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Mirrors > Home > MPE Home > Th. List > subdii | Structured version Visualization version GIF version |
Description: Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 26-Nov-1994.) |
Ref | Expression |
---|---|
mulm1.1 | ⊢ 𝐴 ∈ ℂ |
mulneg.2 | ⊢ 𝐵 ∈ ℂ |
subdi.3 | ⊢ 𝐶 ∈ ℂ |
Ref | Expression |
---|---|
subdii | ⊢ (𝐴 · (𝐵 − 𝐶)) = ((𝐴 · 𝐵) − (𝐴 · 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulm1.1 | . 2 ⊢ 𝐴 ∈ ℂ | |
2 | mulneg.2 | . 2 ⊢ 𝐵 ∈ ℂ | |
3 | subdi.3 | . 2 ⊢ 𝐶 ∈ ℂ | |
4 | subdi 11644 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 − 𝐶)) = ((𝐴 · 𝐵) − (𝐴 · 𝐶))) | |
5 | 1, 2, 3, 4 | mp3an 1462 | 1 ⊢ (𝐴 · (𝐵 − 𝐶)) = ((𝐴 · 𝐵) − (𝐴 · 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 (class class class)co 7406 ℂcc 11105 · cmul 11112 − cmin 11441 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11247 df-mnf 11248 df-ltxr 11250 df-sub 11443 |
This theorem is referenced by: nummac 12719 0.999... 15824 sincos3rdpi 26018 birthday 26449 ax5seglem7 28183 ex-ind-dvds 29704 ip0i 30066 siilem1 30092 sqdeccom12 41199 areaquad 41951 inductionexd 42892 |
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