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Mirrors > Home > MPE Home > Th. List > ltdiv1i | Structured version Visualization version GIF version |
Description: Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.) |
Ref | Expression |
---|---|
ltplus1.1 | ⊢ 𝐴 ∈ ℝ |
prodgt0.2 | ⊢ 𝐵 ∈ ℝ |
ltmul1.3 | ⊢ 𝐶 ∈ ℝ |
Ref | Expression |
---|---|
ltdiv1i | ⊢ (0 < 𝐶 → (𝐴 < 𝐵 ↔ (𝐴 / 𝐶) < (𝐵 / 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltmul1.3 | . 2 ⊢ 𝐶 ∈ ℝ | |
2 | ltplus1.1 | . . 3 ⊢ 𝐴 ∈ ℝ | |
3 | prodgt0.2 | . . 3 ⊢ 𝐵 ∈ ℝ | |
4 | ltdiv1 12065 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 < 𝐵 ↔ (𝐴 / 𝐶) < (𝐵 / 𝐶))) | |
5 | 2, 3, 4 | mp3an12 1452 | . 2 ⊢ ((𝐶 ∈ ℝ ∧ 0 < 𝐶) → (𝐴 < 𝐵 ↔ (𝐴 / 𝐶) < (𝐵 / 𝐶))) |
6 | 1, 5 | mpan 689 | 1 ⊢ (0 < 𝐶 → (𝐴 < 𝐵 ↔ (𝐴 / 𝐶) < (𝐵 / 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∈ wcel 2107 class class class wbr 5144 (class class class)co 7396 ℝcr 11096 0cc0 11097 < clt 11235 / cdiv 11858 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-er 8691 df-en 8928 df-dom 8929 df-sdom 8930 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-div 11859 |
This theorem is referenced by: ltdiv1ii 12130 hgt750lem 33594 |
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