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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 2758 | . . . . . 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 20856 | . . . . 5 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑃 RingHom (𝑆 ↑s (𝐾 ↑m 𝐼)))) |
10 | 1, 2, 3, 9 | syl3anc 1368 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ (𝑃 RingHom (𝑆 ↑s (𝐾 ↑m 𝐼)))) |
11 | rhmrcl1 19547 | . . . 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 498 | . . 3 ⊢ (𝜑 → 𝑀 ∈ 𝐵) |
15 | evlsaddval.n | . . . 4 ⊢ (𝜑 → (𝑁 ∈ 𝐵 ∧ ((𝑄‘𝑁)‘𝐴) = 𝑊)) | |
16 | 15 | simpld 498 | . . 3 ⊢ (𝜑 → 𝑁 ∈ 𝐵) |
17 | evlsaddval.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
18 | evlsmulval.g | . . . 4 ⊢ ∙ = (.r‘𝑃) | |
19 | 17, 18 | ringcl 19387 | . . 3 ⊢ ((𝑃 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝑁 ∈ 𝐵) → (𝑀 ∙ 𝑁) ∈ 𝐵) |
20 | 12, 14, 16, 19 | syl3anc 1368 | . 2 ⊢ (𝜑 → (𝑀 ∙ 𝑁) ∈ 𝐵) |
21 | eqid 2758 | . . . . . . 7 ⊢ (.r‘(𝑆 ↑s (𝐾 ↑m 𝐼))) = (.r‘(𝑆 ↑s (𝐾 ↑m 𝐼))) | |
22 | 17, 18, 21 | rhmmul 19555 | . . . . . 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 1368 | . . . . 5 ⊢ (𝜑 → (𝑄‘(𝑀 ∙ 𝑁)) = ((𝑄‘𝑀)(.r‘(𝑆 ↑s (𝐾 ↑m 𝐼)))(𝑄‘𝑁))) |
24 | eqid 2758 | . . . . . 6 ⊢ (Base‘(𝑆 ↑s (𝐾 ↑m 𝐼))) = (Base‘(𝑆 ↑s (𝐾 ↑m 𝐼))) | |
25 | ovexd 7190 | . . . . . 6 ⊢ (𝜑 → (𝐾 ↑m 𝐼) ∈ V) | |
26 | 17, 24 | rhmf 19554 | . . . . . . . 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 6848 | . . . . . 6 ⊢ (𝜑 → (𝑄‘𝑀) ∈ (Base‘(𝑆 ↑s (𝐾 ↑m 𝐼)))) |
29 | 27, 16 | ffvelrnd 6848 | . . . . . 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 16827 | . . . . 5 ⊢ (𝜑 → ((𝑄‘𝑀)(.r‘(𝑆 ↑s (𝐾 ↑m 𝐼)))(𝑄‘𝑁)) = ((𝑄‘𝑀) ∘f · (𝑄‘𝑁))) |
32 | 23, 31 | eqtrd 2793 | . . . 4 ⊢ (𝜑 → (𝑄‘(𝑀 ∙ 𝑁)) = ((𝑄‘𝑀) ∘f · (𝑄‘𝑁))) |
33 | 32 | fveq1d 6664 | . . 3 ⊢ (𝜑 → ((𝑄‘(𝑀 ∙ 𝑁))‘𝐴) = (((𝑄‘𝑀) ∘f · (𝑄‘𝑁))‘𝐴)) |
34 | 7, 8, 24, 2, 25, 28 | pwselbas 16825 | . . . . 5 ⊢ (𝜑 → (𝑄‘𝑀):(𝐾 ↑m 𝐼)⟶𝐾) |
35 | 34 | ffnd 6503 | . . . 4 ⊢ (𝜑 → (𝑄‘𝑀) Fn (𝐾 ↑m 𝐼)) |
36 | 7, 8, 24, 2, 25, 29 | pwselbas 16825 | . . . . 5 ⊢ (𝜑 → (𝑄‘𝑁):(𝐾 ↑m 𝐼)⟶𝐾) |
37 | 36 | ffnd 6503 | . . . 4 ⊢ (𝜑 → (𝑄‘𝑁) Fn (𝐾 ↑m 𝐼)) |
38 | evlsaddval.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼)) | |
39 | fnfvof 7426 | . . . 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 837 | . . 3 ⊢ (𝜑 → (((𝑄‘𝑀) ∘f · (𝑄‘𝑁))‘𝐴) = (((𝑄‘𝑀)‘𝐴) · ((𝑄‘𝑁)‘𝐴))) |
41 | 13 | simprd 499 | . . . 4 ⊢ (𝜑 → ((𝑄‘𝑀)‘𝐴) = 𝑉) |
42 | 15 | simprd 499 | . . . 4 ⊢ (𝜑 → ((𝑄‘𝑁)‘𝐴) = 𝑊) |
43 | 41, 42 | oveq12d 7173 | . . 3 ⊢ (𝜑 → (((𝑄‘𝑀)‘𝐴) · ((𝑄‘𝑁)‘𝐴)) = (𝑉 · 𝑊)) |
44 | 33, 40, 43 | 3eqtrd 2797 | . 2 ⊢ (𝜑 → ((𝑄‘(𝑀 ∙ 𝑁))‘𝐴) = (𝑉 · 𝑊)) |
45 | 20, 44 | jca 515 | 1 ⊢ (𝜑 → ((𝑀 ∙ 𝑁) ∈ 𝐵 ∧ ((𝑄‘(𝑀 ∙ 𝑁))‘𝐴) = (𝑉 · 𝑊))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3409 Fn wfn 6334 ⟶wf 6335 ‘cfv 6339 (class class class)co 7155 ∘f cof 7408 ↑m cmap 8421 Basecbs 16546 ↾s cress 16547 .rcmulr 16629 ↑s cpws 16783 Ringcrg 19370 CRingccrg 19371 RingHom crh 19540 SubRingcsubrg 19604 mPoly cmpl 20673 evalSub ces 20838 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5159 ax-sep 5172 ax-nul 5179 ax-pow 5237 ax-pr 5301 ax-un 7464 ax-cnex 10636 ax-resscn 10637 ax-1cn 10638 ax-icn 10639 ax-addcl 10640 ax-addrcl 10641 ax-mulcl 10642 ax-mulrcl 10643 ax-mulcom 10644 ax-addass 10645 ax-mulass 10646 ax-distr 10647 ax-i2m1 10648 ax-1ne0 10649 ax-1rid 10650 ax-rnegex 10651 ax-rrecex 10652 ax-cnre 10653 ax-pre-lttri 10654 ax-pre-lttrn 10655 ax-pre-ltadd 10656 ax-pre-mulgt0 10657 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-pss 3879 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4802 df-int 4842 df-iun 4888 df-iin 4889 df-br 5036 df-opab 5098 df-mpt 5116 df-tr 5142 df-id 5433 df-eprel 5438 df-po 5446 df-so 5447 df-fr 5486 df-se 5487 df-we 5488 df-xp 5533 df-rel 5534 df-cnv 5535 df-co 5536 df-dm 5537 df-rn 5538 df-res 5539 df-ima 5540 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 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 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7410 df-ofr 7411 df-om 7585 df-1st 7698 df-2nd 7699 df-supp 7841 df-wrecs 7962 df-recs 8023 df-rdg 8061 df-1o 8117 df-er 8304 df-map 8423 df-pm 8424 df-ixp 8485 df-en 8533 df-dom 8534 df-sdom 8535 df-fin 8536 df-fsupp 8872 df-sup 8944 df-oi 9012 df-card 9406 df-pnf 10720 df-mnf 10721 df-xr 10722 df-ltxr 10723 df-le 10724 df-sub 10915 df-neg 10916 df-nn 11680 df-2 11742 df-3 11743 df-4 11744 df-5 11745 df-6 11746 df-7 11747 df-8 11748 df-9 11749 df-n0 11940 df-z 12026 df-dec 12143 df-uz 12288 df-fz 12945 df-fzo 13088 df-seq 13424 df-hash 13746 df-struct 16548 df-ndx 16549 df-slot 16550 df-base 16552 df-sets 16553 df-ress 16554 df-plusg 16641 df-mulr 16642 df-sca 16644 df-vsca 16645 df-ip 16646 df-tset 16647 df-ple 16648 df-ds 16650 df-hom 16652 df-cco 16653 df-0g 16778 df-gsum 16779 df-prds 16784 df-pws 16786 df-mre 16920 df-mrc 16921 df-acs 16923 df-mgm 17923 df-sgrp 17972 df-mnd 17983 df-mhm 18027 df-submnd 18028 df-grp 18177 df-minusg 18178 df-sbg 18179 df-mulg 18297 df-subg 18348 df-ghm 18428 df-cntz 18519 df-cmn 18980 df-abl 18981 df-mgp 19313 df-ur 19325 df-srg 19329 df-ring 19372 df-cring 19373 df-rnghom 19543 df-subrg 19606 df-lmod 19709 df-lss 19777 df-lsp 19817 df-assa 20623 df-asp 20624 df-ascl 20625 df-psr 20676 df-mvr 20677 df-mpl 20678 df-evls 20840 |
This theorem is referenced by: mhphf 39818 |
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