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Mirrors > Home > MPE Home > Th. List > divdivdivd | Structured version Visualization version GIF version |
Description: Division of two ratios. Theorem I.15 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) |
divdivdivd.7 | ⊢ (𝜑 → 𝐶 ≠ 0) |
Ref | Expression |
---|---|
divdivdivd | ⊢ (𝜑 → ((𝐴 / 𝐵) / (𝐶 / 𝐷)) = ((𝐴 · 𝐷) / (𝐵 · 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | div1d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | divcld.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
3 | divmuldivd.5 | . . 3 ⊢ (𝜑 → 𝐵 ≠ 0) | |
4 | 2, 3 | jca 514 | . 2 ⊢ (𝜑 → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) |
5 | divmuld.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
6 | divdivdivd.7 | . . 3 ⊢ (𝜑 → 𝐶 ≠ 0) | |
7 | 5, 6 | jca 514 | . 2 ⊢ (𝜑 → (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) |
8 | divmuldivd.4 | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℂ) | |
9 | divmuldivd.6 | . . 3 ⊢ (𝜑 → 𝐷 ≠ 0) | |
10 | 8, 9 | jca 514 | . 2 ⊢ (𝜑 → (𝐷 ∈ ℂ ∧ 𝐷 ≠ 0)) |
11 | divdivdiv 11341 | . 2 ⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 ≠ 0))) → ((𝐴 / 𝐵) / (𝐶 / 𝐷)) = ((𝐴 · 𝐷) / (𝐵 · 𝐶))) | |
12 | 1, 4, 7, 10, 11 | syl22anc 836 | 1 ⊢ (𝜑 → ((𝐴 / 𝐵) / (𝐶 / 𝐷)) = ((𝐴 · 𝐷) / (𝐵 · 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 (class class class)co 7156 ℂcc 10535 0cc0 10537 · cmul 10542 / cdiv 11297 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 |
This theorem is referenced by: pcadd 16225 pnt 26190 wallispilem4 42373 stirlinglem4 42382 stirlinglem10 42388 |
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