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Theorem evlsgsummul 22060

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 20092	. . 3 ⊢ 𝐵 = (Base‘𝐺)
4		evlsgsummul.1	. . . 4 ⊢ 1 = (1_r‘𝑊)
5	1, 4	ringidval 20135	. . 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 20526	. . . . . 6 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑈 ∈ CRing)
11	7, 8, 10	syl2anc 582	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ CRing)
12		evlsgsummul.w	. . . . . 6 ⊢ 𝑊 = (𝐼 mPoly 𝑈)
13	12	mplcrng 21983	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑈 ∈ CRing) → 𝑊 ∈ CRing)
14	6, 11, 13	syl2anc 582	. . . 4 ⊢ (𝜑 → 𝑊 ∈ CRing)
15	1	crngmgp 20193	. . . 4 ⊢ (𝑊 ∈ CRing → 𝐺 ∈ CMnd)
16	14, 15	syl 17	. . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd)
17		crngring 20197	. . . . . 6 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring)
18	7, 17	syl 17	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ Ring)
19		ovex 7452	. . . . 5 ⊢ (𝐾 ↑_m 𝐼) ∈ V
20	18, 19	jctir 519	. . . 4 ⊢ (𝜑 → (𝑆 ∈ Ring ∧ (𝐾 ↑_m 𝐼) ∈ V))
21		evlsgsummul.p	. . . . 5 ⊢ 𝑃 = (𝑆 ↑_s (𝐾 ↑_m 𝐼))
22	21	pwsring 20272	. . . 4 ⊢ ((𝑆 ∈ Ring ∧ (𝐾 ↑_m 𝐼) ∈ V) → 𝑃 ∈ Ring)
23		evlsgsummul.h	. . . . 5 ⊢ 𝐻 = (mulGrp‘𝑃)
24	23	ringmgp 20191	. . . 4 ⊢ (𝑃 ∈ Ring → 𝐻 ∈ Mnd)
25	20, 22, 24	3syl 18	. . 3 ⊢ (𝜑 → 𝐻 ∈ Mnd)
26		nn0ex 12511	. . . . 5 ⊢ ℕ₀ ∈ V
27	26	a1i 11	. . . 4 ⊢ (𝜑 → ℕ₀ ∈ V)
28		evlsgsummul.n	. . . 4 ⊢ (𝜑 → 𝑁 ⊆ ℕ₀)
29	27, 28	ssexd 5325	. . 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 22056	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑊 RingHom 𝑃))
33	6, 7, 8, 32	syl3anc 1368	. . . 4 ⊢ (𝜑 → 𝑄 ∈ (𝑊 RingHom 𝑃))
34	1, 23	rhmmhm 20430	. . . 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 19907	. 2 ⊢ (𝜑 → (𝐻 Σ_g (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌))) = (𝑄‘(𝐺 Σ_g (𝑥 ∈ 𝑁 ↦ 𝑌))))
39	38	eqcomd 2731	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 394 = wceq 1533 ∈ wcel 2098 Vcvv 3461 ⊆ wss 3944 class class class wbr 5149 ↦ cmpt 5232 ‘cfv 6549 (class class class)co 7419 ↑_m cmap 8845 finSupp cfsupp 9387 ℕ₀cn0 12505 Basecbs 17183 ↾_s cress 17212 Σ_g cgsu 17425 ↑_s cpws 17431 Mndcmnd 18697 MndHom cmhm 18741 CMndccmn 19747 mulGrpcmgp 20086 1_rcur 20133 Ringcrg 20185 CRingccrg 20186 RingHom crh 20420 SubRingcsubrg 20518 mPoly cmpl 21856 evalSub ces 22038

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 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217

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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-ofr 7686 df-om 7872 df-1st 7994 df-2nd 7995 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9388 df-sup 9467 df-oi 9535 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-fz 13520 df-fzo 13663 df-seq 14003 df-hash 14326 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-sca 17252 df-vsca 17253 df-ip 17254 df-tset 17255 df-ple 17256 df-ds 17258 df-hom 17260 df-cco 17261 df-0g 17426 df-gsum 17427 df-prds 17432 df-pws 17434 df-mre 17569 df-mrc 17570 df-acs 17572 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-mhm 18743 df-submnd 18744 df-grp 18901 df-minusg 18902 df-sbg 18903 df-mulg 19032 df-subg 19086 df-ghm 19176 df-cntz 19280 df-cmn 19749 df-abl 19750 df-mgp 20087 df-rng 20105 df-ur 20134 df-srg 20139 df-ring 20187 df-cring 20188 df-rhm 20423 df-subrng 20495 df-subrg 20520 df-lmod 20757 df-lss 20828 df-lsp 20868 df-assa 21804 df-asp 21805 df-ascl 21806 df-psr 21859 df-mvr 21860 df-mpl 21861 df-evls 22040

This theorem is referenced by: (None)

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