨𝐵, 0R⟩ → (𝑦 ∈ ℝ ↔ ⟨𝐵, 0R⟩ ∈ ℝ))
1918anbi2d 630 . . . . . . . 8 (𝑦 = ⟨𝐵, 0R⟩ → ((⟨𝐴, 0R⟩ ∈ ℝ ∧ 𝑦 ∈ ℝ) ↔ (⟨𝐴, 0R⟩ ∈ ℝ ∧ ⟨𝐵, 0R⟩ ∈ ℝ)))
20 eqeq1 2741 . . . . . . . . . . 11 (𝑦 = ⟨𝐵, 0R⟩ → (𝑦 = ⟨𝑤, 0R⟩ ↔ ⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩))
2120anbi2d 630 . . . . . . . . . 10 (𝑦 = ⟨𝐵, 0R⟩ → ((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ↔ (⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ ⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩)))
2221anbi1d 631 . . . . . . . . 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 1928 . . . . . . . 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 632 . . . . . . 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 11071 . . . . . . 7 < = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∃𝑧𝑤((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤))}
269, 10, 17, 24, 25brab 5505 . . . . . 6 (⟨𝐴, 0R⟩ <𝐵, 0R⟩ ↔ ((⟨𝐴, 0R⟩ ∈ ℝ ∧ ⟨𝐵, 0R⟩ ∈ ℝ) ∧ ∃𝑧𝑤((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ ⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤)))
2726baib 537 . . . . 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 3452 . . . . . . . . . . 11 𝑧 ∈ V
2928eqresr 11080 . . . . . . . . . 10 (⟨𝑧, 0R⟩ = ⟨𝐴, 0R⟩ ↔ 𝑧 = 𝐴)
30 eqcom 2744 . . . . . . . . . 10 (⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ↔ ⟨𝑧, 0R⟩ = ⟨𝐴, 0R⟩)
31 eqcom 2744 . . . . . . . . . 10 (𝐴 = 𝑧𝑧 = 𝐴)
3229, 30, 313bitr4i 303 . . . . . . . . 9 (⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ↔ 𝐴 = 𝑧)
33 vex 3452 . . . . . . . . . . 11 𝑤 ∈ V
3433eqresr 11080 . . . . . . . . . 10 (⟨𝑤, 0R⟩ = ⟨𝐵, 0R⟩ ↔ 𝑤 = 𝐵)
35 eqcom 2744 . . . . . . . . . 10 (⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩ ↔ ⟨𝑤, 0R⟩ = ⟨𝐵, 0R⟩)
36 eqcom 2744 . . . . . . . . . 10 (𝐵 = 𝑤𝑤 = 𝐵)
3734, 35, 363bitr4i 303 . . . . . . . . 9 (⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩ ↔ 𝐵 = 𝑤)
3832, 37anbi12i 628 . . . . . . . 8 ((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ ⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩) ↔ (𝐴 = 𝑧𝐵 = 𝑤))
3928, 33opth2 5442 . . . . . . . 8 (⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ↔ (𝐴 = 𝑧𝐵 = 𝑤))
4038, 39bitr4i 278 . . . . . . 7 ((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ ⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩) ↔ ⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩)
4140anbi1i 625 . . . . . 6 (((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ ⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ 𝑧 <R 𝑤))
42412exbii 1852 . . . . 5 (∃𝑧𝑤((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ ⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤) ↔ ∃𝑧𝑤(⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ 𝑧 <R 𝑤))
4327, 42bitrdi 287 . . . 4 ((⟨𝐴, 0R⟩ ∈ ℝ ∧ ⟨𝐵, 0R⟩ ∈ ℝ) → (⟨𝐴, 0R⟩ <𝐵, 0R⟩ ↔ ∃𝑧𝑤(⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ 𝑧 <R 𝑤)))
443, 4, 43syl2anbr 600 . . 3 ((𝐴R𝐵R) → (⟨𝐴, 0R⟩ <𝐵, 0R⟩ ↔ ∃𝑧𝑤(⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ 𝑧 <R 𝑤)))
45 breq12 5115 . . . 4 ((𝑧 = 𝐴𝑤 = 𝐵) → (𝑧 <R 𝑤𝐴 <R 𝐵))
4645copsex2g 5455 . . 3 ((𝐴R𝐵R) → (∃𝑧𝑤(⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ 𝑧 <R 𝑤) ↔ 𝐴 <R 𝐵))
4744, 46bitrd 279 . 2 ((𝐴R𝐵R) → (⟨𝐴, 0R⟩ <𝐵, 0R⟩ ↔ 𝐴 <R 𝐵))
486, 8, 47pm5.21nii 380 1 (⟨𝐴, 0R⟩ <𝐵, 0R⟩ ↔ 𝐴 <R 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  wex 1782  wcel 2107  cop 4597   class class class wbr 5110  Rcnr 10808  0Rc0r 10809   <R cltr 10814  cr 11057   < cltrr 11062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-inf2 9584
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-oadd 8421  df-omul 8422  df-er 8655  df-ec 8657  df-qs 8661  df-ni 10815  df-pli 10816  df-mi 10817  df-lti 10818  df-plpq 10851  df-mpq 10852  df-ltpq 10853  df-enq 10854  df-nq 10855  df-erq 10856  df-plq 10857  df-mq 10858  df-1nq 10859  df-rq 10860  df-ltnq 10861  df-np 10924  df-1p 10925  df-enr 10998  df-nr 10999  df-ltr 11002  df-0r 11003  df-r 11068  df-lt 11071
This theorem is referenced by:  ltresr2  11084  axpre-lttri  11108  axpre-lttrn  11109  axpre-ltadd  11110  axpre-mulgt0  11111  axpre-sup  11112
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