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Mirrors > Home > MPE Home > Th. List > evl1muld | Structured version Visualization version GIF version |
Description: Polynomial evaluation builder for multiplication of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.) |
Ref | Expression |
---|---|
evl1addd.q | ⊢ 𝑂 = (eval1‘𝑅) |
evl1addd.p | ⊢ 𝑃 = (Poly1‘𝑅) |
evl1addd.b | ⊢ 𝐵 = (Base‘𝑅) |
evl1addd.u | ⊢ 𝑈 = (Base‘𝑃) |
evl1addd.1 | ⊢ (𝜑 → 𝑅 ∈ CRing) |
evl1addd.2 | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
evl1addd.3 | ⊢ (𝜑 → (𝑀 ∈ 𝑈 ∧ ((𝑂‘𝑀)‘𝑌) = 𝑉)) |
evl1addd.4 | ⊢ (𝜑 → (𝑁 ∈ 𝑈 ∧ ((𝑂‘𝑁)‘𝑌) = 𝑊)) |
evl1muld.t | ⊢ ∙ = (.r‘𝑃) |
evl1muld.s | ⊢ · = (.r‘𝑅) |
Ref | Expression |
---|---|
evl1muld | ⊢ (𝜑 → ((𝑀 ∙ 𝑁) ∈ 𝑈 ∧ ((𝑂‘(𝑀 ∙ 𝑁))‘𝑌) = (𝑉 · 𝑊))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evl1addd.1 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
2 | evl1addd.q | . . . . . 6 ⊢ 𝑂 = (eval1‘𝑅) | |
3 | evl1addd.p | . . . . . 6 ⊢ 𝑃 = (Poly1‘𝑅) | |
4 | eqid 2724 | . . . . . 6 ⊢ (𝑅 ↑s 𝐵) = (𝑅 ↑s 𝐵) | |
5 | evl1addd.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
6 | 2, 3, 4, 5 | evl1rhm 22175 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑s 𝐵))) |
7 | 1, 6 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑s 𝐵))) |
8 | rhmrcl1 20370 | . . . 4 ⊢ (𝑂 ∈ (𝑃 RingHom (𝑅 ↑s 𝐵)) → 𝑃 ∈ Ring) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ (𝜑 → 𝑃 ∈ Ring) |
10 | evl1addd.3 | . . . 4 ⊢ (𝜑 → (𝑀 ∈ 𝑈 ∧ ((𝑂‘𝑀)‘𝑌) = 𝑉)) | |
11 | 10 | simpld 494 | . . 3 ⊢ (𝜑 → 𝑀 ∈ 𝑈) |
12 | evl1addd.4 | . . . 4 ⊢ (𝜑 → (𝑁 ∈ 𝑈 ∧ ((𝑂‘𝑁)‘𝑌) = 𝑊)) | |
13 | 12 | simpld 494 | . . 3 ⊢ (𝜑 → 𝑁 ∈ 𝑈) |
14 | evl1addd.u | . . . 4 ⊢ 𝑈 = (Base‘𝑃) | |
15 | evl1muld.t | . . . 4 ⊢ ∙ = (.r‘𝑃) | |
16 | 14, 15 | ringcl 20147 | . . 3 ⊢ ((𝑃 ∈ Ring ∧ 𝑀 ∈ 𝑈 ∧ 𝑁 ∈ 𝑈) → (𝑀 ∙ 𝑁) ∈ 𝑈) |
17 | 9, 11, 13, 16 | syl3anc 1368 | . 2 ⊢ (𝜑 → (𝑀 ∙ 𝑁) ∈ 𝑈) |
18 | eqid 2724 | . . . . . . 7 ⊢ (.r‘(𝑅 ↑s 𝐵)) = (.r‘(𝑅 ↑s 𝐵)) | |
19 | 14, 15, 18 | rhmmul 20380 | . . . . . 6 ⊢ ((𝑂 ∈ (𝑃 RingHom (𝑅 ↑s 𝐵)) ∧ 𝑀 ∈ 𝑈 ∧ 𝑁 ∈ 𝑈) → (𝑂‘(𝑀 ∙ 𝑁)) = ((𝑂‘𝑀)(.r‘(𝑅 ↑s 𝐵))(𝑂‘𝑁))) |
20 | 7, 11, 13, 19 | syl3anc 1368 | . . . . 5 ⊢ (𝜑 → (𝑂‘(𝑀 ∙ 𝑁)) = ((𝑂‘𝑀)(.r‘(𝑅 ↑s 𝐵))(𝑂‘𝑁))) |
21 | eqid 2724 | . . . . . 6 ⊢ (Base‘(𝑅 ↑s 𝐵)) = (Base‘(𝑅 ↑s 𝐵)) | |
22 | 5 | fvexi 6896 | . . . . . . 7 ⊢ 𝐵 ∈ V |
23 | 22 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V) |
24 | 14, 21 | rhmf 20379 | . . . . . . . 8 ⊢ (𝑂 ∈ (𝑃 RingHom (𝑅 ↑s 𝐵)) → 𝑂:𝑈⟶(Base‘(𝑅 ↑s 𝐵))) |
25 | 7, 24 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑂:𝑈⟶(Base‘(𝑅 ↑s 𝐵))) |
26 | 25, 11 | ffvelcdmd 7078 | . . . . . 6 ⊢ (𝜑 → (𝑂‘𝑀) ∈ (Base‘(𝑅 ↑s 𝐵))) |
27 | 25, 13 | ffvelcdmd 7078 | . . . . . 6 ⊢ (𝜑 → (𝑂‘𝑁) ∈ (Base‘(𝑅 ↑s 𝐵))) |
28 | evl1muld.s | . . . . . 6 ⊢ · = (.r‘𝑅) | |
29 | 4, 21, 1, 23, 26, 27, 28, 18 | pwsmulrval 17438 | . . . . 5 ⊢ (𝜑 → ((𝑂‘𝑀)(.r‘(𝑅 ↑s 𝐵))(𝑂‘𝑁)) = ((𝑂‘𝑀) ∘f · (𝑂‘𝑁))) |
30 | 20, 29 | eqtrd 2764 | . . . 4 ⊢ (𝜑 → (𝑂‘(𝑀 ∙ 𝑁)) = ((𝑂‘𝑀) ∘f · (𝑂‘𝑁))) |
31 | 30 | fveq1d 6884 | . . 3 ⊢ (𝜑 → ((𝑂‘(𝑀 ∙ 𝑁))‘𝑌) = (((𝑂‘𝑀) ∘f · (𝑂‘𝑁))‘𝑌)) |
32 | 4, 5, 21, 1, 23, 26 | pwselbas 17436 | . . . . 5 ⊢ (𝜑 → (𝑂‘𝑀):𝐵⟶𝐵) |
33 | 32 | ffnd 6709 | . . . 4 ⊢ (𝜑 → (𝑂‘𝑀) Fn 𝐵) |
34 | 4, 5, 21, 1, 23, 27 | pwselbas 17436 | . . . . 5 ⊢ (𝜑 → (𝑂‘𝑁):𝐵⟶𝐵) |
35 | 34 | ffnd 6709 | . . . 4 ⊢ (𝜑 → (𝑂‘𝑁) Fn 𝐵) |
36 | evl1addd.2 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
37 | fnfvof 7681 | . . . 4 ⊢ ((((𝑂‘𝑀) Fn 𝐵 ∧ (𝑂‘𝑁) Fn 𝐵) ∧ (𝐵 ∈ V ∧ 𝑌 ∈ 𝐵)) → (((𝑂‘𝑀) ∘f · (𝑂‘𝑁))‘𝑌) = (((𝑂‘𝑀)‘𝑌) · ((𝑂‘𝑁)‘𝑌))) | |
38 | 33, 35, 23, 36, 37 | syl22anc 836 | . . 3 ⊢ (𝜑 → (((𝑂‘𝑀) ∘f · (𝑂‘𝑁))‘𝑌) = (((𝑂‘𝑀)‘𝑌) · ((𝑂‘𝑁)‘𝑌))) |
39 | 10 | simprd 495 | . . . 4 ⊢ (𝜑 → ((𝑂‘𝑀)‘𝑌) = 𝑉) |
40 | 12 | simprd 495 | . . . 4 ⊢ (𝜑 → ((𝑂‘𝑁)‘𝑌) = 𝑊) |
41 | 39, 40 | oveq12d 7420 | . . 3 ⊢ (𝜑 → (((𝑂‘𝑀)‘𝑌) · ((𝑂‘𝑁)‘𝑌)) = (𝑉 · 𝑊)) |
42 | 31, 38, 41 | 3eqtrd 2768 | . 2 ⊢ (𝜑 → ((𝑂‘(𝑀 ∙ 𝑁))‘𝑌) = (𝑉 · 𝑊)) |
43 | 17, 42 | jca 511 | 1 ⊢ (𝜑 → ((𝑀 ∙ 𝑁) ∈ 𝑈 ∧ ((𝑂‘(𝑀 ∙ 𝑁))‘𝑌) = (𝑉 · 𝑊))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Vcvv 3466 Fn wfn 6529 ⟶wf 6530 ‘cfv 6534 (class class class)co 7402 ∘f cof 7662 Basecbs 17145 .rcmulr 17199 ↑s cpws 17393 Ringcrg 20130 CRingccrg 20131 RingHom crh 20363 Poly1cpl1 22021 eval1ce1 22157 |
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 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
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 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-iin 4991 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-of 7664 df-ofr 7665 df-om 7850 df-1st 7969 df-2nd 7970 df-supp 8142 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-pm 8820 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-sup 9434 df-oi 9502 df-card 9931 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-nn 12211 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-7 12278 df-8 12279 df-9 12280 df-n0 12471 df-z 12557 df-dec 12676 df-uz 12821 df-fz 13483 df-fzo 13626 df-seq 13965 df-hash 14289 df-struct 17081 df-sets 17098 df-slot 17116 df-ndx 17128 df-base 17146 df-ress 17175 df-plusg 17211 df-mulr 17212 df-sca 17214 df-vsca 17215 df-ip 17216 df-tset 17217 df-ple 17218 df-ds 17220 df-hom 17222 df-cco 17223 df-0g 17388 df-gsum 17389 df-prds 17394 df-pws 17396 df-mre 17531 df-mrc 17532 df-acs 17534 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18705 df-submnd 18706 df-grp 18858 df-minusg 18859 df-sbg 18860 df-mulg 18988 df-subg 19042 df-ghm 19131 df-cntz 19225 df-cmn 19694 df-abl 19695 df-mgp 20032 df-rng 20050 df-ur 20079 df-srg 20084 df-ring 20132 df-cring 20133 df-rhm 20366 df-subrng 20438 df-subrg 20463 df-lmod 20700 df-lss 20771 df-lsp 20811 df-assa 21718 df-asp 21719 df-ascl 21720 df-psr 21773 df-mvr 21774 df-mpl 21775 df-opsr 21777 df-evls 21947 df-evl 21948 df-psr1 22024 df-ply1 22026 df-evl1 22159 |
This theorem is referenced by: evl1vsd 22187 evls1muld 33119 |
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