ltresr2 - Metamath Proof Explorer

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Theorem ltresr2 11050

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
ltresr2	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 <_ℝ 𝐵 ↔ (1^st ‘𝐴) <_R (1^st ‘𝐵)))

**Proof of Theorem ltresr2**
Step	Hyp	Ref	Expression
1		elreal2 11041	. . . 4 ⊢ (𝐴 ∈ ℝ ↔ ((1^st ‘𝐴) ∈ R ∧ 𝐴 = ⟨(1^st ‘𝐴), 0_R⟩))
2	1	simprbi 496	. . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 = ⟨(1^st ‘𝐴), 0_R⟩)
3		elreal2 11041	. . . 4 ⊢ (𝐵 ∈ ℝ ↔ ((1^st ‘𝐵) ∈ R ∧ 𝐵 = ⟨(1^st ‘𝐵), 0_R⟩))
4	3	simprbi 496	. . 3 ⊢ (𝐵 ∈ ℝ → 𝐵 = ⟨(1^st ‘𝐵), 0_R⟩)
5	2, 4	breqan12d 5112	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 <_ℝ 𝐵 ↔ ⟨(1^st ‘𝐴), 0_R⟩ <_ℝ ⟨(1^st ‘𝐵), 0_R⟩))
6		ltresr 11049	. 2 ⊢ (⟨(1^st ‘𝐴), 0_R⟩ <_ℝ ⟨(1^st ‘𝐵), 0_R⟩ ↔ (1^st ‘𝐴) <_R (1^st ‘𝐵))
7	5, 6	bitrdi 287	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 <_ℝ 𝐵 ↔ (1^st ‘𝐴) <_R (1^st ‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ⟨cop 4584 class class class wbr 5096 ‘cfv 6490 1^st c1st 7929 Rcnr 10774 0_Rc0r 10775 <_R cltr 10780 ℝcr 11023 <_ℝ cltrr 11028

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-inf2 9548

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-oadd 8399 df-omul 8400 df-er 8633 df-ec 8635 df-qs 8639 df-ni 10781 df-pli 10782 df-mi 10783 df-lti 10784 df-plpq 10817 df-mpq 10818 df-ltpq 10819 df-enq 10820 df-nq 10821 df-erq 10822 df-plq 10823 df-mq 10824 df-1nq 10825 df-rq 10826 df-ltnq 10827 df-np 10890 df-1p 10891 df-enr 10964 df-nr 10965 df-ltr 10968 df-0r 10969 df-r 11034 df-lt 11037

This theorem is referenced by: axpre-sup 11078

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