ltresr2 - Intuitionistic Logic Explorer

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Theorem ltresr2 7651

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 7641	. . . 4 ⊢ (𝐴 ∈ ℝ ↔ ((1^st ‘𝐴) ∈ R ∧ 𝐴 = ⟨(1^st ‘𝐴), 0_R⟩))
2	1	simprbi 273	. . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 = ⟨(1^st ‘𝐴), 0_R⟩)
3		elreal2 7641	. . . 4 ⊢ (𝐵 ∈ ℝ ↔ ((1^st ‘𝐵) ∈ R ∧ 𝐵 = ⟨(1^st ‘𝐵), 0_R⟩))
4	3	simprbi 273	. . 3 ⊢ (𝐵 ∈ ℝ → 𝐵 = ⟨(1^st ‘𝐵), 0_R⟩)
5	2, 4	breqan12d 3945	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 <_ℝ 𝐵 ↔ ⟨(1^st ‘𝐴), 0_R⟩ <_ℝ ⟨(1^st ‘𝐵), 0_R⟩))
6		ltresr 7650	. 2 ⊢ (⟨(1^st ‘𝐴), 0_R⟩ <_ℝ ⟨(1^st ‘𝐵), 0_R⟩ ↔ (1^st ‘𝐴) <_R (1^st ‘𝐵))
7	5, 6	syl6bb 195	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 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 ⟨cop 3530 class class class wbr 3929 ‘cfv 5123 1^st c1st 6036 Rcnr 7108 0_Rc0r 7109 <_R cltr 7114 ℝcr 7622 <_ℝ cltrr 7627

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502

This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-eprel 4211 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-1o 6313 df-oadd 6317 df-omul 6318 df-er 6429 df-ec 6431 df-qs 6435 df-ni 7115 df-pli 7116 df-mi 7117 df-lti 7118 df-plpq 7155 df-mpq 7156 df-enq 7158 df-nqqs 7159 df-plqqs 7160 df-mqqs 7161 df-1nqqs 7162 df-rq 7163 df-ltnqqs 7164 df-inp 7277 df-i1p 7278 df-enr 7537 df-nr 7538 df-ltr 7541 df-0r 7542 df-r 7633 df-lt 7636

This theorem is referenced by: (None)

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