divge1b - Mathbox for Alexander van der Vekens

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Theorem divge1b 48505

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 12969	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℂ)
2	1	mullidd 11199	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (1 · 𝐴) = 𝐴)
3	2	eqcomd 2736	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 = (1 · 𝐴))
4	3	adantr 480	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ) → 𝐴 = (1 · 𝐴))
5	4	breq1d 5120	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (1 · 𝐴) ≤ 𝐵))
6		1red 11182	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ) → 1 ∈ ℝ)
7		simpr 484	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℝ)
8		rpregt0 12973	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 ∈ ℝ ∧ 0 < 𝐴))
9	8	adantr 480	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ) → (𝐴 ∈ ℝ ∧ 0 < 𝐴))
10		lemuldiv 12070	. . 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 5110 (class class class)co 7390 ℝcr 11074 0cc0 11075 1c1 11076 · cmul 11080 < clt 11215 ≤ cle 11216 / cdiv 11842 ℝ⁺crp 12958

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 2702 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-po 5549 df-so 5550 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-rp 12959

This theorem is referenced by: fldivexpfllog2 48558

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