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Mirrors > Home > MPE Home > Th. List > ltdiv1ii | 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 |
ltmul1i.4 | ⊢ 0 < 𝐶 |
Ref | Expression |
---|---|
ltdiv1ii | ⊢ (𝐴 < 𝐵 ↔ (𝐴 / 𝐶) < (𝐵 / 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltmul1i.4 | . 2 ⊢ 0 < 𝐶 | |
2 | ltplus1.1 | . . 3 ⊢ 𝐴 ∈ ℝ | |
3 | prodgt0.2 | . . 3 ⊢ 𝐵 ∈ ℝ | |
4 | ltmul1.3 | . . 3 ⊢ 𝐶 ∈ ℝ | |
5 | 2, 3, 4 | ltdiv1i 12180 | . 2 ⊢ (0 < 𝐶 → (𝐴 < 𝐵 ↔ (𝐴 / 𝐶) < (𝐵 / 𝐶))) |
6 | 1, 5 | ax-mp 5 | 1 ⊢ (𝐴 < 𝐵 ↔ (𝐴 / 𝐶) < (𝐵 / 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∈ wcel 2098 class class class wbr 5152 (class class class)co 7423 ℝcr 11153 0cc0 11154 < clt 11294 / cdiv 11917 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 ax-pre-mulgt0 11231 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5579 df-po 5593 df-so 5594 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-er 8733 df-en 8974 df-dom 8975 df-sdom 8976 df-pnf 11296 df-mnf 11297 df-xr 11298 df-ltxr 11299 df-le 11300 df-sub 11492 df-neg 11493 df-div 11918 |
This theorem is referenced by: ef01bndlem 16181 cos01gt0 16188 sincos4thpi 26533 pigt3 26537 log2le1 26970 bposlem8 27312 dp2ltsuc 32736 sqsscirc1 33679 problem5 35449 stoweidlem26 45584 |
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