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Theorem evlsgsummul 21302

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 19726	. . 3 ⊢ 𝐵 = (Base‘𝐺)
4		evlsgsummul.1	. . . 4 ⊢ 1 = (1_r‘𝑊)
5	1, 4	ringidval 19739	. . 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 20028	. . . . . 6 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑈 ∈ CRing)
11	7, 8, 10	syl2anc 584	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ CRing)
12		evlsgsummul.w	. . . . . 6 ⊢ 𝑊 = (𝐼 mPoly 𝑈)
13	12	mplcrng 21226	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑈 ∈ CRing) → 𝑊 ∈ CRing)
14	6, 11, 13	syl2anc 584	. . . 4 ⊢ (𝜑 → 𝑊 ∈ CRing)
15	1	crngmgp 19791	. . . 4 ⊢ (𝑊 ∈ CRing → 𝐺 ∈ CMnd)
16	14, 15	syl 17	. . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd)
17		crngring 19795	. . . . . 6 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring)
18	7, 17	syl 17	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ Ring)
19		ovex 7308	. . . . 5 ⊢ (𝐾 ↑_m 𝐼) ∈ V
20	18, 19	jctir 521	. . . 4 ⊢ (𝜑 → (𝑆 ∈ Ring ∧ (𝐾 ↑_m 𝐼) ∈ V))
21		evlsgsummul.p	. . . . 5 ⊢ 𝑃 = (𝑆 ↑_s (𝐾 ↑_m 𝐼))
22	21	pwsring 19854	. . . 4 ⊢ ((𝑆 ∈ Ring ∧ (𝐾 ↑_m 𝐼) ∈ V) → 𝑃 ∈ Ring)
23		evlsgsummul.h	. . . . 5 ⊢ 𝐻 = (mulGrp‘𝑃)
24	23	ringmgp 19789	. . . 4 ⊢ (𝑃 ∈ Ring → 𝐻 ∈ Mnd)
25	20, 22, 24	3syl 18	. . 3 ⊢ (𝜑 → 𝐻 ∈ Mnd)
26		nn0ex 12239	. . . . 5 ⊢ ℕ₀ ∈ V
27	26	a1i 11	. . . 4 ⊢ (𝜑 → ℕ₀ ∈ V)
28		evlsgsummul.n	. . . 4 ⊢ (𝜑 → 𝑁 ⊆ ℕ₀)
29	27, 28	ssexd 5248	. . 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 21298	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑊 RingHom 𝑃))
33	6, 7, 8, 32	syl3anc 1370	. . . 4 ⊢ (𝜑 → 𝑄 ∈ (𝑊 RingHom 𝑃))
34	1, 23	rhmmhm 19966	. . . 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 19541	. 2 ⊢ (𝜑 → (𝐻 Σ_g (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌))) = (𝑄‘(𝐺 Σ_g (𝑥 ∈ 𝑁 ↦ 𝑌))))
39	38	eqcomd 2744	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 396 = wceq 1539 ∈ wcel 2106 Vcvv 3432 ⊆ wss 3887 class class class wbr 5074 ↦ cmpt 5157 ‘cfv 6433 (class class class)co 7275 ↑_m cmap 8615 finSupp cfsupp 9128 ℕ₀cn0 12233 Basecbs 16912 ↾_s cress 16941 Σ_g cgsu 17151 ↑_s cpws 17157 Mndcmnd 18385 MndHom cmhm 18428 CMndccmn 19386 mulGrpcmgp 19720 1_rcur 19737 Ringcrg 19783 CRingccrg 19784 RingHom crh 19956 SubRingcsubrg 20020 mPoly cmpl 21109 evalSub ces 21280

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-ofr 7534 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-sup 9201 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-fz 13240 df-fzo 13383 df-seq 13722 df-hash 14045 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-hom 16986 df-cco 16987 df-0g 17152 df-gsum 17153 df-prds 17158 df-pws 17160 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-mhm 18430 df-submnd 18431 df-grp 18580 df-minusg 18581 df-sbg 18582 df-mulg 18701 df-subg 18752 df-ghm 18832 df-cntz 18923 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-srg 19742 df-ring 19785 df-cring 19786 df-rnghom 19959 df-subrg 20022 df-lmod 20125 df-lss 20194 df-lsp 20234 df-assa 21060 df-asp 21061 df-ascl 21062 df-psr 21112 df-mvr 21113 df-mpl 21114 df-evls 21282

This theorem is referenced by: (None)

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