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Mirrors > Home > MPE Home > Th. List > mul32d | Structured version Visualization version GIF version |
Description: Commutative/associative law that swaps the last two factors in a triple product. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
muld.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
addcomd.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
addcand.3 | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
Ref | Expression |
---|---|
mul32d | ⊢ (𝜑 → ((𝐴 · 𝐵) · 𝐶) = ((𝐴 · 𝐶) · 𝐵)) |
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 | mul32 10794 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = ((𝐴 · 𝐶) · 𝐵)) | |
5 | 1, 2, 3, 4 | syl3anc 1363 | 1 ⊢ (𝜑 → ((𝐴 · 𝐵) · 𝐶) = ((𝐴 · 𝐶) · 𝐵)) |
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-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: conjmul 11345 modmul1 13280 binom3 13573 bernneq 13578 expmulnbnd 13584 discr 13589 bcm1k 13663 bcp1n 13664 reccn2 14941 binomlem 15172 binomfallfaclem2 15382 tanadd 15508 eirrlem 15545 dvds2ln 15630 bezoutlem4 15878 divgcdcoprm0 15997 modprm0 16130 nrginvrcnlem 23227 tcphcphlem2 23766 csbren 23929 radcnvlem1 24928 tanarg 25129 cxpeq 25265 quad2 25344 binom4 25355 dquartlem2 25357 dquart 25358 quart1lem 25360 dvatan 25440 log2cnv 25449 basellem8 25592 bcmono 25780 gausslemma2d 25877 lgsquadlem1 25883 2lgslem3b 25900 2lgslem3c 25901 2lgslem3d 25902 rplogsumlem1 25987 dchrisumlem2 25993 chpdifbndlem1 26056 selberg3lem1 26060 selberg4 26064 selberg3r 26072 pntrlog2bndlem2 26081 pntrlog2bndlem3 26082 pntrlog2bndlem5 26084 pntlemf 26108 pntlemo 26110 ostth2lem1 26121 ostth2lem3 26138 logdivsqrle 31820 circum 32814 jm2.25 39474 jm2.27c 39482 binomcxplemnotnn0 40565 dvasinbx 42081 stirlinglem3 42238 dirkercncflem2 42266 cevathlem1 43001 itschlc0yqe 44675 |
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