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Mirrors > Home > MPE Home > Th. List > mplvscaval | Structured version Visualization version GIF version |
Description: The scalar multiplication operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.) |
Ref | Expression |
---|---|
mplvsca.p | ⊢ 𝑃 = (𝐼 mPoly 𝑅) |
mplvsca.n | ⊢ ∙ = ( ·𝑠 ‘𝑃) |
mplvsca.k | ⊢ 𝐾 = (Base‘𝑅) |
mplvsca.b | ⊢ 𝐵 = (Base‘𝑃) |
mplvsca.m | ⊢ · = (.r‘𝑅) |
mplvsca.d | ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} |
mplvsca.x | ⊢ (𝜑 → 𝑋 ∈ 𝐾) |
mplvsca.f | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
mplvscaval.y | ⊢ (𝜑 → 𝑌 ∈ 𝐷) |
Ref | Expression |
---|---|
mplvscaval | ⊢ (𝜑 → ((𝑋 ∙ 𝐹)‘𝑌) = (𝑋 · (𝐹‘𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mplvsca.p | . . . 4 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
2 | mplvsca.n | . . . 4 ⊢ ∙ = ( ·𝑠 ‘𝑃) | |
3 | mplvsca.k | . . . 4 ⊢ 𝐾 = (Base‘𝑅) | |
4 | mplvsca.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
5 | mplvsca.m | . . . 4 ⊢ · = (.r‘𝑅) | |
6 | mplvsca.d | . . . 4 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
7 | mplvsca.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐾) | |
8 | mplvsca.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | mplvsca 21419 | . . 3 ⊢ (𝜑 → (𝑋 ∙ 𝐹) = ((𝐷 × {𝑋}) ∘f · 𝐹)) |
10 | 9 | fveq1d 6844 | . 2 ⊢ (𝜑 → ((𝑋 ∙ 𝐹)‘𝑌) = (((𝐷 × {𝑋}) ∘f · 𝐹)‘𝑌)) |
11 | mplvscaval.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐷) | |
12 | ovex 7390 | . . . . . 6 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
13 | 6, 12 | rabex2 5291 | . . . . 5 ⊢ 𝐷 ∈ V |
14 | 13 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ V) |
15 | eqid 2736 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
16 | 1, 15, 4, 6, 8 | mplelf 21404 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐷⟶(Base‘𝑅)) |
17 | 16 | ffnd 6669 | . . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐷) |
18 | eqidd 2737 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐷) → (𝐹‘𝑌) = (𝐹‘𝑌)) | |
19 | 14, 7, 17, 18 | ofc1 7643 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐷) → (((𝐷 × {𝑋}) ∘f · 𝐹)‘𝑌) = (𝑋 · (𝐹‘𝑌))) |
20 | 11, 19 | mpdan 685 | . 2 ⊢ (𝜑 → (((𝐷 × {𝑋}) ∘f · 𝐹)‘𝑌) = (𝑋 · (𝐹‘𝑌))) |
21 | 10, 20 | eqtrd 2776 | 1 ⊢ (𝜑 → ((𝑋 ∙ 𝐹)‘𝑌) = (𝑋 · (𝐹‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {crab 3407 Vcvv 3445 {csn 4586 × cxp 5631 ◡ccnv 5632 “ cima 5636 ‘cfv 6496 (class class class)co 7357 ∘f cof 7615 ↑m cmap 8765 Fincfn 8883 ℕcn 12153 ℕ0cn0 12413 Basecbs 17083 .rcmulr 17134 ·𝑠 cvsca 17137 mPoly cmpl 21308 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-uz 12764 df-fz 13425 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-sca 17149 df-vsca 17150 df-tset 17152 df-psr 21311 df-mpl 21313 |
This theorem is referenced by: mhpvscacl 21544 mdegvscale 25440 |
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