mhphf2 - Mathbox for Steven Nguyen

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Theorem mhphf2 40758

Description: A homogeneous polynomial defines a homogeneous function; this is mhphf 40757 with simpler notation in the conclusion in exchange for a complex definition of ∙, which is based on frlmvscafval 21172 but without the finite support restriction (frlmpws 21156, frlmbas 21161) on the assignments 𝐴 from variables to values.

TODO?: Polynomials (df-mpl 21313) 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 2736	. . . . 5 ⊢ ((ringLMod‘𝑆) ↑_s 𝐼) = ((ringLMod‘𝑆) ↑_s 𝐼)
2		eqid 2736	. . . . 5 ⊢ (Base‘((ringLMod‘𝑆) ↑_s 𝐼)) = (Base‘((ringLMod‘𝑆) ↑_s 𝐼))
3		rlmvsca 20671	. . . . 5 ⊢ (._r‘𝑆) = ( ·_𝑠 ‘(ringLMod‘𝑆))
4		mhphf2.b	. . . . 5 ⊢ ∙ = ( ·_𝑠 ‘((ringLMod‘𝑆) ↑_s 𝐼))
5		eqid 2736	. . . . 5 ⊢ (Scalar‘(ringLMod‘𝑆)) = (Scalar‘(ringLMod‘𝑆))
6		eqid 2736	. . . . 5 ⊢ (Base‘(Scalar‘(ringLMod‘𝑆))) = (Base‘(Scalar‘(ringLMod‘𝑆)))
7		fvexd 6857	. . . . 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 20223	. . . . . . . 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 3945	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ 𝐾)
15		mhphf2.s	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ CRing)
16		rlmsca 20669	. . . . . . . . 9 ⊢ (𝑆 ∈ CRing → 𝑆 = (Scalar‘(ringLMod‘𝑆)))
17	15, 16	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑆 = (Scalar‘(ringLMod‘𝑆)))
18	17	fveq2d 6846	. . . . . . 7 ⊢ (𝜑 → (Base‘𝑆) = (Base‘(Scalar‘(ringLMod‘𝑆))))
19	10, 18	eqtrid 2788	. . . . . 6 ⊢ (𝜑 → 𝐾 = (Base‘(Scalar‘(ringLMod‘𝑆))))
20	14, 19	eleqtrd 2840	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ (Base‘(Scalar‘(ringLMod‘𝑆))))
21		mhphf2.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑_m 𝐼))
22	10	oveq1i 7367	. . . . . . 7 ⊢ (𝐾 ↑_m 𝐼) = ((Base‘𝑆) ↑_m 𝐼)
23	21, 22	eleqtrdi 2848	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ((Base‘𝑆) ↑_m 𝐼))
24		rlmbas 20664	. . . . . . . 8 ⊢ (Base‘𝑆) = (Base‘(ringLMod‘𝑆))
25	1, 24	pwsbas 17369	. . . . . . 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 2840	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (Base‘((ringLMod‘𝑆) ↑_s 𝐼)))
28	1, 2, 3, 4, 5, 6, 7, 8, 20, 27	pwsvscafval 17376	. . . 4 ⊢ (𝜑 → (𝐿 ∙ 𝐴) = ((𝐼 × {𝐿}) ∘_f (._r‘𝑆)𝐴))
29		mhphf2.m	. . . . . . 7 ⊢ · = (._r‘𝑆)
30	29	eqcomi 2745	. . . . . 6 ⊢ (._r‘𝑆) = ·
31		ofeq 7620	. . . . . 6 ⊢ ((._r‘𝑆) = · → ∘_f (._r‘𝑆) = ∘_f · )
32	30, 31	mp1i 13	. . . . 5 ⊢ (𝜑 → ∘_f (._r‘𝑆) = ∘_f · )
33	32	oveqd 7374	. . . 4 ⊢ (𝜑 → ((𝐼 × {𝐿}) ∘_f (._r‘𝑆)𝐴) = ((𝐼 × {𝐿}) ∘_f · 𝐴))
34	28, 33	eqtrd 2776	. . 3 ⊢ (𝜑 → (𝐿 ∙ 𝐴) = ((𝐼 × {𝐿}) ∘_f · 𝐴))
35	34	fveq2d 6846	. 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 40757	. 2 ⊢ (𝜑 → ((𝑄‘𝑋)‘((𝐼 × {𝐿}) ∘_f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑋)‘𝐴)))
43	35, 42	eqtrd 2776	1 ⊢ (𝜑 → ((𝑄‘𝑋)‘(𝐿 ∙ 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑋)‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 Vcvv 3445 ⊆ wss 3910 {csn 4586 × cxp 5631 ‘cfv 6496 (class class class)co 7357 ∘_f cof 7615 ↑_m cmap 8765 ℕ₀cn0 12413 Basecbs 17083 ↾_s cress 17112 ._rcmulr 17134 Scalarcsca 17136 ·_𝑠 cvsca 17137 ↑_s cpws 17328 ._gcmg 18872 mulGrpcmgp 19896 CRingccrg 19965 SubRingcsubrg 20218 ringLModcrglmod 20630 evalSub ces 21480 mHomP cmhp 21519

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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-addf 11130 ax-mulf 11131

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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-ofr 7618 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-oadd 8416 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-sup 9378 df-oi 9446 df-dju 9837 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-fz 13425 df-fzo 13568 df-seq 13907 df-hash 14231 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-0g 17323 df-gsum 17324 df-prds 17329 df-pws 17331 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-mhm 18601 df-submnd 18602 df-grp 18751 df-minusg 18752 df-sbg 18753 df-mulg 18873 df-subg 18925 df-ghm 19006 df-cntz 19097 df-cmn 19564 df-abl 19565 df-mgp 19897 df-ur 19914 df-srg 19918 df-ring 19966 df-cring 19967 df-rnghom 20146 df-subrg 20220 df-lmod 20324 df-lss 20393 df-lsp 20433 df-sra 20633 df-rgmod 20634 df-cnfld 20797 df-assa 21259 df-asp 21260 df-ascl 21261 df-psr 21311 df-mvr 21312 df-mpl 21313 df-evls 21482 df-mhp 21523

This theorem is referenced by: (None)

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