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Mirrors > Home > MPE Home > Th. List > divsubdivd | Structured version Visualization version GIF version |
Description: Subtraction of two ratios. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
div1d.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
divcld.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
divmuld.3 | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
divmuldivd.4 | ⊢ (𝜑 → 𝐷 ∈ ℂ) |
divmuldivd.5 | ⊢ (𝜑 → 𝐵 ≠ 0) |
divmuldivd.6 | ⊢ (𝜑 → 𝐷 ≠ 0) |
Ref | Expression |
---|---|
divsubdivd | ⊢ (𝜑 → ((𝐴 / 𝐵) − (𝐶 / 𝐷)) = (((𝐴 · 𝐷) − (𝐶 · 𝐵)) / (𝐵 · 𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | div1d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | divmuld.3 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
3 | divcld.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
4 | divmuldivd.5 | . . 3 ⊢ (𝜑 → 𝐵 ≠ 0) | |
5 | 3, 4 | jca 507 | . 2 ⊢ (𝜑 → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) |
6 | divmuldivd.4 | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℂ) | |
7 | divmuldivd.6 | . . 3 ⊢ (𝜑 → 𝐷 ≠ 0) | |
8 | 6, 7 | jca 507 | . 2 ⊢ (𝜑 → (𝐷 ∈ ℂ ∧ 𝐷 ≠ 0)) |
9 | divsubdiv 11091 | . 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 ≠ 0))) → ((𝐴 / 𝐵) − (𝐶 / 𝐷)) = (((𝐴 · 𝐷) − (𝐶 · 𝐵)) / (𝐵 · 𝐷))) | |
10 | 1, 2, 5, 8, 9 | syl22anc 829 | 1 ⊢ (𝜑 → ((𝐴 / 𝐵) − (𝐶 / 𝐷)) = (((𝐴 · 𝐷) − (𝐶 · 𝐵)) / (𝐵 · 𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2106 ≠ wne 2968 (class class class)co 6922 ℂcc 10270 0cc0 10272 · cmul 10277 − cmin 10606 / 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 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 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 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 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: subrec 11204 reclimc 40775 |
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