coe1f - Metamath Proof Explorer

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Theorem coe1f 21726

Description: Functionality of univariate polynomial coefficient vectors. (Contributed by Stefan O'Rear, 21-Mar-2015.)

**Hypotheses**
Ref	Expression
coe1fval.a	⊢ 𝐴 = (coe₁‘𝐹)
coe1f.b	⊢ 𝐵 = (Base‘𝑃)
coe1f.p	⊢ 𝑃 = (Poly₁‘𝑅)
coe1f.k	⊢ 𝐾 = (Base‘𝑅)

**Assertion**
Ref	Expression
coe1f	⊢ (𝐹 ∈ 𝐵 → 𝐴:ℕ₀⟶𝐾)

**Proof of Theorem coe1f**
Step	Hyp	Ref	Expression
1		coe1f.p	. . 3 ⊢ 𝑃 = (Poly₁‘𝑅)
2		coe1f.b	. . 3 ⊢ 𝐵 = (Base‘𝑃)
3	1, 2	ply1bascl 21718	. 2 ⊢ (𝐹 ∈ 𝐵 → 𝐹 ∈ (Base‘(PwSer₁‘𝑅)))
4		coe1fval.a	. . 3 ⊢ 𝐴 = (coe₁‘𝐹)
5		eqid 2732	. . 3 ⊢ (Base‘(PwSer₁‘𝑅)) = (Base‘(PwSer₁‘𝑅))
6		eqid 2732	. . 3 ⊢ (PwSer₁‘𝑅) = (PwSer₁‘𝑅)
7		coe1f.k	. . 3 ⊢ 𝐾 = (Base‘𝑅)
8	4, 5, 6, 7	coe1f2 21724	. 2 ⊢ (𝐹 ∈ (Base‘(PwSer₁‘𝑅)) → 𝐴:ℕ₀⟶𝐾)
9	3, 8	syl 17	1 ⊢ (𝐹 ∈ 𝐵 → 𝐴:ℕ₀⟶𝐾)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ⟶wf 6536 ‘cfv 6540 ℕ₀cn0 12468 Basecbs 17140 PwSer₁cps1 21690 Poly₁cpl1 21692 coe₁cco1 21693

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 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-sca 17209 df-vsca 17210 df-tset 17212 df-ple 17213 df-psr 21453 df-opsr 21457 df-psr1 21695 df-ply1 21697 df-coe1 21698

This theorem is referenced by: coe1fvalcl 21727 coe1fsupp 21729 coe1addfv 21778 coe1subfv 21779 coe1tmmul2 21789 coe1tmmul 21790 coe1pwmul 21792 coe1sclmul 21795 coe1sclmulfv 21796 coe1sclmul2 21797 ply1coefsupp 21810 ply1coe 21811 coe1fzgsumdlem 21816 mply1topmatcl 22298 cayhamlem3 22380 deg1ldgdomn 25603 coe1mul3 25608 deg1add 25612 deg1sublt 25619 deg1mul2 25623 deg1mul3 25624 deg1mul3le 25625 ply1divex 25645 uc1pmon1p 25660 ply1rem 25672 fta1g 25676 drnguc1p 25679 plypf1 25717 evls1fpws 32634 ply1gsumz 32657 ply1degltdimlem 32695 hbtlem2 41851

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