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Mirrors > Home > ILE Home > Th. List > subdid | GIF version |
Description: Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
mulm1d.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
mulnegd.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
subdid.3 | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
Ref | Expression |
---|---|
subdid | ⊢ (𝜑 → (𝐴 · (𝐵 − 𝐶)) = ((𝐴 · 𝐵) − (𝐴 · 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulm1d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | mulnegd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
3 | subdid.3 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
4 | subdi 8265 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 − 𝐶)) = ((𝐴 · 𝐵) − (𝐴 · 𝐶))) | |
5 | 1, 2, 3, 4 | syl3anc 1220 | 1 ⊢ (𝜑 → (𝐴 · (𝐵 − 𝐶)) = ((𝐴 · 𝐵) − (𝐴 · 𝐶))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1335 ∈ wcel 2128 (class class class)co 5827 ℂcc 7733 · cmul 7740 − cmin 8051 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4085 ax-pow 4138 ax-pr 4172 ax-setind 4499 ax-resscn 7827 ax-1cn 7828 ax-icn 7830 ax-addcl 7831 ax-addrcl 7832 ax-mulcl 7833 ax-addcom 7835 ax-addass 7837 ax-distr 7839 ax-i2m1 7840 ax-0id 7843 ax-rnegex 7844 ax-cnre 7846 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-br 3968 df-opab 4029 df-id 4256 df-xp 4595 df-rel 4596 df-cnv 4597 df-co 4598 df-dm 4599 df-iota 5138 df-fun 5175 df-fv 5181 df-riota 5783 df-ov 5830 df-oprab 5831 df-mpo 5832 df-sub 8053 |
This theorem is referenced by: cru 8482 recextlem1 8530 cju 8838 zneo 9271 lincmb01cmp 9914 iccf1o 9915 intfracq 10229 modqlt 10242 modqdi 10301 modqsubdir 10302 subsq 10535 crre 10769 remullem 10783 mulcn2 11221 fsumparts 11379 geosergap 11415 mertensabs 11446 tanval3ap 11623 tanaddap 11648 eirraplem 11685 bezoutlemnewy 11896 cncongr1 11996 eulerthlemh 12122 prmdiv 12126 prmdiveq 12127 dvmulxxbr 13162 tangtx 13255 qdencn 13695 |
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