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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 512 | . 2 ⊢ (𝜑 → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) |
5 | divmuld.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
6 | divdivdivd.7 | . . 3 ⊢ (𝜑 → 𝐶 ≠ 0) | |
7 | 5, 6 | jca 512 | . 2 ⊢ (𝜑 → (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) |
8 | divmuldivd.4 | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℂ) | |
9 | divmuldivd.6 | . . 3 ⊢ (𝜑 → 𝐷 ≠ 0) | |
10 | 8, 9 | jca 512 | . 2 ⊢ (𝜑 → (𝐷 ∈ ℂ ∧ 𝐷 ≠ 0)) |
11 | divdivdiv 11329 | . 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 834 | 1 ⊢ (𝜑 → ((𝐴 / 𝐵) / (𝐶 / 𝐷)) = ((𝐴 · 𝐷) / (𝐵 · 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 (class class class)co 7145 ℂcc 10523 0cc0 10525 · cmul 10530 / cdiv 11285 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 |
This theorem is referenced by: pcadd 16213 pnt 26117 wallispilem4 42230 stirlinglem4 42239 stirlinglem10 42245 |
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