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Mirrors > Home > MPE Home > Th. List > Mathboxes > evlsmulval | Structured version Visualization version GIF version |
Description: Polynomial evaluation builder for multiplication. (Contributed by SN, 27-Jul-2024.) |
Ref | Expression |
---|---|
evlsaddval.q | ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅) |
evlsaddval.p | ⊢ 𝑃 = (𝐼 mPoly 𝑈) |
evlsaddval.u | ⊢ 𝑈 = (𝑆 ↾s 𝑅) |
evlsaddval.k | ⊢ 𝐾 = (Base‘𝑆) |
evlsaddval.b | ⊢ 𝐵 = (Base‘𝑃) |
evlsaddval.i | ⊢ (𝜑 → 𝐼 ∈ 𝑍) |
evlsaddval.s | ⊢ (𝜑 → 𝑆 ∈ CRing) |
evlsaddval.r | ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) |
evlsaddval.a | ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼)) |
evlsaddval.m | ⊢ (𝜑 → (𝑀 ∈ 𝐵 ∧ ((𝑄‘𝑀)‘𝐴) = 𝑉)) |
evlsaddval.n | ⊢ (𝜑 → (𝑁 ∈ 𝐵 ∧ ((𝑄‘𝑁)‘𝐴) = 𝑊)) |
evlsmulval.g | ⊢ ∙ = (.r‘𝑃) |
evlsmulval.f | ⊢ · = (.r‘𝑆) |
Ref | Expression |
---|---|
evlsmulval | ⊢ (𝜑 → ((𝑀 ∙ 𝑁) ∈ 𝐵 ∧ ((𝑄‘(𝑀 ∙ 𝑁))‘𝐴) = (𝑉 · 𝑊))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evlsaddval.i | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑍) | |
2 | evlsaddval.s | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ CRing) | |
3 | evlsaddval.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) | |
4 | evlsaddval.q | . . . . . 6 ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅) | |
5 | evlsaddval.p | . . . . . 6 ⊢ 𝑃 = (𝐼 mPoly 𝑈) | |
6 | evlsaddval.u | . . . . . 6 ⊢ 𝑈 = (𝑆 ↾s 𝑅) | |
7 | eqid 2738 | . . . . . 6 ⊢ (𝑆 ↑s (𝐾 ↑m 𝐼)) = (𝑆 ↑s (𝐾 ↑m 𝐼)) | |
8 | evlsaddval.k | . . . . . 6 ⊢ 𝐾 = (Base‘𝑆) | |
9 | 4, 5, 6, 7, 8 | evlsrhm 21308 | . . . . 5 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑃 RingHom (𝑆 ↑s (𝐾 ↑m 𝐼)))) |
10 | 1, 2, 3, 9 | syl3anc 1370 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ (𝑃 RingHom (𝑆 ↑s (𝐾 ↑m 𝐼)))) |
11 | rhmrcl1 19973 | . . . 4 ⊢ (𝑄 ∈ (𝑃 RingHom (𝑆 ↑s (𝐾 ↑m 𝐼))) → 𝑃 ∈ Ring) | |
12 | 10, 11 | syl 17 | . . 3 ⊢ (𝜑 → 𝑃 ∈ Ring) |
13 | evlsaddval.m | . . . 4 ⊢ (𝜑 → (𝑀 ∈ 𝐵 ∧ ((𝑄‘𝑀)‘𝐴) = 𝑉)) | |
14 | 13 | simpld 495 | . . 3 ⊢ (𝜑 → 𝑀 ∈ 𝐵) |
15 | evlsaddval.n | . . . 4 ⊢ (𝜑 → (𝑁 ∈ 𝐵 ∧ ((𝑄‘𝑁)‘𝐴) = 𝑊)) | |
16 | 15 | simpld 495 | . . 3 ⊢ (𝜑 → 𝑁 ∈ 𝐵) |
17 | evlsaddval.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
18 | evlsmulval.g | . . . 4 ⊢ ∙ = (.r‘𝑃) | |
19 | 17, 18 | ringcl 19810 | . . 3 ⊢ ((𝑃 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝑁 ∈ 𝐵) → (𝑀 ∙ 𝑁) ∈ 𝐵) |
20 | 12, 14, 16, 19 | syl3anc 1370 | . 2 ⊢ (𝜑 → (𝑀 ∙ 𝑁) ∈ 𝐵) |
21 | eqid 2738 | . . . . . . 7 ⊢ (.r‘(𝑆 ↑s (𝐾 ↑m 𝐼))) = (.r‘(𝑆 ↑s (𝐾 ↑m 𝐼))) | |
22 | 17, 18, 21 | rhmmul 19981 | . . . . . 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))) |
23 | 10, 14, 16, 22 | syl3anc 1370 | . . . . 5 ⊢ (𝜑 → (𝑄‘(𝑀 ∙ 𝑁)) = ((𝑄‘𝑀)(.r‘(𝑆 ↑s (𝐾 ↑m 𝐼)))(𝑄‘𝑁))) |
24 | eqid 2738 | . . . . . 6 ⊢ (Base‘(𝑆 ↑s (𝐾 ↑m 𝐼))) = (Base‘(𝑆 ↑s (𝐾 ↑m 𝐼))) | |
25 | ovexd 7302 | . . . . . 6 ⊢ (𝜑 → (𝐾 ↑m 𝐼) ∈ V) | |
26 | 17, 24 | rhmf 19980 | . . . . . . . 8 ⊢ (𝑄 ∈ (𝑃 RingHom (𝑆 ↑s (𝐾 ↑m 𝐼))) → 𝑄:𝐵⟶(Base‘(𝑆 ↑s (𝐾 ↑m 𝐼)))) |
27 | 10, 26 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑄:𝐵⟶(Base‘(𝑆 ↑s (𝐾 ↑m 𝐼)))) |
28 | 27, 14 | ffvelrnd 6954 | . . . . . 6 ⊢ (𝜑 → (𝑄‘𝑀) ∈ (Base‘(𝑆 ↑s (𝐾 ↑m 𝐼)))) |
29 | 27, 16 | ffvelrnd 6954 | . . . . . 6 ⊢ (𝜑 → (𝑄‘𝑁) ∈ (Base‘(𝑆 ↑s (𝐾 ↑m 𝐼)))) |
30 | evlsmulval.f | . . . . . 6 ⊢ · = (.r‘𝑆) | |
31 | 7, 24, 2, 25, 28, 29, 30, 21 | pwsmulrval 17212 | . . . . 5 ⊢ (𝜑 → ((𝑄‘𝑀)(.r‘(𝑆 ↑s (𝐾 ↑m 𝐼)))(𝑄‘𝑁)) = ((𝑄‘𝑀) ∘f · (𝑄‘𝑁))) |
32 | 23, 31 | eqtrd 2778 | . . . 4 ⊢ (𝜑 → (𝑄‘(𝑀 ∙ 𝑁)) = ((𝑄‘𝑀) ∘f · (𝑄‘𝑁))) |
33 | 32 | fveq1d 6768 | . . 3 ⊢ (𝜑 → ((𝑄‘(𝑀 ∙ 𝑁))‘𝐴) = (((𝑄‘𝑀) ∘f · (𝑄‘𝑁))‘𝐴)) |
34 | 7, 8, 24, 2, 25, 28 | pwselbas 17210 | . . . . 5 ⊢ (𝜑 → (𝑄‘𝑀):(𝐾 ↑m 𝐼)⟶𝐾) |
35 | 34 | ffnd 6593 | . . . 4 ⊢ (𝜑 → (𝑄‘𝑀) Fn (𝐾 ↑m 𝐼)) |
36 | 7, 8, 24, 2, 25, 29 | pwselbas 17210 | . . . . 5 ⊢ (𝜑 → (𝑄‘𝑁):(𝐾 ↑m 𝐼)⟶𝐾) |
37 | 36 | ffnd 6593 | . . . 4 ⊢ (𝜑 → (𝑄‘𝑁) Fn (𝐾 ↑m 𝐼)) |
38 | evlsaddval.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼)) | |
39 | fnfvof 7540 | . . . 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))) | |
40 | 35, 37, 25, 38, 39 | syl22anc 836 | . . 3 ⊢ (𝜑 → (((𝑄‘𝑀) ∘f · (𝑄‘𝑁))‘𝐴) = (((𝑄‘𝑀)‘𝐴) · ((𝑄‘𝑁)‘𝐴))) |
41 | 13 | simprd 496 | . . . 4 ⊢ (𝜑 → ((𝑄‘𝑀)‘𝐴) = 𝑉) |
42 | 15 | simprd 496 | . . . 4 ⊢ (𝜑 → ((𝑄‘𝑁)‘𝐴) = 𝑊) |
43 | 41, 42 | oveq12d 7285 | . . 3 ⊢ (𝜑 → (((𝑄‘𝑀)‘𝐴) · ((𝑄‘𝑁)‘𝐴)) = (𝑉 · 𝑊)) |
44 | 33, 40, 43 | 3eqtrd 2782 | . 2 ⊢ (𝜑 → ((𝑄‘(𝑀 ∙ 𝑁))‘𝐴) = (𝑉 · 𝑊)) |
45 | 20, 44 | jca 512 | 1 ⊢ (𝜑 → ((𝑀 ∙ 𝑁) ∈ 𝐵 ∧ ((𝑄‘(𝑀 ∙ 𝑁))‘𝐴) = (𝑉 · 𝑊))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 Vcvv 3429 Fn wfn 6421 ⟶wf 6422 ‘cfv 6426 (class class class)co 7267 ∘f cof 7521 ↑m cmap 8602 Basecbs 16922 ↾s cress 16951 .rcmulr 16973 ↑s cpws 17167 Ringcrg 19793 CRingccrg 19794 RingHom crh 19966 SubRingcsubrg 20030 mPoly cmpl 21119 evalSub ces 21290 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5208 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-cnex 10937 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 ax-pre-mulgt0 10958 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-se 5540 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-isom 6435 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-of 7523 df-ofr 7524 df-om 7703 df-1st 7820 df-2nd 7821 df-supp 7965 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-1o 8284 df-er 8485 df-map 8604 df-pm 8605 df-ixp 8673 df-en 8721 df-dom 8722 df-sdom 8723 df-fin 8724 df-fsupp 9116 df-sup 9188 df-oi 9256 df-card 9707 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 df-sub 11217 df-neg 11218 df-nn 11984 df-2 12046 df-3 12047 df-4 12048 df-5 12049 df-6 12050 df-7 12051 df-8 12052 df-9 12053 df-n0 12244 df-z 12330 df-dec 12448 df-uz 12593 df-fz 13250 df-fzo 13393 df-seq 13732 df-hash 14055 df-struct 16858 df-sets 16875 df-slot 16893 df-ndx 16905 df-base 16923 df-ress 16952 df-plusg 16985 df-mulr 16986 df-sca 16988 df-vsca 16989 df-ip 16990 df-tset 16991 df-ple 16992 df-ds 16994 df-hom 16996 df-cco 16997 df-0g 17162 df-gsum 17163 df-prds 17168 df-pws 17170 df-mre 17305 df-mrc 17306 df-acs 17308 df-mgm 18336 df-sgrp 18385 df-mnd 18396 df-mhm 18440 df-submnd 18441 df-grp 18590 df-minusg 18591 df-sbg 18592 df-mulg 18711 df-subg 18762 df-ghm 18842 df-cntz 18933 df-cmn 19398 df-abl 19399 df-mgp 19731 df-ur 19748 df-srg 19752 df-ring 19795 df-cring 19796 df-rnghom 19969 df-subrg 20032 df-lmod 20135 df-lss 20204 df-lsp 20244 df-assa 21070 df-asp 21071 df-ascl 21072 df-psr 21122 df-mvr 21123 df-mpl 21124 df-evls 21292 |
This theorem is referenced by: mhphf 40293 |
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