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Mirrors > Home > MPE Home > Th. List > Mathboxes > evlsscaval | Structured version Visualization version GIF version |
Description: Polynomial evaluation builder for a scalar. Compare evl1scad 21107. Note that scalar multiplication by 𝑋 is the same as vector multiplication by (𝐴‘𝑋) by asclmul1 20701. (Contributed by SN, 27-Jul-2024.) |
Ref | Expression |
---|---|
evlsscaval.q | ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅) |
evlsscaval.p | ⊢ 𝑃 = (𝐼 mPoly 𝑈) |
evlsscaval.u | ⊢ 𝑈 = (𝑆 ↾s 𝑅) |
evlsscaval.k | ⊢ 𝐾 = (Base‘𝑆) |
evlsscaval.b | ⊢ 𝐵 = (Base‘𝑃) |
evlsscaval.a | ⊢ 𝐴 = (algSc‘𝑃) |
evlsscaval.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
evlsscaval.s | ⊢ (𝜑 → 𝑆 ∈ CRing) |
evlsscaval.r | ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) |
evlsscaval.x | ⊢ (𝜑 → 𝑋 ∈ 𝑅) |
evlsscaval.l | ⊢ (𝜑 → 𝐿 ∈ (𝐾 ↑m 𝐼)) |
Ref | Expression |
---|---|
evlsscaval | ⊢ (𝜑 → ((𝐴‘𝑋) ∈ 𝐵 ∧ ((𝑄‘(𝐴‘𝑋))‘𝐿) = 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evlsscaval.p | . . . 4 ⊢ 𝑃 = (𝐼 mPoly 𝑈) | |
2 | evlsscaval.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
3 | eqid 2738 | . . . 4 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
4 | evlsscaval.a | . . . 4 ⊢ 𝐴 = (algSc‘𝑃) | |
5 | evlsscaval.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
6 | evlsscaval.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) | |
7 | evlsscaval.u | . . . . . 6 ⊢ 𝑈 = (𝑆 ↾s 𝑅) | |
8 | 7 | subrgring 19659 | . . . . 5 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑈 ∈ Ring) |
9 | 6, 8 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ Ring) |
10 | 1, 2, 3, 4, 5, 9 | mplasclf 20879 | . . 3 ⊢ (𝜑 → 𝐴:(Base‘𝑈)⟶𝐵) |
11 | evlsscaval.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑅) | |
12 | 7 | subrgbas 19665 | . . . . 5 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 = (Base‘𝑈)) |
13 | 6, 12 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑅 = (Base‘𝑈)) |
14 | 11, 13 | eleqtrd 2835 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑈)) |
15 | 10, 14 | ffvelrnd 6864 | . 2 ⊢ (𝜑 → (𝐴‘𝑋) ∈ 𝐵) |
16 | evlsscaval.q | . . . . 5 ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅) | |
17 | evlsscaval.k | . . . . 5 ⊢ 𝐾 = (Base‘𝑆) | |
18 | evlsscaval.s | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ CRing) | |
19 | 16, 1, 7, 17, 4, 5, 18, 6, 11 | evlssca 20905 | . . . 4 ⊢ (𝜑 → (𝑄‘(𝐴‘𝑋)) = ((𝐾 ↑m 𝐼) × {𝑋})) |
20 | 19 | fveq1d 6678 | . . 3 ⊢ (𝜑 → ((𝑄‘(𝐴‘𝑋))‘𝐿) = (((𝐾 ↑m 𝐼) × {𝑋})‘𝐿)) |
21 | evlsscaval.l | . . . 4 ⊢ (𝜑 → 𝐿 ∈ (𝐾 ↑m 𝐼)) | |
22 | fvconst2g 6976 | . . . 4 ⊢ ((𝑋 ∈ 𝑅 ∧ 𝐿 ∈ (𝐾 ↑m 𝐼)) → (((𝐾 ↑m 𝐼) × {𝑋})‘𝐿) = 𝑋) | |
23 | 11, 21, 22 | syl2anc 587 | . . 3 ⊢ (𝜑 → (((𝐾 ↑m 𝐼) × {𝑋})‘𝐿) = 𝑋) |
24 | 20, 23 | eqtrd 2773 | . 2 ⊢ (𝜑 → ((𝑄‘(𝐴‘𝑋))‘𝐿) = 𝑋) |
25 | 15, 24 | jca 515 | 1 ⊢ (𝜑 → ((𝐴‘𝑋) ∈ 𝐵 ∧ ((𝑄‘(𝐴‘𝑋))‘𝐿) = 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 {csn 4516 × cxp 5523 ‘cfv 6339 (class class class)co 7172 ↑m cmap 8439 Basecbs 16588 ↾s cress 16589 Ringcrg 19418 CRingccrg 19419 SubRingcsubrg 19652 algSccascl 20670 mPoly cmpl 20721 evalSub ces 20886 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7481 ax-cnex 10673 ax-resscn 10674 ax-1cn 10675 ax-icn 10676 ax-addcl 10677 ax-addrcl 10678 ax-mulcl 10679 ax-mulrcl 10680 ax-mulcom 10681 ax-addass 10682 ax-mulass 10683 ax-distr 10684 ax-i2m1 10685 ax-1ne0 10686 ax-1rid 10687 ax-rnegex 10688 ax-rrecex 10689 ax-cnre 10690 ax-pre-lttri 10691 ax-pre-lttrn 10692 ax-pre-ltadd 10693 ax-pre-mulgt0 10694 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-se 5484 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-isom 6348 df-riota 7129 df-ov 7175 df-oprab 7176 df-mpo 7177 df-of 7427 df-ofr 7428 df-om 7602 df-1st 7716 df-2nd 7717 df-supp 7859 df-wrecs 7978 df-recs 8039 df-rdg 8077 df-1o 8133 df-er 8322 df-map 8441 df-pm 8442 df-ixp 8510 df-en 8558 df-dom 8559 df-sdom 8560 df-fin 8561 df-fsupp 8909 df-sup 8981 df-oi 9049 df-card 9443 df-pnf 10757 df-mnf 10758 df-xr 10759 df-ltxr 10760 df-le 10761 df-sub 10952 df-neg 10953 df-nn 11719 df-2 11781 df-3 11782 df-4 11783 df-5 11784 df-6 11785 df-7 11786 df-8 11787 df-9 11788 df-n0 11979 df-z 12065 df-dec 12182 df-uz 12327 df-fz 12984 df-fzo 13127 df-seq 13463 df-hash 13785 df-struct 16590 df-ndx 16591 df-slot 16592 df-base 16594 df-sets 16595 df-ress 16596 df-plusg 16683 df-mulr 16684 df-sca 16686 df-vsca 16687 df-ip 16688 df-tset 16689 df-ple 16690 df-ds 16692 df-hom 16694 df-cco 16695 df-0g 16820 df-gsum 16821 df-prds 16826 df-pws 16828 df-mre 16962 df-mrc 16963 df-acs 16965 df-mgm 17970 df-sgrp 18019 df-mnd 18030 df-mhm 18074 df-submnd 18075 df-grp 18224 df-minusg 18225 df-sbg 18226 df-mulg 18345 df-subg 18396 df-ghm 18476 df-cntz 18567 df-cmn 19028 df-abl 19029 df-mgp 19361 df-ur 19373 df-srg 19377 df-ring 19420 df-cring 19421 df-rnghom 19591 df-subrg 19654 df-lmod 19757 df-lss 19825 df-lsp 19865 df-assa 20671 df-asp 20672 df-ascl 20673 df-psr 20724 df-mvr 20725 df-mpl 20726 df-evls 20888 |
This theorem is referenced by: mhphf 39886 |
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