mhphf2 - Mathbox for Steven Nguyen

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Theorem mhphf2 41862

Description: A homogeneous polynomial defines a homogeneous function; this is mhphf 41861 with simpler notation in the conclusion in exchange for a complex definition of ∙, which is based on frlmvscafval 21707 but without the finite support restriction (frlmpws 21691, frlmbas 21696) on the assignments 𝐴 from variables to values.

TODO?: Polynomials (df-mpl 21851) 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 2728	. . . . 5 ⊢ ((ringLMod‘𝑆) ↑_s 𝐼) = ((ringLMod‘𝑆) ↑_s 𝐼)
2		eqid 2728	. . . . 5 ⊢ (Base‘((ringLMod‘𝑆) ↑_s 𝐼)) = (Base‘((ringLMod‘𝑆) ↑_s 𝐼))
3		rlmvsca 21100	. . . . 5 ⊢ (._r‘𝑆) = ( ·_𝑠 ‘(ringLMod‘𝑆))
4		mhphf2.b	. . . . 5 ⊢ ∙ = ( ·_𝑠 ‘((ringLMod‘𝑆) ↑_s 𝐼))
5		eqid 2728	. . . . 5 ⊢ (Scalar‘(ringLMod‘𝑆)) = (Scalar‘(ringLMod‘𝑆))
6		eqid 2728	. . . . 5 ⊢ (Base‘(Scalar‘(ringLMod‘𝑆))) = (Base‘(Scalar‘(ringLMod‘𝑆)))
7		fvexd 6917	. . . . 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 20518	. . . . . . . 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 3983	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ 𝐾)
15		mhphf2.s	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ CRing)
16		rlmsca 21098	. . . . . . . . 9 ⊢ (𝑆 ∈ CRing → 𝑆 = (Scalar‘(ringLMod‘𝑆)))
17	15, 16	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑆 = (Scalar‘(ringLMod‘𝑆)))
18	17	fveq2d 6906	. . . . . . 7 ⊢ (𝜑 → (Base‘𝑆) = (Base‘(Scalar‘(ringLMod‘𝑆))))
19	10, 18	eqtrid 2780	. . . . . 6 ⊢ (𝜑 → 𝐾 = (Base‘(Scalar‘(ringLMod‘𝑆))))
20	14, 19	eleqtrd 2831	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ (Base‘(Scalar‘(ringLMod‘𝑆))))
21		mhphf2.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑_m 𝐼))
22	10	oveq1i 7436	. . . . . . 7 ⊢ (𝐾 ↑_m 𝐼) = ((Base‘𝑆) ↑_m 𝐼)
23	21, 22	eleqtrdi 2839	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ((Base‘𝑆) ↑_m 𝐼))
24		rlmbas 21093	. . . . . . . 8 ⊢ (Base‘𝑆) = (Base‘(ringLMod‘𝑆))
25	1, 24	pwsbas 17476	. . . . . . 7 ⊢ (((ringLMod‘𝑆) ∈ V ∧ 𝐼 ∈ 𝑉) → ((Base‘𝑆) ↑_m 𝐼) = (Base‘((ringLMod‘𝑆) ↑_s 𝐼)))
26	7, 8, 25	syl2anc 582	. . . . . 6 ⊢ (𝜑 → ((Base‘𝑆) ↑_m 𝐼) = (Base‘((ringLMod‘𝑆) ↑_s 𝐼)))
27	23, 26	eleqtrd 2831	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (Base‘((ringLMod‘𝑆) ↑_s 𝐼)))
28	1, 2, 3, 4, 5, 6, 7, 8, 20, 27	pwsvscafval 17483	. . . 4 ⊢ (𝜑 → (𝐿 ∙ 𝐴) = ((𝐼 × {𝐿}) ∘_f (._r‘𝑆)𝐴))
29		mhphf2.m	. . . . . . 7 ⊢ · = (._r‘𝑆)
30	29	eqcomi 2737	. . . . . 6 ⊢ (._r‘𝑆) = ·
31		ofeq 7694	. . . . . 6 ⊢ ((._r‘𝑆) = · → ∘_f (._r‘𝑆) = ∘_f · )
32	30, 31	mp1i 13	. . . . 5 ⊢ (𝜑 → ∘_f (._r‘𝑆) = ∘_f · )
33	32	oveqd 7443	. . . 4 ⊢ (𝜑 → ((𝐼 × {𝐿}) ∘_f (._r‘𝑆)𝐴) = ((𝐼 × {𝐿}) ∘_f · 𝐴))
34	28, 33	eqtrd 2768	. . 3 ⊢ (𝜑 → (𝐿 ∙ 𝐴) = ((𝐼 × {𝐿}) ∘_f · 𝐴))
35	34	fveq2d 6906	. 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 41861	. 2 ⊢ (𝜑 → ((𝑄‘𝑋)‘((𝐼 × {𝐿}) ∘_f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑋)‘𝐴)))
43	35, 42	eqtrd 2768	1 ⊢ (𝜑 → ((𝑄‘𝑋)‘(𝐿 ∙ 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑋)‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3473 ⊆ wss 3949 {csn 4632 × cxp 5680 ‘cfv 6553 (class class class)co 7426 ∘_f cof 7689 ↑_m cmap 8851 ℕ₀cn0 12510 Basecbs 17187 ↾_s cress 17216 ._rcmulr 17241 Scalarcsca 17243 ·_𝑠 cvsca 17244 ↑_s cpws 17435 ._gcmg 19030 mulGrpcmgp 20081 CRingccrg 20181 SubRingcsubrg 20513 ringLModcrglmod 21064 evalSub ces 22023 mHomP cmhp 22062

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-addf 11225

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-ofr 7692 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-pm 8854 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-sup 9473 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-fz 13525 df-fzo 13668 df-seq 14007 df-hash 14330 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-starv 17255 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-hom 17264 df-cco 17265 df-0g 17430 df-gsum 17431 df-prds 17436 df-pws 17438 df-mre 17573 df-mrc 17574 df-acs 17576 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-mhm 18747 df-submnd 18748 df-grp 18900 df-minusg 18901 df-sbg 18902 df-mulg 19031 df-subg 19085 df-ghm 19175 df-cntz 19275 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-srg 20134 df-ring 20182 df-cring 20183 df-rhm 20418 df-subrng 20490 df-subrg 20515 df-lmod 20752 df-lss 20823 df-lsp 20863 df-sra 21065 df-rgmod 21066 df-cnfld 21287 df-assa 21794 df-asp 21795 df-ascl 21796 df-psr 21849 df-mvr 21850 df-mpl 21851 df-evls 22025 df-mhp 22069

This theorem is referenced by: (None)

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