mhphf2 - Mathbox for Steven Nguyen

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Theorem mhphf2 40181

Description: A homogeneous polynomial defines a homogeneous function; this is mhphf 40180 with simpler notation in the conclusion in exchange for a complex definition of ∙, which is based on frlmvscafval 20858 but without the finite support restriction (frlmpws 20842, frlmbas 20847) on the assignments 𝐴 from variables to values.

TODO?: Polynomials (df-mpl 20999) 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 2739	. . . . 5 ⊢ ((ringLMod‘𝑆) ↑_s 𝐼) = ((ringLMod‘𝑆) ↑_s 𝐼)
2		eqid 2739	. . . . 5 ⊢ (Base‘((ringLMod‘𝑆) ↑_s 𝐼)) = (Base‘((ringLMod‘𝑆) ↑_s 𝐼))
3		rlmvsca 20360	. . . . 5 ⊢ (._r‘𝑆) = ( ·_𝑠 ‘(ringLMod‘𝑆))
4		mhphf2.b	. . . . 5 ⊢ ∙ = ( ·_𝑠 ‘((ringLMod‘𝑆) ↑_s 𝐼))
5		eqid 2739	. . . . 5 ⊢ (Scalar‘(ringLMod‘𝑆)) = (Scalar‘(ringLMod‘𝑆))
6		eqid 2739	. . . . 5 ⊢ (Base‘(Scalar‘(ringLMod‘𝑆))) = (Base‘(Scalar‘(ringLMod‘𝑆)))
7		fvexd 6768	. . . . 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 19915	. . . . . . . 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 3919	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ 𝐾)
15		mhphf2.s	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ CRing)
16		rlmsca 20358	. . . . . . . . 9 ⊢ (𝑆 ∈ CRing → 𝑆 = (Scalar‘(ringLMod‘𝑆)))
17	15, 16	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑆 = (Scalar‘(ringLMod‘𝑆)))
18	17	fveq2d 6757	. . . . . . 7 ⊢ (𝜑 → (Base‘𝑆) = (Base‘(Scalar‘(ringLMod‘𝑆))))
19	10, 18	syl5eq 2792	. . . . . 6 ⊢ (𝜑 → 𝐾 = (Base‘(Scalar‘(ringLMod‘𝑆))))
20	14, 19	eleqtrd 2842	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ (Base‘(Scalar‘(ringLMod‘𝑆))))
21		mhphf2.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑_m 𝐼))
22	10	oveq1i 7262	. . . . . . 7 ⊢ (𝐾 ↑_m 𝐼) = ((Base‘𝑆) ↑_m 𝐼)
23	21, 22	eleqtrdi 2850	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ((Base‘𝑆) ↑_m 𝐼))
24		rlmbas 20353	. . . . . . . 8 ⊢ (Base‘𝑆) = (Base‘(ringLMod‘𝑆))
25	1, 24	pwsbas 17090	. . . . . . 7 ⊢ (((ringLMod‘𝑆) ∈ V ∧ 𝐼 ∈ 𝑉) → ((Base‘𝑆) ↑_m 𝐼) = (Base‘((ringLMod‘𝑆) ↑_s 𝐼)))
26	7, 8, 25	syl2anc 587	. . . . . 6 ⊢ (𝜑 → ((Base‘𝑆) ↑_m 𝐼) = (Base‘((ringLMod‘𝑆) ↑_s 𝐼)))
27	23, 26	eleqtrd 2842	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (Base‘((ringLMod‘𝑆) ↑_s 𝐼)))
28	1, 2, 3, 4, 5, 6, 7, 8, 20, 27	pwsvscafval 17097	. . . 4 ⊢ (𝜑 → (𝐿 ∙ 𝐴) = ((𝐼 × {𝐿}) ∘_f (._r‘𝑆)𝐴))
29		mhphf2.m	. . . . . . 7 ⊢ · = (._r‘𝑆)
30	29	eqcomi 2748	. . . . . 6 ⊢ (._r‘𝑆) = ·
31		ofeq 7511	. . . . . 6 ⊢ ((._r‘𝑆) = · → ∘_f (._r‘𝑆) = ∘_f · )
32	30, 31	mp1i 13	. . . . 5 ⊢ (𝜑 → ∘_f (._r‘𝑆) = ∘_f · )
33	32	oveqd 7269	. . . 4 ⊢ (𝜑 → ((𝐼 × {𝐿}) ∘_f (._r‘𝑆)𝐴) = ((𝐼 × {𝐿}) ∘_f · 𝐴))
34	28, 33	eqtrd 2779	. . 3 ⊢ (𝜑 → (𝐿 ∙ 𝐴) = ((𝐼 × {𝐿}) ∘_f · 𝐴))
35	34	fveq2d 6757	. 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 40180	. 2 ⊢ (𝜑 → ((𝑄‘𝑋)‘((𝐼 × {𝐿}) ∘_f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑋)‘𝐴)))
43	35, 42	eqtrd 2779	1 ⊢ (𝜑 → ((𝑄‘𝑋)‘(𝐿 ∙ 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑋)‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 Vcvv 3423 ⊆ wss 3884 {csn 4558 × cxp 5577 ‘cfv 6415 (class class class)co 7252 ∘_f cof 7506 ↑_m cmap 8550 ℕ₀cn0 12138 Basecbs 16815 ↾_s cress 16842 ._rcmulr 16864 Scalarcsca 16866 ·_𝑠 cvsca 16867 ↑_s cpws 17049 ._gcmg 18590 mulGrpcmgp 19610 CRingccrg 19674 SubRingcsubrg 19910 ringLModcrglmod 20321 evalSub ces 21165 mHomP cmhp 21204

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-rep 5203 ax-sep 5216 ax-nul 5223 ax-pow 5282 ax-pr 5346 ax-un 7563 ax-cnex 10833 ax-resscn 10834 ax-1cn 10835 ax-icn 10836 ax-addcl 10837 ax-addrcl 10838 ax-mulcl 10839 ax-mulrcl 10840 ax-mulcom 10841 ax-addass 10842 ax-mulass 10843 ax-distr 10844 ax-i2m1 10845 ax-1ne0 10846 ax-1rid 10847 ax-rnegex 10848 ax-rrecex 10849 ax-cnre 10850 ax-pre-lttri 10851 ax-pre-lttrn 10852 ax-pre-ltadd 10853 ax-pre-mulgt0 10854 ax-addf 10856 ax-mulf 10857

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3425 df-sbc 3713 df-csb 3830 df-dif 3887 df-un 3889 df-in 3891 df-ss 3901 df-pss 3903 df-nul 4255 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5153 df-tr 5186 df-id 5479 df-eprel 5485 df-po 5493 df-so 5494 df-fr 5534 df-se 5535 df-we 5536 df-xp 5585 df-rel 5586 df-cnv 5587 df-co 5588 df-dm 5589 df-rn 5590 df-res 5591 df-ima 5592 df-pred 6189 df-ord 6251 df-on 6252 df-lim 6253 df-suc 6254 df-iota 6373 df-fun 6417 df-fn 6418 df-f 6419 df-f1 6420 df-fo 6421 df-f1o 6422 df-fv 6423 df-isom 6424 df-riota 7209 df-ov 7255 df-oprab 7256 df-mpo 7257 df-of 7508 df-ofr 7509 df-om 7685 df-1st 7801 df-2nd 7802 df-supp 7946 df-wrecs 8089 df-recs 8150 df-rdg 8188 df-1o 8244 df-oadd 8248 df-er 8433 df-map 8552 df-pm 8553 df-ixp 8621 df-en 8669 df-dom 8670 df-sdom 8671 df-fin 8672 df-fsupp 9034 df-sup 9106 df-oi 9174 df-dju 9565 df-card 9603 df-pnf 10917 df-mnf 10918 df-xr 10919 df-ltxr 10920 df-le 10921 df-sub 11112 df-neg 11113 df-nn 11879 df-2 11941 df-3 11942 df-4 11943 df-5 11944 df-6 11945 df-7 11946 df-8 11947 df-9 11948 df-n0 12139 df-z 12225 df-dec 12342 df-uz 12487 df-fz 13144 df-fzo 13287 df-seq 13625 df-hash 13948 df-struct 16751 df-sets 16768 df-slot 16786 df-ndx 16798 df-base 16816 df-ress 16843 df-plusg 16876 df-mulr 16877 df-starv 16878 df-sca 16879 df-vsca 16880 df-ip 16881 df-tset 16882 df-ple 16883 df-ds 16885 df-unif 16886 df-hom 16887 df-cco 16888 df-0g 17044 df-gsum 17045 df-prds 17050 df-pws 17052 df-mre 17187 df-mrc 17188 df-acs 17190 df-mgm 18216 df-sgrp 18265 df-mnd 18276 df-mhm 18320 df-submnd 18321 df-grp 18470 df-minusg 18471 df-sbg 18472 df-mulg 18591 df-subg 18642 df-ghm 18722 df-cntz 18813 df-cmn 19278 df-abl 19279 df-mgp 19611 df-ur 19628 df-srg 19632 df-ring 19675 df-cring 19676 df-rnghom 19849 df-subrg 19912 df-lmod 20015 df-lss 20084 df-lsp 20124 df-sra 20324 df-rgmod 20325 df-cnfld 20486 df-assa 20945 df-asp 20946 df-ascl 20947 df-psr 20997 df-mvr 20998 df-mpl 20999 df-evls 21167 df-mhp 21208

This theorem is referenced by: (None)

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