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Mirrors > Home > MPE Home > Th. List > ltmul2dd | Structured version Visualization version GIF version |
Description: Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by Mario Carneiro, 30-May-2016.) |
Ref | Expression |
---|---|
ltmul1d.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ltmul1d.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
ltmul1d.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ+) |
ltdiv1dd.4 | ⊢ (𝜑 → 𝐴 < 𝐵) |
Ref | Expression |
---|---|
ltmul2dd | ⊢ (𝜑 → (𝐶 · 𝐴) < (𝐶 · 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltdiv1dd.4 | . 2 ⊢ (𝜑 → 𝐴 < 𝐵) | |
2 | ltmul1d.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
3 | ltmul1d.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
4 | ltmul1d.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ+) | |
5 | 2, 3, 4 | ltmul2d 13114 | . 2 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐶 · 𝐴) < (𝐶 · 𝐵))) |
6 | 1, 5 | mpbid 231 | 1 ⊢ (𝜑 → (𝐶 · 𝐴) < (𝐶 · 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 class class class wbr 5155 (class class class)co 7426 ℝcr 11159 · cmul 11165 < clt 11300 ℝ+crp 13030 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-resscn 11217 ax-1cn 11218 ax-icn 11219 ax-addcl 11220 ax-addrcl 11221 ax-mulcl 11222 ax-mulrcl 11223 ax-mulcom 11224 ax-addass 11225 ax-mulass 11226 ax-distr 11227 ax-i2m1 11228 ax-1ne0 11229 ax-1rid 11230 ax-rnegex 11231 ax-rrecex 11232 ax-cnre 11233 ax-pre-lttri 11234 ax-pre-lttrn 11235 ax-pre-ltadd 11236 ax-pre-mulgt0 11237 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4916 df-br 5156 df-opab 5218 df-mpt 5239 df-id 5582 df-po 5596 df-so 5597 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-er 8736 df-en 8977 df-dom 8978 df-sdom 8979 df-pnf 11302 df-mnf 11303 df-ltxr 11305 df-sub 11498 df-neg 11499 df-rp 13031 |
This theorem is referenced by: mul2lt0bi 13136 reccn2 15601 mertenslem1 15890 nrginvrcnlem 24702 nmoleub2lem3 25136 bclbnd 27312 pntlemb 27629 hgt750lemd 34496 knoppndvlem12 36228 itg2addnclem2 37375 cntotbnd 37499 aks4d1p8d2 41786 2ap1caineq 41845 fltnltalem 42332 fltnlta 42333 sqrlearg 45189 0ellimcdiv 45288 stirlinglem5 45717 2itscp 48187 |
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