mhphf2 - Mathbox for Steven Nguyen

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Theorem mhphf2 40286

Description: A homogeneous polynomial defines a homogeneous function; this is mhphf 40285 with simpler notation in the conclusion in exchange for a complex definition of ∙, which is based on frlmvscafval 20973 but without the finite support restriction (frlmpws 20957, frlmbas 20962) on the assignments 𝐴 from variables to values.

TODO?: Polynomials (df-mpl 21114) are defined to have a finite amount of terms (of finite degree). As such, any assignment may be replaced by an assignment with finite support (as only a finite amount of variables matter in a given polynomial, even if the set of variables is infinite). So the finite support restriction can be assumed without loss of generality. (Contributed by SN, 11-Nov-2024.)

**Hypotheses**
Ref	Expression
mhphf2.q	⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)
mhphf2.h	⊢ 𝐻 = (𝐼 mHomP 𝑈)
mhphf2.u	⊢ 𝑈 = (𝑆 ↾_s 𝑅)
mhphf2.k	⊢ 𝐾 = (Base‘𝑆)
mhphf2.b	⊢ ∙ = ( ·_𝑠 ‘((ringLMod‘𝑆) ↑_s 𝐼))
mhphf2.m	⊢ · = (._r‘𝑆)
mhphf2.e	⊢ ↑ = (._g‘(mulGrp‘𝑆))
mhphf2.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
mhphf2.s	⊢ (𝜑 → 𝑆 ∈ CRing)
mhphf2.r	⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))
mhphf2.l	⊢ (𝜑 → 𝐿 ∈ 𝑅)
mhphf2.n	⊢ (𝜑 → 𝑁 ∈ ℕ₀)
mhphf2.x	⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁))
mhphf2.a	⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑_m 𝐼))

**Assertion**
Ref	Expression
mhphf2	⊢ (𝜑 → ((𝑄‘𝑋)‘(𝐿 ∙ 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑋)‘𝐴)))

**Proof of Theorem mhphf2**
Step	Hyp	Ref	Expression
1		eqid 2738	. . . . 5 ⊢ ((ringLMod‘𝑆) ↑_s 𝐼) = ((ringLMod‘𝑆) ↑_s 𝐼)
2		eqid 2738	. . . . 5 ⊢ (Base‘((ringLMod‘𝑆) ↑_s 𝐼)) = (Base‘((ringLMod‘𝑆) ↑_s 𝐼))
3		rlmvsca 20472	. . . . 5 ⊢ (._r‘𝑆) = ( ·_𝑠 ‘(ringLMod‘𝑆))
4		mhphf2.b	. . . . 5 ⊢ ∙ = ( ·_𝑠 ‘((ringLMod‘𝑆) ↑_s 𝐼))
5		eqid 2738	. . . . 5 ⊢ (Scalar‘(ringLMod‘𝑆)) = (Scalar‘(ringLMod‘𝑆))
6		eqid 2738	. . . . 5 ⊢ (Base‘(Scalar‘(ringLMod‘𝑆))) = (Base‘(Scalar‘(ringLMod‘𝑆)))
7		fvexd 6789	. . . . 5 ⊢ (𝜑 → (ringLMod‘𝑆) ∈ V)
8		mhphf2.i	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
9		mhphf2.r	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))
10		mhphf2.k	. . . . . . . . 9 ⊢ 𝐾 = (Base‘𝑆)
11	10	subrgss 20025	. . . . . . . 8 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ⊆ 𝐾)
12	9, 11	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑅 ⊆ 𝐾)
13		mhphf2.l	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ 𝑅)
14	12, 13	sseldd 3922	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ 𝐾)
15		mhphf2.s	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ CRing)
16		rlmsca 20470	. . . . . . . . 9 ⊢ (𝑆 ∈ CRing → 𝑆 = (Scalar‘(ringLMod‘𝑆)))
17	15, 16	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑆 = (Scalar‘(ringLMod‘𝑆)))
18	17	fveq2d 6778	. . . . . . 7 ⊢ (𝜑 → (Base‘𝑆) = (Base‘(Scalar‘(ringLMod‘𝑆))))
19	10, 18	eqtrid 2790	. . . . . 6 ⊢ (𝜑 → 𝐾 = (Base‘(Scalar‘(ringLMod‘𝑆))))
20	14, 19	eleqtrd 2841	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ (Base‘(Scalar‘(ringLMod‘𝑆))))
21		mhphf2.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑_m 𝐼))
22	10	oveq1i 7285	. . . . . . 7 ⊢ (𝐾 ↑_m 𝐼) = ((Base‘𝑆) ↑_m 𝐼)
23	21, 22	eleqtrdi 2849	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ((Base‘𝑆) ↑_m 𝐼))
24		rlmbas 20465	. . . . . . . 8 ⊢ (Base‘𝑆) = (Base‘(ringLMod‘𝑆))
25	1, 24	pwsbas 17198	. . . . . . 7 ⊢ (((ringLMod‘𝑆) ∈ V ∧ 𝐼 ∈ 𝑉) → ((Base‘𝑆) ↑_m 𝐼) = (Base‘((ringLMod‘𝑆) ↑_s 𝐼)))
26	7, 8, 25	syl2anc 584	. . . . . 6 ⊢ (𝜑 → ((Base‘𝑆) ↑_m 𝐼) = (Base‘((ringLMod‘𝑆) ↑_s 𝐼)))
27	23, 26	eleqtrd 2841	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (Base‘((ringLMod‘𝑆) ↑_s 𝐼)))
28	1, 2, 3, 4, 5, 6, 7, 8, 20, 27	pwsvscafval 17205	. . . 4 ⊢ (𝜑 → (𝐿 ∙ 𝐴) = ((𝐼 × {𝐿}) ∘_f (._r‘𝑆)𝐴))
29		mhphf2.m	. . . . . . 7 ⊢ · = (._r‘𝑆)
30	29	eqcomi 2747	. . . . . 6 ⊢ (._r‘𝑆) = ·
31		ofeq 7536	. . . . . 6 ⊢ ((._r‘𝑆) = · → ∘_f (._r‘𝑆) = ∘_f · )
32	30, 31	mp1i 13	. . . . 5 ⊢ (𝜑 → ∘_f (._r‘𝑆) = ∘_f · )
33	32	oveqd 7292	. . . 4 ⊢ (𝜑 → ((𝐼 × {𝐿}) ∘_f (._r‘𝑆)𝐴) = ((𝐼 × {𝐿}) ∘_f · 𝐴))
34	28, 33	eqtrd 2778	. . 3 ⊢ (𝜑 → (𝐿 ∙ 𝐴) = ((𝐼 × {𝐿}) ∘_f · 𝐴))
35	34	fveq2d 6778	. 2 ⊢ (𝜑 → ((𝑄‘𝑋)‘(𝐿 ∙ 𝐴)) = ((𝑄‘𝑋)‘((𝐼 × {𝐿}) ∘_f · 𝐴)))
36		mhphf2.q	. . 3 ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)
37		mhphf2.h	. . 3 ⊢ 𝐻 = (𝐼 mHomP 𝑈)
38		mhphf2.u	. . 3 ⊢ 𝑈 = (𝑆 ↾_s 𝑅)
39		mhphf2.e	. . 3 ⊢ ↑ = (._g‘(mulGrp‘𝑆))
40		mhphf2.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
41		mhphf2.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁))
42	36, 37, 38, 10, 29, 39, 8, 15, 9, 13, 40, 41, 21	mhphf 40285	. 2 ⊢ (𝜑 → ((𝑄‘𝑋)‘((𝐼 × {𝐿}) ∘_f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑋)‘𝐴)))
43	35, 42	eqtrd 2778	1 ⊢ (𝜑 → ((𝑄‘𝑋)‘(𝐿 ∙ 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑋)‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 Vcvv 3432 ⊆ wss 3887 {csn 4561 × cxp 5587 ‘cfv 6433 (class class class)co 7275 ∘_f cof 7531 ↑_m cmap 8615 ℕ₀cn0 12233 Basecbs 16912 ↾_s cress 16941 ._rcmulr 16963 Scalarcsca 16965 ·_𝑠 cvsca 16966 ↑_s cpws 17157 ._gcmg 18700 mulGrpcmgp 19720 CRingccrg 19784 SubRingcsubrg 20020 ringLModcrglmod 20431 evalSub ces 21280 mHomP cmhp 21319

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 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-addf 10950 ax-mulf 10951

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-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 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 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-ofr 7534 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-oadd 8301 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-sup 9201 df-oi 9269 df-dju 9659 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-fz 13240 df-fzo 13383 df-seq 13722 df-hash 14045 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-0g 17152 df-gsum 17153 df-prds 17158 df-pws 17160 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-mhm 18430 df-submnd 18431 df-grp 18580 df-minusg 18581 df-sbg 18582 df-mulg 18701 df-subg 18752 df-ghm 18832 df-cntz 18923 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-srg 19742 df-ring 19785 df-cring 19786 df-rnghom 19959 df-subrg 20022 df-lmod 20125 df-lss 20194 df-lsp 20234 df-sra 20434 df-rgmod 20435 df-cnfld 20598 df-assa 21060 df-asp 21061 df-ascl 21062 df-psr 21112 df-mvr 21113 df-mpl 21114 df-evls 21282 df-mhp 21323

This theorem is referenced by: (None)

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