ltresr2 - Intuitionistic Logic Explorer

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Theorem ltresr2 7842

Description: Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.)

**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 7832	. . . 4 ⊢ (𝐴 ∈ ℝ ↔ ((1^st ‘𝐴) ∈ R ∧ 𝐴 = ⟨(1^st ‘𝐴), 0_R⟩))
2	1	simprbi 275	. . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 = ⟨(1^st ‘𝐴), 0_R⟩)
3		elreal2 7832	. . . 4 ⊢ (𝐵 ∈ ℝ ↔ ((1^st ‘𝐵) ∈ R ∧ 𝐵 = ⟨(1^st ‘𝐵), 0_R⟩))
4	3	simprbi 275	. . 3 ⊢ (𝐵 ∈ ℝ → 𝐵 = ⟨(1^st ‘𝐵), 0_R⟩)
5	2, 4	breqan12d 4021	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 <_ℝ 𝐵 ↔ ⟨(1^st ‘𝐴), 0_R⟩ <_ℝ ⟨(1^st ‘𝐵), 0_R⟩))
6		ltresr 7841	. 2 ⊢ (⟨(1^st ‘𝐴), 0_R⟩ <_ℝ ⟨(1^st ‘𝐵), 0_R⟩ ↔ (1^st ‘𝐴) <_R (1^st ‘𝐵))
7	5, 6	bitrdi 196	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 <_ℝ 𝐵 ↔ (1^st ‘𝐴) <_R (1^st ‘𝐵)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 ⟨cop 3597 class class class wbr 4005 ‘cfv 5218 1^st c1st 6142 Rcnr 7299 0_Rc0r 7300 <_R cltr 7305 ℝcr 7813 <_ℝ cltrr 7818

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-eprel 4291 df-id 4295 df-po 4298 df-iso 4299 df-iord 4368 df-on 4370 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-ov 5881 df-oprab 5882 df-mpo 5883 df-1st 6144 df-2nd 6145 df-recs 6309 df-irdg 6374 df-1o 6420 df-oadd 6424 df-omul 6425 df-er 6538 df-ec 6540 df-qs 6544 df-ni 7306 df-pli 7307 df-mi 7308 df-lti 7309 df-plpq 7346 df-mpq 7347 df-enq 7349 df-nqqs 7350 df-plqqs 7351 df-mqqs 7352 df-1nqqs 7353 df-rq 7354 df-ltnqqs 7355 df-inp 7468 df-i1p 7469 df-enr 7728 df-nr 7729 df-ltr 7732 df-0r 7733 df-r 7824 df-lt 7827

This theorem is referenced by: (None)

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