ltresr2 - Intuitionistic Logic Explorer

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Theorem ltresr2 8213

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 8203	. . . 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 8203	. . . 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 4147	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 <_ℝ 𝐵 ↔ ⟨(1^st ‘𝐴), 0_R⟩ <_ℝ ⟨(1^st ‘𝐵), 0_R⟩))
6		ltresr 8212	. 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

This proof depends on syntax axioms: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1402 ∈ wcel 2209 ⟨cop 3712 class class class wbr 4131 ‘cfv 5380 1^st c1st 6376 Rcnr 7670 0_Rc0r 7671 <_R cltr 7676 ℝcr 8184 <_ℝ cltrr 8189

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4248 ax-sep 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-iinf 4737

This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4132 df-opab 4194 df-mpt 4195 df-tr 4232 df-eprel 4436 df-id 4440 df-po 4443 df-iso 4444 df-iord 4513 df-on 4515 df-suc 4518 df-iom 4740 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-ov 6091 df-oprab 6092 df-mpo 6093 df-1st 6378 df-2nd 6379 df-recs 6580 df-irdg 6645 df-1o 6691 df-oadd 6695 df-omul 6696 df-er 6811 df-ec 6813 df-qs 6817 df-ni 7677 df-pli 7678 df-mi 7679 df-lti 7680 df-plpq 7717 df-mpq 7718 df-enq 7720 df-nqqs 7721 df-plqqs 7722 df-mqqs 7723 df-1nqqs 7724 df-rq 7725 df-ltnqqs 7726 df-inp 7839 df-i1p 7840 df-enr 8099 df-nr 8100 df-ltr 8103 df-0r 8104 df-r 8195 df-lt 8198

This theorem is used by: (None)

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