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Mirrors > Home > MPE Home > Th. List > divdivdivi | Structured version Visualization version GIF version |
Description: Division of two ratios. Theorem I.15 of [Apostol] p. 18. (Contributed by NM, 22-Feb-1995.) |
Ref | Expression |
---|---|
divclz.1 | ⊢ 𝐴 ∈ ℂ |
divclz.2 | ⊢ 𝐵 ∈ ℂ |
divmulz.3 | ⊢ 𝐶 ∈ ℂ |
divmuldiv.4 | ⊢ 𝐷 ∈ ℂ |
divmuldiv.5 | ⊢ 𝐵 ≠ 0 |
divmuldiv.6 | ⊢ 𝐷 ≠ 0 |
divdivdiv.7 | ⊢ 𝐶 ≠ 0 |
Ref | Expression |
---|---|
divdivdivi | ⊢ ((𝐴 / 𝐵) / (𝐶 / 𝐷)) = ((𝐴 · 𝐷) / (𝐵 · 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | divclz.1 | . 2 ⊢ 𝐴 ∈ ℂ | |
2 | divclz.2 | . . 3 ⊢ 𝐵 ∈ ℂ | |
3 | divmuldiv.5 | . . 3 ⊢ 𝐵 ≠ 0 | |
4 | 2, 3 | pm3.2i 473 | . 2 ⊢ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) |
5 | divmulz.3 | . . 3 ⊢ 𝐶 ∈ ℂ | |
6 | divdivdiv.7 | . . 3 ⊢ 𝐶 ≠ 0 | |
7 | 5, 6 | pm3.2i 473 | . 2 ⊢ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0) |
8 | divmuldiv.4 | . . 3 ⊢ 𝐷 ∈ ℂ | |
9 | divmuldiv.6 | . . 3 ⊢ 𝐷 ≠ 0 | |
10 | 8, 9 | pm3.2i 473 | . 2 ⊢ (𝐷 ∈ ℂ ∧ 𝐷 ≠ 0) |
11 | divdivdiv 11335 | . 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 | mp4an 691 | 1 ⊢ ((𝐴 / 𝐵) / (𝐶 / 𝐷)) = ((𝐴 · 𝐷) / (𝐵 · 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 (class class class)co 7150 ℂcc 10529 0cc0 10531 · cmul 10536 / cdiv 11291 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3497 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 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-po 5469 df-so 5470 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 |
This theorem is referenced by: log2tlbnd 25517 |
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