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Mirrors > Home > MPE Home > Th. List > divmuldivd | Structured version Visualization version GIF version |
Description: Multiplication of two ratios. Theorem I.14 of [Apostol] p. 18. (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 |
---|---|
divmuldivd | ⊢ (𝜑 → ((𝐴 / 𝐵) · (𝐶 / 𝐷)) = ((𝐴 · 𝐶) / (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 553 | . 2 ⊢ (𝜑 → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) |
6 | divmuldivd.4 | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℂ) | |
7 | divmuldivd.6 | . . 3 ⊢ (𝜑 → 𝐷 ≠ 0) | |
8 | 6, 7 | jca 553 | . 2 ⊢ (𝜑 → (𝐷 ∈ ℂ ∧ 𝐷 ≠ 0)) |
9 | divmuldiv 10763 | . 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 ≠ 0))) → ((𝐴 / 𝐵) · (𝐶 / 𝐷)) = ((𝐴 · 𝐶) / (𝐵 · 𝐷))) | |
10 | 1, 2, 5, 8, 9 | syl22anc 1367 | 1 ⊢ (𝜑 → ((𝐴 / 𝐵) · (𝐶 / 𝐷)) = ((𝐴 · 𝐶) / (𝐵 · 𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 (class class class)co 6690 ℂcc 9972 0cc0 9974 · cmul 9979 / cdiv 10722 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-po 5064 df-so 5065 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 |
This theorem is referenced by: prodfrec 14671 efcllem 14852 efaddlem 14867 tanaddlem 14940 isprm5 15466 pcpremul 15595 pcqmul 15605 mul4sqlem 15704 dvcnsqrt 24530 mcubic 24619 cubic2 24620 quart1lem 24627 log2tlbnd 24717 basellem5 24856 basellem8 24859 dchrinvcl 25023 dchrmusum2 25228 ttgcontlem1 25810 qqhrhm 30161 faclim2 31760 radcnvrat 38830 bccp1k 38857 dvnprodlem2 40480 wallispilem4 40603 wallispi2lem1 40606 wallispi2lem2 40607 stirlinglem1 40609 stirlinglem3 40611 stirlinglem4 40612 stirlinglem6 40614 stirlinglem10 40618 |
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