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Mirrors > Home > MPE Home > Th. List > evl1vsd | Structured version Visualization version GIF version |
Description: Polynomial evaluation builder for scalar 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)) |
evl1vsd.4 | ⊢ (𝜑 → 𝑁 ∈ 𝐵) |
evl1vsd.s | ⊢ ∙ = ( ·𝑠 ‘𝑃) |
evl1vsd.t | ⊢ · = (.r‘𝑅) |
Ref | Expression |
---|---|
evl1vsd | ⊢ (𝜑 → ((𝑁 ∙ 𝑀) ∈ 𝑈 ∧ ((𝑂‘(𝑁 ∙ 𝑀))‘𝑌) = (𝑁 · 𝑉))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evl1addd.q | . . 3 ⊢ 𝑂 = (eval1‘𝑅) | |
2 | evl1addd.p | . . 3 ⊢ 𝑃 = (Poly1‘𝑅) | |
3 | evl1addd.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
4 | evl1addd.u | . . 3 ⊢ 𝑈 = (Base‘𝑃) | |
5 | evl1addd.1 | . . 3 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
6 | evl1addd.2 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
7 | eqid 2731 | . . . 4 ⊢ (algSc‘𝑃) = (algSc‘𝑃) | |
8 | evl1vsd.4 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ 𝐵) | |
9 | 1, 2, 3, 7, 4, 5, 8, 6 | evl1scad 22174 | . . 3 ⊢ (𝜑 → (((algSc‘𝑃)‘𝑁) ∈ 𝑈 ∧ ((𝑂‘((algSc‘𝑃)‘𝑁))‘𝑌) = 𝑁)) |
10 | evl1addd.3 | . . 3 ⊢ (𝜑 → (𝑀 ∈ 𝑈 ∧ ((𝑂‘𝑀)‘𝑌) = 𝑉)) | |
11 | eqid 2731 | . . 3 ⊢ (.r‘𝑃) = (.r‘𝑃) | |
12 | evl1vsd.t | . . 3 ⊢ · = (.r‘𝑅) | |
13 | 1, 2, 3, 4, 5, 6, 9, 10, 11, 12 | evl1muld 22182 | . 2 ⊢ (𝜑 → ((((algSc‘𝑃)‘𝑁)(.r‘𝑃)𝑀) ∈ 𝑈 ∧ ((𝑂‘(((algSc‘𝑃)‘𝑁)(.r‘𝑃)𝑀))‘𝑌) = (𝑁 · 𝑉))) |
14 | 2 | ply1assa 22042 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ AssAlg) |
15 | 5, 14 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ AssAlg) |
16 | 2 | ply1sca 22095 | . . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 = (Scalar‘𝑃)) |
17 | 5, 16 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃)) |
18 | 17 | fveq2d 6895 | . . . . . . 7 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑃))) |
19 | 3, 18 | eqtrid 2783 | . . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘(Scalar‘𝑃))) |
20 | 8, 19 | eleqtrd 2834 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (Base‘(Scalar‘𝑃))) |
21 | 10 | simpld 494 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ 𝑈) |
22 | eqid 2731 | . . . . . 6 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃) | |
23 | eqid 2731 | . . . . . 6 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃)) | |
24 | evl1vsd.s | . . . . . 6 ⊢ ∙ = ( ·𝑠 ‘𝑃) | |
25 | 7, 22, 23, 4, 11, 24 | asclmul1 21750 | . . . . 5 ⊢ ((𝑃 ∈ AssAlg ∧ 𝑁 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑀 ∈ 𝑈) → (((algSc‘𝑃)‘𝑁)(.r‘𝑃)𝑀) = (𝑁 ∙ 𝑀)) |
26 | 15, 20, 21, 25 | syl3anc 1370 | . . . 4 ⊢ (𝜑 → (((algSc‘𝑃)‘𝑁)(.r‘𝑃)𝑀) = (𝑁 ∙ 𝑀)) |
27 | 26 | eleq1d 2817 | . . 3 ⊢ (𝜑 → ((((algSc‘𝑃)‘𝑁)(.r‘𝑃)𝑀) ∈ 𝑈 ↔ (𝑁 ∙ 𝑀) ∈ 𝑈)) |
28 | 26 | fveq2d 6895 | . . . . 5 ⊢ (𝜑 → (𝑂‘(((algSc‘𝑃)‘𝑁)(.r‘𝑃)𝑀)) = (𝑂‘(𝑁 ∙ 𝑀))) |
29 | 28 | fveq1d 6893 | . . . 4 ⊢ (𝜑 → ((𝑂‘(((algSc‘𝑃)‘𝑁)(.r‘𝑃)𝑀))‘𝑌) = ((𝑂‘(𝑁 ∙ 𝑀))‘𝑌)) |
30 | 29 | eqeq1d 2733 | . . 3 ⊢ (𝜑 → (((𝑂‘(((algSc‘𝑃)‘𝑁)(.r‘𝑃)𝑀))‘𝑌) = (𝑁 · 𝑉) ↔ ((𝑂‘(𝑁 ∙ 𝑀))‘𝑌) = (𝑁 · 𝑉))) |
31 | 27, 30 | anbi12d 630 | . 2 ⊢ (𝜑 → (((((algSc‘𝑃)‘𝑁)(.r‘𝑃)𝑀) ∈ 𝑈 ∧ ((𝑂‘(((algSc‘𝑃)‘𝑁)(.r‘𝑃)𝑀))‘𝑌) = (𝑁 · 𝑉)) ↔ ((𝑁 ∙ 𝑀) ∈ 𝑈 ∧ ((𝑂‘(𝑁 ∙ 𝑀))‘𝑌) = (𝑁 · 𝑉)))) |
32 | 13, 31 | mpbid 231 | 1 ⊢ (𝜑 → ((𝑁 ∙ 𝑀) ∈ 𝑈 ∧ ((𝑂‘(𝑁 ∙ 𝑀))‘𝑌) = (𝑁 · 𝑉))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ‘cfv 6543 (class class class)co 7412 Basecbs 17151 .rcmulr 17205 Scalarcsca 17207 ·𝑠 cvsca 17208 CRingccrg 20135 AssAlgcasa 21715 algSccascl 21717 Poly1cpl1 22020 eval1ce1 22153 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-ofr 7675 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8152 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-pm 8829 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9368 df-sup 9443 df-oi 9511 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-fz 13492 df-fzo 13635 df-seq 13974 df-hash 14298 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-hom 17228 df-cco 17229 df-0g 17394 df-gsum 17395 df-prds 17400 df-pws 17402 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-mhm 18711 df-submnd 18712 df-grp 18864 df-minusg 18865 df-sbg 18866 df-mulg 18994 df-subg 19046 df-ghm 19135 df-cntz 19229 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-srg 20088 df-ring 20136 df-cring 20137 df-rhm 20370 df-subrng 20442 df-subrg 20467 df-lmod 20704 df-lss 20775 df-lsp 20815 df-assa 21718 df-asp 21719 df-ascl 21720 df-psr 21772 df-mvr 21773 df-mpl 21774 df-opsr 21776 df-evls 21946 df-evl 21947 df-psr1 22023 df-ply1 22025 df-evl1 22155 |
This theorem is referenced by: evl1scvarpwval 22203 fta1blem 26024 plypf1 26064 evls1vsca 33090 |
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