![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > mulsubfacd | Structured version Visualization version GIF version |
Description: Multiplication followed by the subtraction of a factor. (Contributed by Alexander van der Vekens, 28-Aug-2018.) |
Ref | Expression |
---|---|
muls1d.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
muls1d.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
Ref | Expression |
---|---|
mulsubfacd | ⊢ (𝜑 → ((𝐴 · 𝐵) − 𝐵) = ((𝐴 − 1) · 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | muls1d.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | 1cnd 10436 | . . 3 ⊢ (𝜑 → 1 ∈ ℂ) | |
3 | muls1d.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
4 | 1, 2, 3 | subdird 10900 | . 2 ⊢ (𝜑 → ((𝐴 − 1) · 𝐵) = ((𝐴 · 𝐵) − (1 · 𝐵))) |
5 | 3 | mulid2d 10460 | . . 3 ⊢ (𝜑 → (1 · 𝐵) = 𝐵) |
6 | 5 | oveq2d 6994 | . 2 ⊢ (𝜑 → ((𝐴 · 𝐵) − (1 · 𝐵)) = ((𝐴 · 𝐵) − 𝐵)) |
7 | 4, 6 | eqtr2d 2815 | 1 ⊢ (𝜑 → ((𝐴 · 𝐵) − 𝐵) = ((𝐴 − 1) · 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2050 (class class class)co 6978 ℂcc 10335 1c1 10338 · cmul 10342 − cmin 10672 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-resscn 10394 ax-1cn 10395 ax-icn 10396 ax-addcl 10397 ax-addrcl 10398 ax-mulcl 10399 ax-mulrcl 10400 ax-mulcom 10401 ax-addass 10402 ax-mulass 10403 ax-distr 10404 ax-i2m1 10405 ax-1ne0 10406 ax-1rid 10407 ax-rnegex 10408 ax-rrecex 10409 ax-cnre 10410 ax-pre-lttri 10411 ax-pre-lttrn 10412 ax-pre-ltadd 10413 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-op 4449 df-uni 4714 df-br 4931 df-opab 4993 df-mpt 5010 df-id 5313 df-po 5327 df-so 5328 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-riota 6939 df-ov 6981 df-oprab 6982 df-mpo 6983 df-er 8091 df-en 8309 df-dom 8310 df-sdom 8311 df-pnf 10478 df-mnf 10479 df-ltxr 10481 df-sub 10674 |
This theorem is referenced by: subhalfhalf 11684 gausslemma2dlem1a 25646 numclwwlk3lem1 27942 dirkertrigeqlem2 41816 fourierdlem42 41866 fourierdlem48 41871 fpprwpprb 43274 m1modmmod 43950 |
Copyright terms: Public domain | W3C validator |