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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > evls1muld | Structured version Visualization version GIF version |
Description: Univariate polynomial evaluation of a product of polynomials. (Contributed by Thierry Arnoux, 24-Jan-2025.) |
Ref | Expression |
---|---|
ressply1evl.q | ⊢ 𝑄 = (𝑆 evalSub1 𝑅) |
ressply1evl.k | ⊢ 𝐾 = (Base‘𝑆) |
ressply1evl.w | ⊢ 𝑊 = (Poly1‘𝑈) |
ressply1evl.u | ⊢ 𝑈 = (𝑆 ↾s 𝑅) |
ressply1evl.b | ⊢ 𝐵 = (Base‘𝑊) |
evls1muld.1 | ⊢ × = (.r‘𝑊) |
evls1muld.2 | ⊢ · = (.r‘𝑆) |
evls1muld.s | ⊢ (𝜑 → 𝑆 ∈ CRing) |
evls1muld.r | ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) |
evls1muld.m | ⊢ (𝜑 → 𝑀 ∈ 𝐵) |
evls1muld.n | ⊢ (𝜑 → 𝑁 ∈ 𝐵) |
evls1muld.c | ⊢ (𝜑 → 𝐶 ∈ 𝐾) |
Ref | Expression |
---|---|
evls1muld | ⊢ (𝜑 → ((𝑄‘(𝑀 × 𝑁))‘𝐶) = (((𝑄‘𝑀)‘𝐶) · ((𝑄‘𝑁)‘𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . . . 6 ⊢ (𝜑 → 𝜑) | |
2 | evls1muld.m | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ 𝐵) | |
3 | evls1muld.n | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ 𝐵) | |
4 | eqid 2737 | . . . . . . 7 ⊢ (Poly1‘𝑆) = (Poly1‘𝑆) | |
5 | ressply1evl.u | . . . . . . 7 ⊢ 𝑈 = (𝑆 ↾s 𝑅) | |
6 | ressply1evl.w | . . . . . . 7 ⊢ 𝑊 = (Poly1‘𝑈) | |
7 | ressply1evl.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑊) | |
8 | evls1muld.r | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) | |
9 | eqid 2737 | . . . . . . 7 ⊢ ((Poly1‘𝑆) ↾s 𝐵) = ((Poly1‘𝑆) ↾s 𝐵) | |
10 | 4, 5, 6, 7, 8, 9 | ressply1mul 21553 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑀 ∈ 𝐵 ∧ 𝑁 ∈ 𝐵)) → (𝑀(.r‘𝑊)𝑁) = (𝑀(.r‘((Poly1‘𝑆) ↾s 𝐵))𝑁)) |
11 | 1, 2, 3, 10 | syl12anc 835 | . . . . 5 ⊢ (𝜑 → (𝑀(.r‘𝑊)𝑁) = (𝑀(.r‘((Poly1‘𝑆) ↾s 𝐵))𝑁)) |
12 | evls1muld.1 | . . . . . 6 ⊢ × = (.r‘𝑊) | |
13 | 12 | oveqi 7364 | . . . . 5 ⊢ (𝑀 × 𝑁) = (𝑀(.r‘𝑊)𝑁) |
14 | 7 | fvexi 6853 | . . . . . . 7 ⊢ 𝐵 ∈ V |
15 | eqid 2737 | . . . . . . . 8 ⊢ (.r‘(Poly1‘𝑆)) = (.r‘(Poly1‘𝑆)) | |
16 | 9, 15 | ressmulr 17147 | . . . . . . 7 ⊢ (𝐵 ∈ V → (.r‘(Poly1‘𝑆)) = (.r‘((Poly1‘𝑆) ↾s 𝐵))) |
17 | 14, 16 | ax-mp 5 | . . . . . 6 ⊢ (.r‘(Poly1‘𝑆)) = (.r‘((Poly1‘𝑆) ↾s 𝐵)) |
18 | 17 | oveqi 7364 | . . . . 5 ⊢ (𝑀(.r‘(Poly1‘𝑆))𝑁) = (𝑀(.r‘((Poly1‘𝑆) ↾s 𝐵))𝑁) |
19 | 11, 13, 18 | 3eqtr4g 2802 | . . . 4 ⊢ (𝜑 → (𝑀 × 𝑁) = (𝑀(.r‘(Poly1‘𝑆))𝑁)) |
20 | 19 | fveq2d 6843 | . . 3 ⊢ (𝜑 → ((eval1‘𝑆)‘(𝑀 × 𝑁)) = ((eval1‘𝑆)‘(𝑀(.r‘(Poly1‘𝑆))𝑁))) |
21 | 20 | fveq1d 6841 | . 2 ⊢ (𝜑 → (((eval1‘𝑆)‘(𝑀 × 𝑁))‘𝐶) = (((eval1‘𝑆)‘(𝑀(.r‘(Poly1‘𝑆))𝑁))‘𝐶)) |
22 | ressply1evl.q | . . . . . 6 ⊢ 𝑄 = (𝑆 evalSub1 𝑅) | |
23 | ressply1evl.k | . . . . . 6 ⊢ 𝐾 = (Base‘𝑆) | |
24 | eqid 2737 | . . . . . 6 ⊢ (eval1‘𝑆) = (eval1‘𝑆) | |
25 | evls1muld.s | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ CRing) | |
26 | 22, 23, 6, 5, 7, 24, 25, 8 | ressply1evl 32089 | . . . . 5 ⊢ (𝜑 → 𝑄 = ((eval1‘𝑆) ↾ 𝐵)) |
27 | 26 | fveq1d 6841 | . . . 4 ⊢ (𝜑 → (𝑄‘(𝑀 × 𝑁)) = (((eval1‘𝑆) ↾ 𝐵)‘(𝑀 × 𝑁))) |
28 | 5 | subrgring 20177 | . . . . . . 7 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑈 ∈ Ring) |
29 | 6 | ply1ring 21570 | . . . . . . 7 ⊢ (𝑈 ∈ Ring → 𝑊 ∈ Ring) |
30 | 8, 28, 29 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ Ring) |
31 | 7, 12, 30, 2, 3 | ringcld 19939 | . . . . 5 ⊢ (𝜑 → (𝑀 × 𝑁) ∈ 𝐵) |
32 | 31 | fvresd 6859 | . . . 4 ⊢ (𝜑 → (((eval1‘𝑆) ↾ 𝐵)‘(𝑀 × 𝑁)) = ((eval1‘𝑆)‘(𝑀 × 𝑁))) |
33 | 27, 32 | eqtr2d 2778 | . . 3 ⊢ (𝜑 → ((eval1‘𝑆)‘(𝑀 × 𝑁)) = (𝑄‘(𝑀 × 𝑁))) |
34 | 33 | fveq1d 6841 | . 2 ⊢ (𝜑 → (((eval1‘𝑆)‘(𝑀 × 𝑁))‘𝐶) = ((𝑄‘(𝑀 × 𝑁))‘𝐶)) |
35 | eqid 2737 | . . . 4 ⊢ (Base‘(Poly1‘𝑆)) = (Base‘(Poly1‘𝑆)) | |
36 | evls1muld.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝐾) | |
37 | eqid 2737 | . . . . . . . 8 ⊢ (PwSer1‘𝑈) = (PwSer1‘𝑈) | |
38 | eqid 2737 | . . . . . . . 8 ⊢ (Base‘(PwSer1‘𝑈)) = (Base‘(PwSer1‘𝑈)) | |
39 | 4, 5, 6, 7, 8, 37, 38, 35 | ressply1bas2 21550 | . . . . . . 7 ⊢ (𝜑 → 𝐵 = ((Base‘(PwSer1‘𝑈)) ∩ (Base‘(Poly1‘𝑆)))) |
40 | inss2 4187 | . . . . . . 7 ⊢ ((Base‘(PwSer1‘𝑈)) ∩ (Base‘(Poly1‘𝑆))) ⊆ (Base‘(Poly1‘𝑆)) | |
41 | 39, 40 | eqsstrdi 3996 | . . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ (Base‘(Poly1‘𝑆))) |
42 | 41, 2 | sseldd 3943 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ (Base‘(Poly1‘𝑆))) |
43 | 26 | fveq1d 6841 | . . . . . . 7 ⊢ (𝜑 → (𝑄‘𝑀) = (((eval1‘𝑆) ↾ 𝐵)‘𝑀)) |
44 | 2 | fvresd 6859 | . . . . . . 7 ⊢ (𝜑 → (((eval1‘𝑆) ↾ 𝐵)‘𝑀) = ((eval1‘𝑆)‘𝑀)) |
45 | 43, 44 | eqtr2d 2778 | . . . . . 6 ⊢ (𝜑 → ((eval1‘𝑆)‘𝑀) = (𝑄‘𝑀)) |
46 | 45 | fveq1d 6841 | . . . . 5 ⊢ (𝜑 → (((eval1‘𝑆)‘𝑀)‘𝐶) = ((𝑄‘𝑀)‘𝐶)) |
47 | 42, 46 | jca 512 | . . . 4 ⊢ (𝜑 → (𝑀 ∈ (Base‘(Poly1‘𝑆)) ∧ (((eval1‘𝑆)‘𝑀)‘𝐶) = ((𝑄‘𝑀)‘𝐶))) |
48 | 41, 3 | sseldd 3943 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (Base‘(Poly1‘𝑆))) |
49 | 26 | fveq1d 6841 | . . . . . . 7 ⊢ (𝜑 → (𝑄‘𝑁) = (((eval1‘𝑆) ↾ 𝐵)‘𝑁)) |
50 | 3 | fvresd 6859 | . . . . . . 7 ⊢ (𝜑 → (((eval1‘𝑆) ↾ 𝐵)‘𝑁) = ((eval1‘𝑆)‘𝑁)) |
51 | 49, 50 | eqtr2d 2778 | . . . . . 6 ⊢ (𝜑 → ((eval1‘𝑆)‘𝑁) = (𝑄‘𝑁)) |
52 | 51 | fveq1d 6841 | . . . . 5 ⊢ (𝜑 → (((eval1‘𝑆)‘𝑁)‘𝐶) = ((𝑄‘𝑁)‘𝐶)) |
53 | 48, 52 | jca 512 | . . . 4 ⊢ (𝜑 → (𝑁 ∈ (Base‘(Poly1‘𝑆)) ∧ (((eval1‘𝑆)‘𝑁)‘𝐶) = ((𝑄‘𝑁)‘𝐶))) |
54 | evls1muld.2 | . . . 4 ⊢ · = (.r‘𝑆) | |
55 | 24, 4, 23, 35, 25, 36, 47, 53, 15, 54 | evl1muld 21660 | . . 3 ⊢ (𝜑 → ((𝑀(.r‘(Poly1‘𝑆))𝑁) ∈ (Base‘(Poly1‘𝑆)) ∧ (((eval1‘𝑆)‘(𝑀(.r‘(Poly1‘𝑆))𝑁))‘𝐶) = (((𝑄‘𝑀)‘𝐶) · ((𝑄‘𝑁)‘𝐶)))) |
56 | 55 | simprd 496 | . 2 ⊢ (𝜑 → (((eval1‘𝑆)‘(𝑀(.r‘(Poly1‘𝑆))𝑁))‘𝐶) = (((𝑄‘𝑀)‘𝐶) · ((𝑄‘𝑁)‘𝐶))) |
57 | 21, 34, 56 | 3eqtr3d 2785 | 1 ⊢ (𝜑 → ((𝑄‘(𝑀 × 𝑁))‘𝐶) = (((𝑄‘𝑀)‘𝐶) · ((𝑄‘𝑁)‘𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 Vcvv 3443 ∩ cin 3907 ↾ cres 5633 ‘cfv 6493 (class class class)co 7351 Basecbs 17042 ↾s cress 17071 .rcmulr 17093 Ringcrg 19917 CRingccrg 19918 SubRingcsubrg 20170 PwSer1cps1 21497 Poly1cpl1 21499 evalSub1 ces1 21630 eval1ce1 21631 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-iin 4955 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-of 7609 df-ofr 7610 df-om 7795 df-1st 7913 df-2nd 7914 df-supp 8085 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-er 8606 df-map 8725 df-pm 8726 df-ixp 8794 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-fsupp 9264 df-sup 9336 df-oi 9404 df-card 9833 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-2 12174 df-3 12175 df-4 12176 df-5 12177 df-6 12178 df-7 12179 df-8 12180 df-9 12181 df-n0 12372 df-z 12458 df-dec 12577 df-uz 12722 df-fz 13379 df-fzo 13522 df-seq 13861 df-hash 14184 df-struct 16978 df-sets 16995 df-slot 17013 df-ndx 17025 df-base 17043 df-ress 17072 df-plusg 17105 df-mulr 17106 df-sca 17108 df-vsca 17109 df-ip 17110 df-tset 17111 df-ple 17112 df-ds 17114 df-hom 17116 df-cco 17117 df-0g 17282 df-gsum 17283 df-prds 17288 df-pws 17290 df-mre 17425 df-mrc 17426 df-acs 17428 df-mgm 18456 df-sgrp 18505 df-mnd 18516 df-mhm 18560 df-submnd 18561 df-grp 18710 df-minusg 18711 df-sbg 18712 df-mulg 18831 df-subg 18883 df-ghm 18964 df-cntz 19055 df-cmn 19522 df-abl 19523 df-mgp 19855 df-ur 19872 df-srg 19876 df-ring 19919 df-cring 19920 df-rnghom 20098 df-subrg 20172 df-lmod 20276 df-lss 20345 df-lsp 20385 df-assa 21211 df-asp 21212 df-ascl 21213 df-psr 21263 df-mvr 21264 df-mpl 21265 df-opsr 21267 df-evls 21433 df-evl 21434 df-psr1 21502 df-vr1 21503 df-ply1 21504 df-coe1 21505 df-evls1 21632 df-evl1 21633 |
This theorem is referenced by: minplyeulem 32176 |
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