divge1b - Mathbox for Alexander van der Vekens

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Theorem divge1b 48511

Description: The ratio of a real number to a positive real number is greater than or equal to 1 iff the divisor (the positive real number) is less than or equal to the dividend (the real number). (Contributed by AV, 26-May-2020.)

**Assertion**
Ref	Expression
divge1b	⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ 1 ≤ (𝐵 / 𝐴)))

**Proof of Theorem divge1b**
Step	Hyp	Ref	Expression
1		rpcn 12892	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℂ)
2	1	mullidd 11121	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (1 · 𝐴) = 𝐴)
3	2	eqcomd 2735	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 = (1 · 𝐴))
4	3	adantr 480	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ) → 𝐴 = (1 · 𝐴))
5	4	breq1d 5098	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (1 · 𝐴) ≤ 𝐵))
6		1red 11104	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ) → 1 ∈ ℝ)
7		simpr 484	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℝ)
8		rpregt0 12896	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 ∈ ℝ ∧ 0 < 𝐴))
9	8	adantr 480	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ) → (𝐴 ∈ ℝ ∧ 0 < 𝐴))
10		lemuldiv 11993	. . 3 ⊢ ((1 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) → ((1 · 𝐴) ≤ 𝐵 ↔ 1 ≤ (𝐵 / 𝐴)))
11	6, 7, 9, 10	syl3anc 1373	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ) → ((1 · 𝐴) ≤ 𝐵 ↔ 1 ≤ (𝐵 / 𝐴)))
12	5, 11	bitrd 279	1 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ 1 ≤ (𝐵 / 𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 class class class wbr 5088 (class class class)co 7340 ℝcr 10996 0cc0 10997 1c1 10998 · cmul 11002 < clt 11137 ≤ cle 11138 / cdiv 11765 ℝ⁺crp 12881

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5367 ax-un 7662 ax-resscn 11054 ax-1cn 11055 ax-icn 11056 ax-addcl 11057 ax-addrcl 11058 ax-mulcl 11059 ax-mulrcl 11060 ax-mulcom 11061 ax-addass 11062 ax-mulass 11063 ax-distr 11064 ax-i2m1 11065 ax-1ne0 11066 ax-1rid 11067 ax-rnegex 11068 ax-rrecex 11069 ax-cnre 11070 ax-pre-lttri 11071 ax-pre-lttrn 11072 ax-pre-ltadd 11073 ax-pre-mulgt0 11074

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3393 df-v 3435 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-br 5089 df-opab 5151 df-mpt 5170 df-id 5508 df-po 5521 df-so 5522 df-xp 5619 df-rel 5620 df-cnv 5621 df-co 5622 df-dm 5623 df-rn 5624 df-res 5625 df-ima 5626 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-riota 7297 df-ov 7343 df-oprab 7344 df-mpo 7345 df-er 8616 df-en 8864 df-dom 8865 df-sdom 8866 df-pnf 11139 df-mnf 11140 df-xr 11141 df-ltxr 11142 df-le 11143 df-sub 11337 df-neg 11338 df-div 11766 df-rp 12882

This theorem is referenced by: fldivexpfllog2 48564

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