mhphf2 - Mathbox for Steven Nguyen

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Theorem mhphf2 40295

Description: A homogeneous polynomial defines a homogeneous function; this is mhphf 40294 with simpler notation in the conclusion in exchange for a complex definition of ∙, which is based on frlmvscafval 20984 but without the finite support restriction (frlmpws 20968, frlmbas 20973) on the assignments 𝐴 from variables to values.

TODO?: Polynomials (df-mpl 21125) 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 2740	. . . . 5 ⊢ ((ringLMod‘𝑆) ↑_s 𝐼) = ((ringLMod‘𝑆) ↑_s 𝐼)
2		eqid 2740	. . . . 5 ⊢ (Base‘((ringLMod‘𝑆) ↑_s 𝐼)) = (Base‘((ringLMod‘𝑆) ↑_s 𝐼))
3		rlmvsca 20483	. . . . 5 ⊢ (._r‘𝑆) = ( ·_𝑠 ‘(ringLMod‘𝑆))
4		mhphf2.b	. . . . 5 ⊢ ∙ = ( ·_𝑠 ‘((ringLMod‘𝑆) ↑_s 𝐼))
5		eqid 2740	. . . . 5 ⊢ (Scalar‘(ringLMod‘𝑆)) = (Scalar‘(ringLMod‘𝑆))
6		eqid 2740	. . . . 5 ⊢ (Base‘(Scalar‘(ringLMod‘𝑆))) = (Base‘(Scalar‘(ringLMod‘𝑆)))
7		fvexd 6786	. . . . 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 20036	. . . . . . . 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 3927	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ 𝐾)
15		mhphf2.s	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ CRing)
16		rlmsca 20481	. . . . . . . . 9 ⊢ (𝑆 ∈ CRing → 𝑆 = (Scalar‘(ringLMod‘𝑆)))
17	15, 16	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑆 = (Scalar‘(ringLMod‘𝑆)))
18	17	fveq2d 6775	. . . . . . 7 ⊢ (𝜑 → (Base‘𝑆) = (Base‘(Scalar‘(ringLMod‘𝑆))))
19	10, 18	eqtrid 2792	. . . . . 6 ⊢ (𝜑 → 𝐾 = (Base‘(Scalar‘(ringLMod‘𝑆))))
20	14, 19	eleqtrd 2843	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ (Base‘(Scalar‘(ringLMod‘𝑆))))
21		mhphf2.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑_m 𝐼))
22	10	oveq1i 7282	. . . . . . 7 ⊢ (𝐾 ↑_m 𝐼) = ((Base‘𝑆) ↑_m 𝐼)
23	21, 22	eleqtrdi 2851	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ((Base‘𝑆) ↑_m 𝐼))
24		rlmbas 20476	. . . . . . . 8 ⊢ (Base‘𝑆) = (Base‘(ringLMod‘𝑆))
25	1, 24	pwsbas 17209	. . . . . . 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 2843	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (Base‘((ringLMod‘𝑆) ↑_s 𝐼)))
28	1, 2, 3, 4, 5, 6, 7, 8, 20, 27	pwsvscafval 17216	. . . 4 ⊢ (𝜑 → (𝐿 ∙ 𝐴) = ((𝐼 × {𝐿}) ∘_f (._r‘𝑆)𝐴))
29		mhphf2.m	. . . . . . 7 ⊢ · = (._r‘𝑆)
30	29	eqcomi 2749	. . . . . 6 ⊢ (._r‘𝑆) = ·
31		ofeq 7531	. . . . . 6 ⊢ ((._r‘𝑆) = · → ∘_f (._r‘𝑆) = ∘_f · )
32	30, 31	mp1i 13	. . . . 5 ⊢ (𝜑 → ∘_f (._r‘𝑆) = ∘_f · )
33	32	oveqd 7289	. . . 4 ⊢ (𝜑 → ((𝐼 × {𝐿}) ∘_f (._r‘𝑆)𝐴) = ((𝐼 × {𝐿}) ∘_f · 𝐴))
34	28, 33	eqtrd 2780	. . 3 ⊢ (𝜑 → (𝐿 ∙ 𝐴) = ((𝐼 × {𝐿}) ∘_f · 𝐴))
35	34	fveq2d 6775	. 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 40294	. 2 ⊢ (𝜑 → ((𝑄‘𝑋)‘((𝐼 × {𝐿}) ∘_f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑋)‘𝐴)))
43	35, 42	eqtrd 2780	1 ⊢ (𝜑 → ((𝑄‘𝑋)‘(𝐿 ∙ 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑋)‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2110 Vcvv 3431 ⊆ wss 3892 {csn 4567 × cxp 5588 ‘cfv 6432 (class class class)co 7272 ∘_f cof 7526 ↑_m cmap 8607 ℕ₀cn0 12244 Basecbs 16923 ↾_s cress 16952 ._rcmulr 16974 Scalarcsca 16976 ·_𝑠 cvsca 16977 ↑_s cpws 17168 ._gcmg 18711 mulGrpcmgp 19731 CRingccrg 19795 SubRingcsubrg 20031 ringLModcrglmod 20442 evalSub ces 21291 mHomP cmhp 21330

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7583 ax-cnex 10938 ax-resscn 10939 ax-1cn 10940 ax-icn 10941 ax-addcl 10942 ax-addrcl 10943 ax-mulcl 10944 ax-mulrcl 10945 ax-mulcom 10946 ax-addass 10947 ax-mulass 10948 ax-distr 10949 ax-i2m1 10950 ax-1ne0 10951 ax-1rid 10952 ax-rnegex 10953 ax-rrecex 10954 ax-cnre 10955 ax-pre-lttri 10956 ax-pre-lttrn 10957 ax-pre-ltadd 10958 ax-pre-mulgt0 10959 ax-addf 10961 ax-mulf 10962

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-isom 6441 df-riota 7229 df-ov 7275 df-oprab 7276 df-mpo 7277 df-of 7528 df-ofr 7529 df-om 7708 df-1st 7825 df-2nd 7826 df-supp 7970 df-frecs 8089 df-wrecs 8120 df-recs 8194 df-rdg 8233 df-1o 8289 df-oadd 8293 df-er 8490 df-map 8609 df-pm 8610 df-ixp 8678 df-en 8726 df-dom 8727 df-sdom 8728 df-fin 8729 df-fsupp 9117 df-sup 9189 df-oi 9257 df-dju 9670 df-card 9708 df-pnf 11022 df-mnf 11023 df-xr 11024 df-ltxr 11025 df-le 11026 df-sub 11218 df-neg 11219 df-nn 11985 df-2 12047 df-3 12048 df-4 12049 df-5 12050 df-6 12051 df-7 12052 df-8 12053 df-9 12054 df-n0 12245 df-z 12331 df-dec 12449 df-uz 12594 df-fz 13251 df-fzo 13394 df-seq 13733 df-hash 14056 df-struct 16859 df-sets 16876 df-slot 16894 df-ndx 16906 df-base 16924 df-ress 16953 df-plusg 16986 df-mulr 16987 df-starv 16988 df-sca 16989 df-vsca 16990 df-ip 16991 df-tset 16992 df-ple 16993 df-ds 16995 df-unif 16996 df-hom 16997 df-cco 16998 df-0g 17163 df-gsum 17164 df-prds 17169 df-pws 17171 df-mre 17306 df-mrc 17307 df-acs 17309 df-mgm 18337 df-sgrp 18386 df-mnd 18397 df-mhm 18441 df-submnd 18442 df-grp 18591 df-minusg 18592 df-sbg 18593 df-mulg 18712 df-subg 18763 df-ghm 18843 df-cntz 18934 df-cmn 19399 df-abl 19400 df-mgp 19732 df-ur 19749 df-srg 19753 df-ring 19796 df-cring 19797 df-rnghom 19970 df-subrg 20033 df-lmod 20136 df-lss 20205 df-lsp 20245 df-sra 20445 df-rgmod 20446 df-cnfld 20609 df-assa 21071 df-asp 21072 df-ascl 21073 df-psr 21123 df-mvr 21124 df-mpl 21125 df-evls 21293 df-mhp 21334

This theorem is referenced by: (None)

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