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Mirrors > Home > MPE Home > Th. List > divcan5rd | Structured version Visualization version GIF version |
Description: Cancellation of common factor in a ratio. (Contributed by Mario Carneiro, 1-Jan-2017.) |
Ref | Expression |
---|---|
div1d.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
divcld.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
divmuld.3 | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
divmuld.4 | ⊢ (𝜑 → 𝐵 ≠ 0) |
divdiv23d.5 | ⊢ (𝜑 → 𝐶 ≠ 0) |
Ref | Expression |
---|---|
divcan5rd | ⊢ (𝜑 → ((𝐴 · 𝐶) / (𝐵 · 𝐶)) = (𝐴 / 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | div1d.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | divmuld.3 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
3 | 1, 2 | mulcomd 10398 | . . 3 ⊢ (𝜑 → (𝐴 · 𝐶) = (𝐶 · 𝐴)) |
4 | divcld.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
5 | 4, 2 | mulcomd 10398 | . . 3 ⊢ (𝜑 → (𝐵 · 𝐶) = (𝐶 · 𝐵)) |
6 | 3, 5 | oveq12d 6940 | . 2 ⊢ (𝜑 → ((𝐴 · 𝐶) / (𝐵 · 𝐶)) = ((𝐶 · 𝐴) / (𝐶 · 𝐵))) |
7 | divmuld.4 | . . 3 ⊢ (𝜑 → 𝐵 ≠ 0) | |
8 | divdiv23d.5 | . . 3 ⊢ (𝜑 → 𝐶 ≠ 0) | |
9 | 1, 4, 2, 7, 8 | divcan5d 11177 | . 2 ⊢ (𝜑 → ((𝐶 · 𝐴) / (𝐶 · 𝐵)) = (𝐴 / 𝐵)) |
10 | 6, 9 | eqtrd 2814 | 1 ⊢ (𝜑 → ((𝐴 · 𝐶) / (𝐵 · 𝐶)) = (𝐴 / 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 (class class class)co 6922 ℂcc 10270 0cc0 10272 · cmul 10277 / cdiv 11032 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-po 5274 df-so 5275 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 |
This theorem is referenced by: dvmptdiv 24174 dvtaylp 24561 chordthmlem2 25011 itg2addnclem 34088 stirlinglem1 41222 dirkertrigeqlem2 41247 dirkercncflem2 41252 sigardiv 41981 |
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