divgt1b - Mathbox for Alexander van der Vekens

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Theorem divgt1b 45380

Description: The ratio of a real number to a positive real number is greater than 1 iff the divisor (the positive real number) is less than the dividend (the real number). (Contributed by AV, 30-May-2020.)

**Assertion**
Ref	Expression
divgt1b	⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 1 < (𝐵 / 𝐴)))

**Proof of Theorem divgt1b**
Step	Hyp	Ref	Expression
1		rpcn 12483	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℂ)
2	1	adantr 484	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ) → 𝐴 ∈ ℂ)
3	2	mulid2d 10738	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ) → (1 · 𝐴) = 𝐴)
4	3	eqcomd 2744	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ) → 𝐴 = (1 · 𝐴))
5	4	breq1d 5041	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (1 · 𝐴) < 𝐵))
6		1red 10721	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ) → 1 ∈ ℝ)
7		simpr 488	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℝ)
8		rpregt0 12487	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 ∈ ℝ ∧ 0 < 𝐴))
9	8	adantr 484	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ) → (𝐴 ∈ ℝ ∧ 0 < 𝐴))
10		ltmuldiv 11592	. . 3 ⊢ ((1 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) → ((1 · 𝐴) < 𝐵 ↔ 1 < (𝐵 / 𝐴)))
11	6, 7, 9, 10	syl3anc 1372	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ) → ((1 · 𝐴) < 𝐵 ↔ 1 < (𝐵 / 𝐴)))
12	5, 11	bitrd 282	1 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 1 < (𝐵 / 𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2113 class class class wbr 5031 (class class class)co 7171 ℂcc 10614 ℝcr 10615 0cc0 10616 1c1 10617 · cmul 10621 < clt 10754 / cdiv 11376 ℝ⁺crp 12473

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7480 ax-resscn 10673 ax-1cn 10674 ax-icn 10675 ax-addcl 10676 ax-addrcl 10677 ax-mulcl 10678 ax-mulrcl 10679 ax-mulcom 10680 ax-addass 10681 ax-mulass 10682 ax-distr 10683 ax-i2m1 10684 ax-1ne0 10685 ax-1rid 10686 ax-rnegex 10687 ax-rrecex 10688 ax-cnre 10689 ax-pre-lttri 10690 ax-pre-lttrn 10691 ax-pre-ltadd 10692 ax-pre-mulgt0 10693

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3683 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-op 4524 df-uni 4798 df-br 5032 df-opab 5094 df-mpt 5112 df-id 5430 df-po 5443 df-so 5444 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7128 df-ov 7174 df-oprab 7175 df-mpo 7176 df-er 8321 df-en 8557 df-dom 8558 df-sdom 8559 df-pnf 10756 df-mnf 10757 df-xr 10758 df-ltxr 10759 df-le 10760 df-sub 10951 df-neg 10952 df-div 11377 df-rp 12474

This theorem is referenced by: pw2m1lepw2m1 45387

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