Theorem ltresr 11121

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⟩ <ℝ ⟨𝐵, 0R⟩ ↔ 𝐴 <R 𝐵)

Proof of Theorem ltresr

Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltrelre 11115	. . . 4 ⊢ <ℝ ⊆ (ℝ × ℝ)
2	1	brel 5724	. . 3 ⊢ (⟨𝐴, 0R⟩ <ℝ ⟨𝐵, 0R⟩ → (⟨𝐴, 0R⟩ ∈ ℝ ∧ ⟨𝐵, 0R⟩ ∈ ℝ))
3		opelreal 11111	. . . 4 ⊢ (⟨𝐴, 0R⟩ ∈ ℝ ↔ 𝐴 ∈ R)
4		opelreal 11111	. . . 4 ⊢ (⟨𝐵, 0R⟩ ∈ ℝ ↔ 𝐵 ∈ R)
5	3, 4	anbi12i 639	. . 3 ⊢ ((⟨𝐴, 0R⟩ ∈ ℝ ∧ ⟨𝐵, 0R⟩ ∈ ℝ) ↔ (𝐴 ∈ R ∧ 𝐵 ∈ R))
6	2, 5	sylib 221	. 2 ⊢ (⟨𝐴, 0R⟩ <ℝ ⟨𝐵, 0R⟩ → (𝐴 ∈ R ∧ 𝐵 ∈ R))
7		ltrelsr 11049	. . 3 ⊢ <R ⊆ (R × R)
8	7	brel 5724	. 2 ⊢ (𝐴 <R 𝐵 → (𝐴 ∈ R ∧ 𝐵 ∈ R))
9		opex 5443	. . . . . . 7 ⊢ ⟨𝐴, 0R⟩ ∈ V
10		opex 5443	. . . . . . 7 ⊢ ⟨𝐵, 0R⟩ ∈ V
11		eleq1 2857	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝐴, 0R⟩ → (𝑥 ∈ ℝ ↔ ⟨𝐴, 0R⟩ ∈ ℝ))
12	11	anbi1d 642	. . . . . . . 8 ⊢ (𝑥 = ⟨𝐴, 0R⟩ → ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ↔ (⟨𝐴, 0R⟩ ∈ ℝ ∧ 𝑦 ∈ ℝ)))
13		eqeq1 2773	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨𝐴, 0R⟩ → (𝑥 = ⟨𝑧, 0R⟩ ↔ ⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩))
14	13	anbi1d 642	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝐴, 0R⟩ → ((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ↔ (⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩)))
15	14	anbi1d 642	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝐴, 0R⟩ → (((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤) ↔ ((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤)))
16	15	2exbidv 1951	. . . . . . . 8 ⊢ (𝑥 = ⟨𝐴, 0R⟩ → (∃𝑧∃𝑤((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤) ↔ ∃𝑧∃𝑤((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤)))
17	12, 16	anbi12d 643	. . . . . . 7 ⊢ (𝑥 = ⟨𝐴, 0R⟩ → (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∃𝑧∃𝑤((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤)) ↔ ((⟨𝐴, 0R⟩ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∃𝑧∃𝑤((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤))))
18		eleq1 2857	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝐵, 0R⟩ → (𝑦 ∈ ℝ ↔ ⟨𝐵, 0R⟩ ∈ ℝ))
19	18	anbi2d 641	. . . . . . . 8 ⊢ (𝑦 = ⟨𝐵, 0R⟩ → ((⟨𝐴, 0R⟩ ∈ ℝ ∧ 𝑦 ∈ ℝ) ↔ (⟨𝐴, 0R⟩ ∈ ℝ ∧ ⟨𝐵, 0R⟩ ∈ ℝ)))
20		eqeq1 2773	. . . . . . . . . . 11 ⊢ (𝑦 = ⟨𝐵, 0R⟩ → (𝑦 = ⟨𝑤, 0R⟩ ↔ ⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩))
21	20	anbi2d 641	. . . . . . . . . 10 ⊢ (𝑦 = ⟨𝐵, 0R⟩ → ((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ↔ (⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ ⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩)))
22	21	anbi1d 642	. . . . . . . . 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)))
23	22	2exbidv 1951	. . . . . . . 8 ⊢ (𝑦 = ⟨𝐵, 0R⟩ → (∃𝑧∃𝑤((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤) ↔ ∃𝑧∃𝑤((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ ⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤)))
24	19, 23	anbi12d 643	. . . . . . 7 ⊢ (𝑦 = ⟨𝐵, 0R⟩ → (((⟨𝐴, 0R⟩ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∃𝑧∃𝑤((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤)) ↔ ((⟨𝐴, 0R⟩ ∈ ℝ ∧ ⟨𝐵, 0R⟩ ∈ ℝ) ∧ ∃𝑧∃𝑤((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ ⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤))))
25		df-lt 11109	. . . . . . 7 ⊢ <ℝ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∃𝑧∃𝑤((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤))}
26	9, 10, 17, 24, 25	brab 5526	. . . . . 6 ⊢ (⟨𝐴, 0R⟩ <ℝ ⟨𝐵, 0R⟩ ↔ ((⟨𝐴, 0R⟩ ∈ ℝ ∧ ⟨𝐵, 0R⟩ ∈ ℝ) ∧ ∃𝑧∃𝑤((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ ⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤)))
27	26	baib 544	. . . . 5 ⊢ ((⟨𝐴, 0R⟩ ∈ ℝ ∧ ⟨𝐵, 0R⟩ ∈ ℝ) → (⟨𝐴, 0R⟩ <ℝ ⟨𝐵, 0R⟩ ↔ ∃𝑧∃𝑤((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ ⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤)))
28		vex 3467	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
29	28	eqresr 11118	. . . . . . . . . 10 ⊢ (⟨𝑧, 0R⟩ = ⟨𝐴, 0R⟩ ↔ 𝑧 = 𝐴)
30		eqcom 2776	. . . . . . . . . 10 ⊢ (⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ↔ ⟨𝑧, 0R⟩ = ⟨𝐴, 0R⟩)
31		eqcom 2776	. . . . . . . . . 10 ⊢ (𝐴 = 𝑧 ↔ 𝑧 = 𝐴)
32	29, 30, 31	3bitr4i 306	. . . . . . . . 9 ⊢ (⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ↔ 𝐴 = 𝑧)
33		vex 3467	. . . . . . . . . . 11 ⊢ 𝑤 ∈ V
34	33	eqresr 11118	. . . . . . . . . 10 ⊢ (⟨𝑤, 0R⟩ = ⟨𝐵, 0R⟩ ↔ 𝑤 = 𝐵)
35		eqcom 2776	. . . . . . . . . 10 ⊢ (⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩ ↔ ⟨𝑤, 0R⟩ = ⟨𝐵, 0R⟩)
36		eqcom 2776	. . . . . . . . . 10 ⊢ (𝐵 = 𝑤 ↔ 𝑤 = 𝐵)
37	34, 35, 36	3bitr4i 306	. . . . . . . . 9 ⊢ (⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩ ↔ 𝐵 = 𝑤)
38	32, 37	anbi12i 639	. . . . . . . 8 ⊢ ((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ ⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩) ↔ (𝐴 = 𝑧 ∧ 𝐵 = 𝑤))
39	28, 33	opth2 5460	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ↔ (𝐴 = 𝑧 ∧ 𝐵 = 𝑤))
40	38, 39	bitr4i 281	. . . . . . 7 ⊢ ((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ ⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩) ↔ ⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩)
41	40	anbi1i 635	. . . . . 6 ⊢ (((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ ⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ 𝑧 <R 𝑤))
42	41	2exbii 1876	. . . . 5 ⊢ (∃𝑧∃𝑤((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ ⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤) ↔ ∃𝑧∃𝑤(⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ 𝑧 <R 𝑤))
43	27, 42	bitrdi 290	. . . 4 ⊢ ((⟨𝐴, 0R⟩ ∈ ℝ ∧ ⟨𝐵, 0R⟩ ∈ ℝ) → (⟨𝐴, 0R⟩ <ℝ ⟨𝐵, 0R⟩ ↔ ∃𝑧∃𝑤(⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ 𝑧 <R 𝑤)))
44	3, 4, 43	syl2anbr 610	. . 3 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (⟨𝐴, 0R⟩ <ℝ ⟨𝐵, 0R⟩ ↔ ∃𝑧∃𝑤(⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ 𝑧 <R 𝑤)))
45		breq12 5115	. . . 4 ⊢ ((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → (𝑧 <R 𝑤 ↔ 𝐴 <R 𝐵))
46	45	copsex2g 5474	. . 3 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (∃𝑧∃𝑤(⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ 𝑧 <R 𝑤) ↔ 𝐴 <R 𝐵))
47	44, 46	bitrd 282	. 2 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (⟨𝐴, 0R⟩ <ℝ ⟨𝐵, 0R⟩ ↔ 𝐴 <R 𝐵))
48	6, 8, 47	pm5.21nii 381	1 ⊢ (⟨𝐴, 0R⟩ <ℝ ⟨𝐵, 0R⟩ ↔ 𝐴 <R 𝐵)

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149  ⟨cop 4597   class class class wbr 5110  Rcnr 10846  0_Rc0r 10847   <_R cltr 10852  ℝcr 11095   <_ℝ cltrr 11100

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730  ax-inf2 9606

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-oadd 8453  df-omul 8454  df-er 8690  df-ec 8692  df-qs 8696  df-ni 10853  df-pli 10854  df-mi 10855  df-lti 10856  df-plpq 10889  df-mpq 10890  df-ltpq 10891  df-enq 10892  df-nq 10893  df-erq 10894  df-plq 10895  df-mq 10896  df-1nq 10897  df-rq 10898  df-ltnq 10899  df-np 10962  df-1p 10963  df-enr 11036  df-nr 11037  df-ltr 11040  df-0r 11041  df-r 11106  df-lt 11109

This theorem is referenced by:  ltresr2  11122  axpre-lttri  11146  axpre-lttrn  11147  axpre-ltadd  11148  axpre-mulgt0  11149  axpre-sup  11150