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Mirrors > Home > MPE Home > Th. List > Mathboxes > evlmulval | Structured version Visualization version GIF version |
Description: Polynomial evaluation builder for multiplication. (Contributed by SN, 18-Feb-2025.) |
Ref | Expression |
---|---|
evlmulval.q | ⊢ 𝑄 = (𝐼 eval 𝑆) |
evlmulval.p | ⊢ 𝑃 = (𝐼 mPoly 𝑆) |
evlmulval.k | ⊢ 𝐾 = (Base‘𝑆) |
evlmulval.b | ⊢ 𝐵 = (Base‘𝑃) |
evlmulval.g | ⊢ ∙ = (.r‘𝑃) |
evlmulval.f | ⊢ · = (.r‘𝑆) |
evlmulval.i | ⊢ (𝜑 → 𝐼 ∈ 𝑍) |
evlmulval.s | ⊢ (𝜑 → 𝑆 ∈ CRing) |
evlmulval.a | ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼)) |
evlmulval.m | ⊢ (𝜑 → (𝑀 ∈ 𝐵 ∧ ((𝑄‘𝑀)‘𝐴) = 𝑉)) |
evlmulval.n | ⊢ (𝜑 → (𝑁 ∈ 𝐵 ∧ ((𝑄‘𝑁)‘𝐴) = 𝑊)) |
Ref | Expression |
---|---|
evlmulval | ⊢ (𝜑 → ((𝑀 ∙ 𝑁) ∈ 𝐵 ∧ ((𝑄‘(𝑀 ∙ 𝑁))‘𝐴) = (𝑉 · 𝑊))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evlmulval.b | . . 3 ⊢ 𝐵 = (Base‘𝑃) | |
2 | evlmulval.g | . . 3 ⊢ ∙ = (.r‘𝑃) | |
3 | evlmulval.i | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑍) | |
4 | evlmulval.s | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ CRing) | |
5 | evlmulval.q | . . . . . 6 ⊢ 𝑄 = (𝐼 eval 𝑆) | |
6 | evlmulval.k | . . . . . 6 ⊢ 𝐾 = (Base‘𝑆) | |
7 | evlmulval.p | . . . . . 6 ⊢ 𝑃 = (𝐼 mPoly 𝑆) | |
8 | eqid 2724 | . . . . . 6 ⊢ (𝑆 ↑s (𝐾 ↑m 𝐼)) = (𝑆 ↑s (𝐾 ↑m 𝐼)) | |
9 | 5, 6, 7, 8 | evlrhm 21969 | . . . . 5 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing) → 𝑄 ∈ (𝑃 RingHom (𝑆 ↑s (𝐾 ↑m 𝐼)))) |
10 | 3, 4, 9 | syl2anc 583 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ (𝑃 RingHom (𝑆 ↑s (𝐾 ↑m 𝐼)))) |
11 | rhmrcl1 20368 | . . . 4 ⊢ (𝑄 ∈ (𝑃 RingHom (𝑆 ↑s (𝐾 ↑m 𝐼))) → 𝑃 ∈ Ring) | |
12 | 10, 11 | syl 17 | . . 3 ⊢ (𝜑 → 𝑃 ∈ Ring) |
13 | evlmulval.m | . . . 4 ⊢ (𝜑 → (𝑀 ∈ 𝐵 ∧ ((𝑄‘𝑀)‘𝐴) = 𝑉)) | |
14 | 13 | simpld 494 | . . 3 ⊢ (𝜑 → 𝑀 ∈ 𝐵) |
15 | evlmulval.n | . . . 4 ⊢ (𝜑 → (𝑁 ∈ 𝐵 ∧ ((𝑄‘𝑁)‘𝐴) = 𝑊)) | |
16 | 15 | simpld 494 | . . 3 ⊢ (𝜑 → 𝑁 ∈ 𝐵) |
17 | 1, 2, 12, 14, 16 | ringcld 20152 | . 2 ⊢ (𝜑 → (𝑀 ∙ 𝑁) ∈ 𝐵) |
18 | eqid 2724 | . . . . . . 7 ⊢ (.r‘(𝑆 ↑s (𝐾 ↑m 𝐼))) = (.r‘(𝑆 ↑s (𝐾 ↑m 𝐼))) | |
19 | 1, 2, 18 | rhmmul 20378 | . . . . . 6 ⊢ ((𝑄 ∈ (𝑃 RingHom (𝑆 ↑s (𝐾 ↑m 𝐼))) ∧ 𝑀 ∈ 𝐵 ∧ 𝑁 ∈ 𝐵) → (𝑄‘(𝑀 ∙ 𝑁)) = ((𝑄‘𝑀)(.r‘(𝑆 ↑s (𝐾 ↑m 𝐼)))(𝑄‘𝑁))) |
20 | 10, 14, 16, 19 | syl3anc 1368 | . . . . 5 ⊢ (𝜑 → (𝑄‘(𝑀 ∙ 𝑁)) = ((𝑄‘𝑀)(.r‘(𝑆 ↑s (𝐾 ↑m 𝐼)))(𝑄‘𝑁))) |
21 | eqid 2724 | . . . . . 6 ⊢ (Base‘(𝑆 ↑s (𝐾 ↑m 𝐼))) = (Base‘(𝑆 ↑s (𝐾 ↑m 𝐼))) | |
22 | ovexd 7436 | . . . . . 6 ⊢ (𝜑 → (𝐾 ↑m 𝐼) ∈ V) | |
23 | 1, 21 | rhmf 20377 | . . . . . . . 8 ⊢ (𝑄 ∈ (𝑃 RingHom (𝑆 ↑s (𝐾 ↑m 𝐼))) → 𝑄:𝐵⟶(Base‘(𝑆 ↑s (𝐾 ↑m 𝐼)))) |
24 | 10, 23 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑄:𝐵⟶(Base‘(𝑆 ↑s (𝐾 ↑m 𝐼)))) |
25 | 24, 14 | ffvelcdmd 7077 | . . . . . 6 ⊢ (𝜑 → (𝑄‘𝑀) ∈ (Base‘(𝑆 ↑s (𝐾 ↑m 𝐼)))) |
26 | 24, 16 | ffvelcdmd 7077 | . . . . . 6 ⊢ (𝜑 → (𝑄‘𝑁) ∈ (Base‘(𝑆 ↑s (𝐾 ↑m 𝐼)))) |
27 | evlmulval.f | . . . . . 6 ⊢ · = (.r‘𝑆) | |
28 | 8, 21, 4, 22, 25, 26, 27, 18 | pwsmulrval 17436 | . . . . 5 ⊢ (𝜑 → ((𝑄‘𝑀)(.r‘(𝑆 ↑s (𝐾 ↑m 𝐼)))(𝑄‘𝑁)) = ((𝑄‘𝑀) ∘f · (𝑄‘𝑁))) |
29 | 20, 28 | eqtrd 2764 | . . . 4 ⊢ (𝜑 → (𝑄‘(𝑀 ∙ 𝑁)) = ((𝑄‘𝑀) ∘f · (𝑄‘𝑁))) |
30 | 29 | fveq1d 6883 | . . 3 ⊢ (𝜑 → ((𝑄‘(𝑀 ∙ 𝑁))‘𝐴) = (((𝑄‘𝑀) ∘f · (𝑄‘𝑁))‘𝐴)) |
31 | 8, 6, 21, 4, 22, 25 | pwselbas 17434 | . . . . 5 ⊢ (𝜑 → (𝑄‘𝑀):(𝐾 ↑m 𝐼)⟶𝐾) |
32 | 31 | ffnd 6708 | . . . 4 ⊢ (𝜑 → (𝑄‘𝑀) Fn (𝐾 ↑m 𝐼)) |
33 | 8, 6, 21, 4, 22, 26 | pwselbas 17434 | . . . . 5 ⊢ (𝜑 → (𝑄‘𝑁):(𝐾 ↑m 𝐼)⟶𝐾) |
34 | 33 | ffnd 6708 | . . . 4 ⊢ (𝜑 → (𝑄‘𝑁) Fn (𝐾 ↑m 𝐼)) |
35 | evlmulval.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼)) | |
36 | fnfvof 7680 | . . . 4 ⊢ ((((𝑄‘𝑀) Fn (𝐾 ↑m 𝐼) ∧ (𝑄‘𝑁) Fn (𝐾 ↑m 𝐼)) ∧ ((𝐾 ↑m 𝐼) ∈ V ∧ 𝐴 ∈ (𝐾 ↑m 𝐼))) → (((𝑄‘𝑀) ∘f · (𝑄‘𝑁))‘𝐴) = (((𝑄‘𝑀)‘𝐴) · ((𝑄‘𝑁)‘𝐴))) | |
37 | 32, 34, 22, 35, 36 | syl22anc 836 | . . 3 ⊢ (𝜑 → (((𝑄‘𝑀) ∘f · (𝑄‘𝑁))‘𝐴) = (((𝑄‘𝑀)‘𝐴) · ((𝑄‘𝑁)‘𝐴))) |
38 | 13 | simprd 495 | . . . 4 ⊢ (𝜑 → ((𝑄‘𝑀)‘𝐴) = 𝑉) |
39 | 15 | simprd 495 | . . . 4 ⊢ (𝜑 → ((𝑄‘𝑁)‘𝐴) = 𝑊) |
40 | 38, 39 | oveq12d 7419 | . . 3 ⊢ (𝜑 → (((𝑄‘𝑀)‘𝐴) · ((𝑄‘𝑁)‘𝐴)) = (𝑉 · 𝑊)) |
41 | 30, 37, 40 | 3eqtrd 2768 | . 2 ⊢ (𝜑 → ((𝑄‘(𝑀 ∙ 𝑁))‘𝐴) = (𝑉 · 𝑊)) |
42 | 17, 41 | jca 511 | 1 ⊢ (𝜑 → ((𝑀 ∙ 𝑁) ∈ 𝐵 ∧ ((𝑄‘(𝑀 ∙ 𝑁))‘𝐴) = (𝑉 · 𝑊))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Vcvv 3466 Fn wfn 6528 ⟶wf 6529 ‘cfv 6533 (class class class)co 7401 ∘f cof 7661 ↑m cmap 8816 Basecbs 17143 .rcmulr 17197 ↑s cpws 17391 Ringcrg 20128 CRingccrg 20129 RingHom crh 20361 mPoly cmpl 21768 eval cevl 21944 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-ofr 7664 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-sup 9433 df-oi 9501 df-card 9930 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-fz 13482 df-fzo 13625 df-seq 13964 df-hash 14288 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-hom 17220 df-cco 17221 df-0g 17386 df-gsum 17387 df-prds 17392 df-pws 17394 df-mre 17529 df-mrc 17530 df-acs 17532 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-mhm 18703 df-submnd 18704 df-grp 18856 df-minusg 18857 df-sbg 18858 df-mulg 18986 df-subg 19040 df-ghm 19129 df-cntz 19223 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-srg 20082 df-ring 20130 df-cring 20131 df-rhm 20364 df-subrng 20436 df-subrg 20461 df-lmod 20698 df-lss 20769 df-lsp 20809 df-assa 21716 df-asp 21717 df-ascl 21718 df-psr 21771 df-mvr 21772 df-mpl 21773 df-evls 21945 df-evl 21946 |
This theorem is referenced by: selvmul 41650 |
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