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Theorem evlsgsummul 21283

Description: Polynomial evaluation maps (multiplicative) group sums to group sums. (Contributed by SN, 13-Feb-2024.)

**Hypotheses**
Ref	Expression
evlsgsummul.q	⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)
evlsgsummul.w	⊢ 𝑊 = (𝐼 mPoly 𝑈)
evlsgsummul.g	⊢ 𝐺 = (mulGrp‘𝑊)
evlsgsummul.1	⊢ 1 = (1_r‘𝑊)
evlsgsummul.u	⊢ 𝑈 = (𝑆 ↾_s 𝑅)
evlsgsummul.p	⊢ 𝑃 = (𝑆 ↑_s (𝐾 ↑_m 𝐼))
evlsgsummul.h	⊢ 𝐻 = (mulGrp‘𝑃)
evlsgsummul.k	⊢ 𝐾 = (Base‘𝑆)
evlsgsummul.b	⊢ 𝐵 = (Base‘𝑊)
evlsgsummul.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
evlsgsummul.s	⊢ (𝜑 → 𝑆 ∈ CRing)
evlsgsummul.r	⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))
evlsgsummul.y	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝑌 ∈ 𝐵)
evlsgsummul.n	⊢ (𝜑 → 𝑁 ⊆ ℕ₀)
evlsgsummul.f	⊢ (𝜑 → (𝑥 ∈ 𝑁 ↦ 𝑌) finSupp 1 )

**Assertion**
Ref	Expression
evlsgsummul	⊢ (𝜑 → (𝑄‘(𝐺 Σ_g (𝑥 ∈ 𝑁 ↦ 𝑌))) = (𝐻 Σ_g (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌))))

Distinct variable groups: 𝑥,𝐵 𝑥,𝑁 𝑥,𝑄 𝜑,𝑥 Allowed substitution hints: 𝑃(𝑥) 𝑅(𝑥) 𝑆(𝑥) 𝑈(𝑥) 1 (𝑥) 𝐺(𝑥) 𝐻(𝑥) 𝐼(𝑥) 𝐾(𝑥) 𝑉(𝑥) 𝑊(𝑥) 𝑌(𝑥)

**Proof of Theorem evlsgsummul**
Step	Hyp	Ref	Expression
1		evlsgsummul.g	. . . 4 ⊢ 𝐺 = (mulGrp‘𝑊)
2		evlsgsummul.b	. . . 4 ⊢ 𝐵 = (Base‘𝑊)
3	1, 2	mgpbas 19707	. . 3 ⊢ 𝐵 = (Base‘𝐺)
4		evlsgsummul.1	. . . 4 ⊢ 1 = (1_r‘𝑊)
5	1, 4	ringidval 19720	. . 3 ⊢ 1 = (0_g‘𝐺)
6		evlsgsummul.i	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
7		evlsgsummul.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ CRing)
8		evlsgsummul.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))
9		evlsgsummul.u	. . . . . . 7 ⊢ 𝑈 = (𝑆 ↾_s 𝑅)
10	9	subrgcrng 20009	. . . . . 6 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑈 ∈ CRing)
11	7, 8, 10	syl2anc 583	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ CRing)
12		evlsgsummul.w	. . . . . 6 ⊢ 𝑊 = (𝐼 mPoly 𝑈)
13	12	mplcrng 21207	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑈 ∈ CRing) → 𝑊 ∈ CRing)
14	6, 11, 13	syl2anc 583	. . . 4 ⊢ (𝜑 → 𝑊 ∈ CRing)
15	1	crngmgp 19772	. . . 4 ⊢ (𝑊 ∈ CRing → 𝐺 ∈ CMnd)
16	14, 15	syl 17	. . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd)
17		crngring 19776	. . . . . 6 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring)
18	7, 17	syl 17	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ Ring)
19		ovex 7301	. . . . 5 ⊢ (𝐾 ↑_m 𝐼) ∈ V
20	18, 19	jctir 520	. . . 4 ⊢ (𝜑 → (𝑆 ∈ Ring ∧ (𝐾 ↑_m 𝐼) ∈ V))
21		evlsgsummul.p	. . . . 5 ⊢ 𝑃 = (𝑆 ↑_s (𝐾 ↑_m 𝐼))
22	21	pwsring 19835	. . . 4 ⊢ ((𝑆 ∈ Ring ∧ (𝐾 ↑_m 𝐼) ∈ V) → 𝑃 ∈ Ring)
23		evlsgsummul.h	. . . . 5 ⊢ 𝐻 = (mulGrp‘𝑃)
24	23	ringmgp 19770	. . . 4 ⊢ (𝑃 ∈ Ring → 𝐻 ∈ Mnd)
25	20, 22, 24	3syl 18	. . 3 ⊢ (𝜑 → 𝐻 ∈ Mnd)
26		nn0ex 12222	. . . . 5 ⊢ ℕ₀ ∈ V
27	26	a1i 11	. . . 4 ⊢ (𝜑 → ℕ₀ ∈ V)
28		evlsgsummul.n	. . . 4 ⊢ (𝜑 → 𝑁 ⊆ ℕ₀)
29	27, 28	ssexd 5251	. . 3 ⊢ (𝜑 → 𝑁 ∈ V)
30		evlsgsummul.q	. . . . . 6 ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)
31		evlsgsummul.k	. . . . . 6 ⊢ 𝐾 = (Base‘𝑆)
32	30, 12, 9, 21, 31	evlsrhm 21279	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑊 RingHom 𝑃))
33	6, 7, 8, 32	syl3anc 1369	. . . 4 ⊢ (𝜑 → 𝑄 ∈ (𝑊 RingHom 𝑃))
34	1, 23	rhmmhm 19947	. . . 4 ⊢ (𝑄 ∈ (𝑊 RingHom 𝑃) → 𝑄 ∈ (𝐺 MndHom 𝐻))
35	33, 34	syl 17	. . 3 ⊢ (𝜑 → 𝑄 ∈ (𝐺 MndHom 𝐻))
36		evlsgsummul.y	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝑌 ∈ 𝐵)
37		evlsgsummul.f	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑁 ↦ 𝑌) finSupp 1 )
38	3, 5, 16, 25, 29, 35, 36, 37	gsummptmhm 19522	. 2 ⊢ (𝜑 → (𝐻 Σ_g (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌))) = (𝑄‘(𝐺 Σ_g (𝑥 ∈ 𝑁 ↦ 𝑌))))
39	38	eqcomd 2745	1 ⊢ (𝜑 → (𝑄‘(𝐺 Σ_g (𝑥 ∈ 𝑁 ↦ 𝑌))) = (𝐻 Σ_g (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2109 Vcvv 3430 ⊆ wss 3891 class class class wbr 5078 ↦ cmpt 5161 ‘cfv 6430 (class class class)co 7268 ↑_m cmap 8589 finSupp cfsupp 9089 ℕ₀cn0 12216 Basecbs 16893 ↾_s cress 16922 Σ_g cgsu 17132 ↑_s cpws 17138 Mndcmnd 18366 MndHom cmhm 18409 CMndccmn 19367 mulGrpcmgp 19701 1_rcur 19718 Ringcrg 19764 CRingccrg 19765 RingHom crh 19937 SubRingcsubrg 20001 mPoly cmpl 21090 evalSub ces 21261

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-int 4885 df-iun 4931 df-iin 4932 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-se 5544 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-isom 6439 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-of 7524 df-ofr 7525 df-om 7701 df-1st 7817 df-2nd 7818 df-supp 7962 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-er 8472 df-map 8591 df-pm 8592 df-ixp 8660 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-fsupp 9090 df-sup 9162 df-oi 9230 df-card 9681 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-nn 11957 df-2 12019 df-3 12020 df-4 12021 df-5 12022 df-6 12023 df-7 12024 df-8 12025 df-9 12026 df-n0 12217 df-z 12303 df-dec 12420 df-uz 12565 df-fz 13222 df-fzo 13365 df-seq 13703 df-hash 14026 df-struct 16829 df-sets 16846 df-slot 16864 df-ndx 16876 df-base 16894 df-ress 16923 df-plusg 16956 df-mulr 16957 df-sca 16959 df-vsca 16960 df-ip 16961 df-tset 16962 df-ple 16963 df-ds 16965 df-hom 16967 df-cco 16968 df-0g 17133 df-gsum 17134 df-prds 17139 df-pws 17141 df-mre 17276 df-mrc 17277 df-acs 17279 df-mgm 18307 df-sgrp 18356 df-mnd 18367 df-mhm 18411 df-submnd 18412 df-grp 18561 df-minusg 18562 df-sbg 18563 df-mulg 18682 df-subg 18733 df-ghm 18813 df-cntz 18904 df-cmn 19369 df-abl 19370 df-mgp 19702 df-ur 19719 df-srg 19723 df-ring 19766 df-cring 19767 df-rnghom 19940 df-subrg 20003 df-lmod 20106 df-lss 20175 df-lsp 20215 df-assa 21041 df-asp 21042 df-ascl 21043 df-psr 21093 df-mvr 21094 df-mpl 21095 df-evls 21263

This theorem is referenced by: (None)

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