Theorem ltresr 11063

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 11057	. . . 4 ⊢ <ℝ ⊆ (ℝ × ℝ)
2	1	brel 5697	. . 3 ⊢ (⟨𝐴, 0R⟩ <ℝ ⟨𝐵, 0R⟩ → (⟨𝐴, 0R⟩ ∈ ℝ ∧ ⟨𝐵, 0R⟩ ∈ ℝ))
3		opelreal 11053	. . . 4 ⊢ (⟨𝐴, 0R⟩ ∈ ℝ ↔ 𝐴 ∈ R)
4		opelreal 11053	. . . 4 ⊢ (⟨𝐵, 0R⟩ ∈ ℝ ↔ 𝐵 ∈ R)
5	3, 4	anbi12i 629	. . 3 ⊢ ((⟨𝐴, 0R⟩ ∈ ℝ ∧ ⟨𝐵, 0R⟩ ∈ ℝ) ↔ (𝐴 ∈ R ∧ 𝐵 ∈ R))
6	2, 5	sylib 218	. 2 ⊢ (⟨𝐴, 0R⟩ <ℝ ⟨𝐵, 0R⟩ → (𝐴 ∈ R ∧ 𝐵 ∈ R))
7		ltrelsr 10991	. . 3 ⊢ <R ⊆ (R × R)
8	7	brel 5697	. 2 ⊢ (𝐴 <R 𝐵 → (𝐴 ∈ R ∧ 𝐵 ∈ R))
9		opex 5419	. . . . . . 7 ⊢ ⟨𝐴, 0R⟩ ∈ V
10		opex 5419	. . . . . . 7 ⊢ ⟨𝐵, 0R⟩ ∈ V
11		eleq1 2825	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝐴, 0R⟩ → (𝑥 ∈ ℝ ↔ ⟨𝐴, 0R⟩ ∈ ℝ))
12	11	anbi1d 632	. . . . . . . 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⟩))
14	13	anbi1d 632	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝐴, 0R⟩ → ((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ↔ (⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩)))
15	14	anbi1d 632	. . . . . . . . 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 1926	. . . . . . . 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 633	. . . . . . 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 2825	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝐵, 0R⟩ → (𝑦 ∈ ℝ ↔ ⟨𝐵, 0R⟩ ∈ ℝ))
19	18	anbi2d 631	. . . . . . . 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⟩))
21	20	anbi2d 631	. . . . . . . . . 10 ⊢ (𝑦 = ⟨𝐵, 0R⟩ → ((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ↔ (⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ ⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩)))
22	21	anbi1d 632	. . . . . . . . 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 1926	. . . . . . . 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 633	. . . . . . 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 11051	. . . . . . 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 5499	. . . . . 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 535	. . . . 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 3446	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
29	28	eqresr 11060	. . . . . . . . . 10 ⊢ (⟨𝑧, 0R⟩ = ⟨𝐴, 0R⟩ ↔ 𝑧 = 𝐴)
30		eqcom 2744	. . . . . . . . . 10 ⊢ (⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ↔ ⟨𝑧, 0R⟩ = ⟨𝐴, 0R⟩)
31		eqcom 2744	. . . . . . . . . 10 ⊢ (𝐴 = 𝑧 ↔ 𝑧 = 𝐴)
32	29, 30, 31	3bitr4i 303	. . . . . . . . 9 ⊢ (⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ↔ 𝐴 = 𝑧)
33		vex 3446	. . . . . . . . . . 11 ⊢ 𝑤 ∈ V
34	33	eqresr 11060	. . . . . . . . . 10 ⊢ (⟨𝑤, 0R⟩ = ⟨𝐵, 0R⟩ ↔ 𝑤 = 𝐵)
35		eqcom 2744	. . . . . . . . . 10 ⊢ (⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩ ↔ ⟨𝑤, 0R⟩ = ⟨𝐵, 0R⟩)
36		eqcom 2744	. . . . . . . . . 10 ⊢ (𝐵 = 𝑤 ↔ 𝑤 = 𝐵)
37	34, 35, 36	3bitr4i 303	. . . . . . . . 9 ⊢ (⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩ ↔ 𝐵 = 𝑤)
38	32, 37	anbi12i 629	. . . . . . . 8 ⊢ ((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ ⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩) ↔ (𝐴 = 𝑧 ∧ 𝐵 = 𝑤))
39	28, 33	opth2 5436	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ↔ (𝐴 = 𝑧 ∧ 𝐵 = 𝑤))
40	38, 39	bitr4i 278	. . . . . . 7 ⊢ ((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ ⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩) ↔ ⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩)
41	40	anbi1i 625	. . . . . 6 ⊢ (((⟨𝐴, 0R⟩ = ⟨𝑧, 0R⟩ ∧ ⟨𝐵, 0R⟩ = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ 𝑧 <R 𝑤))
42	41	2exbii 1851	. . . . 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 287	. . . 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 600	. . 3 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (⟨𝐴, 0R⟩ <ℝ ⟨𝐵, 0R⟩ ↔ ∃𝑧∃𝑤(⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ 𝑧 <R 𝑤)))
45		breq12 5105	. . . 4 ⊢ ((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → (𝑧 <R 𝑤 ↔ 𝐴 <R 𝐵))
46	45	copsex2g 5449	. . 3 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (∃𝑧∃𝑤(⟨𝐴, 𝐵⟩ = ⟨𝑧, 𝑤⟩ ∧ 𝑧 <R 𝑤) ↔ 𝐴 <R 𝐵))
47	44, 46	bitrd 279	. 2 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (⟨𝐴, 0R⟩ <ℝ ⟨𝐵, 0R⟩ ↔ 𝐴 <R 𝐵))
48	6, 8, 47	pm5.21nii 378	1 ⊢ (⟨𝐴, 0R⟩ <ℝ ⟨𝐵, 0R⟩ ↔ 𝐴 <R 𝐵)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ⟨cop 4588   class class class wbr 5100  Rcnr 10788  0_Rc0r 10789   <_R cltr 10794  ℝcr 11037   <_ℝ cltrr 11042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-inf2 9562

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-oadd 8411  df-omul 8412  df-er 8645  df-ec 8647  df-qs 8651  df-ni 10795  df-pli 10796  df-mi 10797  df-lti 10798  df-plpq 10831  df-mpq 10832  df-ltpq 10833  df-enq 10834  df-nq 10835  df-erq 10836  df-plq 10837  df-mq 10838  df-1nq 10839  df-rq 10840  df-ltnq 10841  df-np 10904  df-1p 10905  df-enr 10978  df-nr 10979  df-ltr 10982  df-0r 10983  df-r 11048  df-lt 11051

This theorem is referenced by:  ltresr2  11064  axpre-lttri  11088  axpre-lttrn  11089  axpre-ltadd  11090  axpre-mulgt0  11091  axpre-sup  11092