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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 20230 | . . 3 ⊢ (𝜑 → (𝑋 ∙ 𝐹) = ((𝐷 × {𝑋}) ∘f · 𝐹)) |
10 | 9 | fveq1d 6675 | . 2 ⊢ (𝜑 → ((𝑋 ∙ 𝐹)‘𝑌) = (((𝐷 × {𝑋}) ∘f · 𝐹)‘𝑌)) |
11 | mplvscaval.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐷) | |
12 | ovex 7192 | . . . . . 6 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
13 | 6, 12 | rabex2 5240 | . . . . 5 ⊢ 𝐷 ∈ V |
14 | 13 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ V) |
15 | eqid 2824 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
16 | 1, 15, 4, 6, 8 | mplelf 20216 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐷⟶(Base‘𝑅)) |
17 | 16 | ffnd 6518 | . . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐷) |
18 | eqidd 2825 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐷) → (𝐹‘𝑌) = (𝐹‘𝑌)) | |
19 | 14, 7, 17, 18 | ofc1 7435 | . . 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 2859 | 1 ⊢ (𝜑 → ((𝑋 ∙ 𝐹)‘𝑌) = (𝑋 · (𝐹‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 {crab 3145 Vcvv 3497 {csn 4570 × cxp 5556 ◡ccnv 5557 “ cima 5561 ‘cfv 6358 (class class class)co 7159 ∘f cof 7410 ↑m cmap 8409 Fincfn 8512 ℕcn 11641 ℕ0cn0 11900 Basecbs 16486 .rcmulr 16569 ·𝑠 cvsca 16572 mPoly cmpl 20136 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-om 7584 df-1st 7692 df-2nd 7693 df-supp 7834 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fsupp 8837 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-sca 16584 df-vsca 16585 df-tset 16587 df-psr 20139 df-mpl 20141 |
This theorem is referenced by: mhpvscacl 20344 mdegvscale 24672 |
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