evlsgsumadd - Metamath Proof Explorer

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Theorem evlsgsumadd 22102

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

**Hypotheses**
Ref	Expression
evlsgsumadd.q	⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)
evlsgsumadd.w	⊢ 𝑊 = (𝐼 mPoly 𝑈)
evlsgsumadd.0	⊢ 0 = (0_g‘𝑊)
evlsgsumadd.u	⊢ 𝑈 = (𝑆 ↾_s 𝑅)
evlsgsumadd.p	⊢ 𝑃 = (𝑆 ↑_s (𝐾 ↑_m 𝐼))
evlsgsumadd.k	⊢ 𝐾 = (Base‘𝑆)
evlsgsumadd.b	⊢ 𝐵 = (Base‘𝑊)
evlsgsumadd.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
evlsgsumadd.s	⊢ (𝜑 → 𝑆 ∈ CRing)
evlsgsumadd.r	⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))
evlsgsumadd.y	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝑌 ∈ 𝐵)
evlsgsumadd.n	⊢ (𝜑 → 𝑁 ⊆ ℕ₀)
evlsgsumadd.f	⊢ (𝜑 → (𝑥 ∈ 𝑁 ↦ 𝑌) finSupp 0 )

**Assertion**
Ref	Expression
evlsgsumadd	⊢ (𝜑 → (𝑄‘(𝑊 Σ_g (𝑥 ∈ 𝑁 ↦ 𝑌))) = (𝑃 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) 𝐼(𝑥) 𝐾(𝑥) 𝑉(𝑥) 𝑊(𝑥) 𝑌(𝑥) 0 (𝑥)

**Proof of Theorem evlsgsumadd**
Step	Hyp	Ref	Expression
1		evlsgsumadd.b	. . 3 ⊢ 𝐵 = (Base‘𝑊)
2		evlsgsumadd.0	. . 3 ⊢ 0 = (0_g‘𝑊)
3		evlsgsumadd.w	. . . . 5 ⊢ 𝑊 = (𝐼 mPoly 𝑈)
4		evlsgsumadd.i	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
5		evlsgsumadd.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))
6		evlsgsumadd.u	. . . . . . 7 ⊢ 𝑈 = (𝑆 ↾_s 𝑅)
7	6	subrgring 20554	. . . . . 6 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑈 ∈ Ring)
8	5, 7	syl 17	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ Ring)
9	3, 4, 8	mplringd 22028	. . . 4 ⊢ (𝜑 → 𝑊 ∈ Ring)
10		ringcmn 20257	. . . 4 ⊢ (𝑊 ∈ Ring → 𝑊 ∈ CMnd)
11	9, 10	syl 17	. . 3 ⊢ (𝜑 → 𝑊 ∈ CMnd)
12		evlsgsumadd.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ CRing)
13		crngring 20224	. . . . . 6 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring)
14	12, 13	syl 17	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ Ring)
15		ovex 7449	. . . . 5 ⊢ (𝐾 ↑_m 𝐼) ∈ V
16	14, 15	jctir 519	. . . 4 ⊢ (𝜑 → (𝑆 ∈ Ring ∧ (𝐾 ↑_m 𝐼) ∈ V))
17		evlsgsumadd.p	. . . . 5 ⊢ 𝑃 = (𝑆 ↑_s (𝐾 ↑_m 𝐼))
18	17	pwsring 20299	. . . 4 ⊢ ((𝑆 ∈ Ring ∧ (𝐾 ↑_m 𝐼) ∈ V) → 𝑃 ∈ Ring)
19		ringmnd 20222	. . . 4 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ Mnd)
20	16, 18, 19	3syl 18	. . 3 ⊢ (𝜑 → 𝑃 ∈ Mnd)
21		nn0ex 12524	. . . . 5 ⊢ ℕ₀ ∈ V
22	21	a1i 11	. . . 4 ⊢ (𝜑 → ℕ₀ ∈ V)
23		evlsgsumadd.n	. . . 4 ⊢ (𝜑 → 𝑁 ⊆ ℕ₀)
24	22, 23	ssexd 5321	. . 3 ⊢ (𝜑 → 𝑁 ∈ V)
25		evlsgsumadd.q	. . . . . 6 ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)
26		evlsgsumadd.k	. . . . . 6 ⊢ 𝐾 = (Base‘𝑆)
27	25, 3, 6, 17, 26	evlsrhm 22099	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑊 RingHom 𝑃))
28	4, 12, 5, 27	syl3anc 1368	. . . 4 ⊢ (𝜑 → 𝑄 ∈ (𝑊 RingHom 𝑃))
29		rhmghm 20462	. . . 4 ⊢ (𝑄 ∈ (𝑊 RingHom 𝑃) → 𝑄 ∈ (𝑊 GrpHom 𝑃))
30		ghmmhm 19216	. . . 4 ⊢ (𝑄 ∈ (𝑊 GrpHom 𝑃) → 𝑄 ∈ (𝑊 MndHom 𝑃))
31	28, 29, 30	3syl 18	. . 3 ⊢ (𝜑 → 𝑄 ∈ (𝑊 MndHom 𝑃))
32		evlsgsumadd.y	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝑌 ∈ 𝐵)
33		evlsgsumadd.f	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑁 ↦ 𝑌) finSupp 0 )
34	1, 2, 11, 20, 24, 31, 32, 33	gsummptmhm 19934	. 2 ⊢ (𝜑 → (𝑃 Σ_g (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌))) = (𝑄‘(𝑊 Σ_g (𝑥 ∈ 𝑁 ↦ 𝑌))))
35	34	eqcomd 2732	1 ⊢ (𝜑 → (𝑄‘(𝑊 Σ_g (𝑥 ∈ 𝑁 ↦ 𝑌))) = (𝑃 Σ_g (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 Vcvv 3462 ⊆ wss 3946 class class class wbr 5145 ↦ cmpt 5228 ‘cfv 6546 (class class class)co 7416 ↑_m cmap 8847 finSupp cfsupp 9398 ℕ₀cn0 12518 Basecbs 17208 ↾_s cress 17237 0_gc0g 17449 Σ_g cgsu 17450 ↑_s cpws 17456 Mndcmnd 18722 MndHom cmhm 18766 GrpHom cghm 19202 CMndccmn 19774 Ringcrg 20212 CRingccrg 20213 RingHom crh 20447 SubRingcsubrg 20547 mPoly cmpl 21899 evalSub ces 22081

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-iin 4996 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-se 5630 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-isom 6555 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-ofr 7683 df-om 7869 df-1st 7995 df-2nd 7996 df-supp 8167 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-2o 8489 df-er 8726 df-map 8849 df-pm 8850 df-ixp 8919 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-fsupp 9399 df-sup 9478 df-oi 9546 df-card 9975 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-nn 12259 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12519 df-z 12605 df-dec 12724 df-uz 12869 df-fz 13533 df-fzo 13676 df-seq 14016 df-hash 14343 df-struct 17144 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-mulr 17275 df-sca 17277 df-vsca 17278 df-ip 17279 df-tset 17280 df-ple 17281 df-ds 17283 df-hom 17285 df-cco 17286 df-0g 17451 df-gsum 17452 df-prds 17457 df-pws 17459 df-mre 17594 df-mrc 17595 df-acs 17597 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-mhm 18768 df-submnd 18769 df-grp 18926 df-minusg 18927 df-sbg 18928 df-mulg 19058 df-subg 19113 df-ghm 19203 df-cntz 19307 df-cmn 19776 df-abl 19777 df-mgp 20114 df-rng 20132 df-ur 20161 df-srg 20166 df-ring 20214 df-cring 20215 df-rhm 20450 df-subrng 20524 df-subrg 20549 df-lmod 20834 df-lss 20905 df-lsp 20945 df-assa 21847 df-asp 21848 df-ascl 21849 df-psr 21902 df-mvr 21903 df-mpl 21904 df-evls 22083

This theorem is referenced by: (None)

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