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Mirrors > Home > MPE Home > Th. List > mul4d | Structured version Visualization version GIF version |
Description: Rearrangement of 4 factors. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
muld.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
addcomd.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
addcand.3 | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
mul4d.4 | ⊢ (𝜑 → 𝐷 ∈ ℂ) |
Ref | Expression |
---|---|
mul4d | ⊢ (𝜑 → ((𝐴 · 𝐵) · (𝐶 · 𝐷)) = ((𝐴 · 𝐶) · (𝐵 · 𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | muld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | addcomd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
3 | addcand.3 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
4 | mul4d.4 | . 2 ⊢ (𝜑 → 𝐷 ∈ ℂ) | |
5 | mul4 10796 | . 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 · 𝐵) · (𝐶 · 𝐷)) = ((𝐴 · 𝐶) · (𝐵 · 𝐷))) | |
6 | 1, 2, 3, 4, 5 | syl22anc 834 | 1 ⊢ (𝜑 → ((𝐴 · 𝐵) · (𝐶 · 𝐷)) = ((𝐴 · 𝐶) · (𝐵 · 𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 (class class class)co 7145 ℂcc 10523 · cmul 10530 |
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-mulcl 10587 ax-mulcom 10589 ax-mulass 10591 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 df-ov 7148 |
This theorem is referenced by: remullem 14475 absmul 14642 binomrisefac 15384 cosadd 15506 tanadd 15508 eulerthlem2 16107 mul4sqlem 16277 odadd2 18898 itgmulc2 24361 plymullem1 24731 chordthmlem4 25340 heron 25343 quartlem1 25362 dchrmulcl 25752 bposlem9 25795 lgsdir 25835 lgsdi 25837 lgsquad2lem1 25887 chtppilimlem1 25976 rplogsumlem1 25987 dchrvmasumlem1 25998 dchrvmasum2lem 25999 chpdifbndlem1 26056 pntlemf 26108 brbtwn2 26618 colinearalglem4 26622 madjusmdetlem4 30994 hgt750lemf 31823 hgt750leme 31828 circum 32814 itgmulc2nc 34841 pellexlem6 39309 pell1234qrmulcl 39330 rmxyadd 39396 wallispi2lem2 42234 dirkertrigeqlem3 42262 cevathlem1 43001 itsclc0xyqsolr 44684 |
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