evlsgsummul - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > evlsgsummul		Structured version Visualization version GIF version

Theorem evlsgsummul 22043

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 20071	. . 3 ⊢ 𝐵 = (Base‘𝐺)
4		evlsgsummul.1	. . . 4 ⊢ 1 = (1_r‘𝑊)
5	1, 4	ringidval 20109	. . 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 20499	. . . . . 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 21967	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑈 ∈ CRing) → 𝑊 ∈ CRing)
14	6, 11, 13	syl2anc 584	. . . 4 ⊢ (𝜑 → 𝑊 ∈ CRing)
15	1	crngmgp 20167	. . . 4 ⊢ (𝑊 ∈ CRing → 𝐺 ∈ CMnd)
16	14, 15	syl 17	. . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd)
17		crngring 20171	. . . . . 6 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring)
18	7, 17	syl 17	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ Ring)
19		ovex 7388	. . . . 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 20250	. . . 4 ⊢ ((𝑆 ∈ Ring ∧ (𝐾 ↑_m 𝐼) ∈ V) → 𝑃 ∈ Ring)
23		evlsgsummul.h	. . . . 5 ⊢ 𝐻 = (mulGrp‘𝑃)
24	23	ringmgp 20165	. . . 4 ⊢ (𝑃 ∈ Ring → 𝐻 ∈ Mnd)
25	20, 22, 24	3syl 18	. . 3 ⊢ (𝜑 → 𝐻 ∈ Mnd)
26		nn0ex 12398	. . . . 5 ⊢ ℕ₀ ∈ V
27	26	a1i 11	. . . 4 ⊢ (𝜑 → ℕ₀ ∈ V)
28		evlsgsummul.n	. . . 4 ⊢ (𝜑 → 𝑁 ⊆ ℕ₀)
29	27, 28	ssexd 5266	. . 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 22034	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑊 RingHom 𝑃))
33	6, 7, 8, 32	syl3anc 1373	. . . 4 ⊢ (𝜑 → 𝑄 ∈ (𝑊 RingHom 𝑃))
34	1, 23	rhmmhm 20406	. . . 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 19860	. 2 ⊢ (𝜑 → (𝐻 Σ_g (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌))) = (𝑄‘(𝐺 Σ_g (𝑥 ∈ 𝑁 ↦ 𝑌))))
39	38	eqcomd 2739	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 2113 Vcvv 3437 ⊆ wss 3898 class class class wbr 5095 ↦ cmpt 5176 ‘cfv 6489 (class class class)co 7355 ↑_m cmap 8759 finSupp cfsupp 9256 ℕ₀cn0 12392 Basecbs 17127 ↾_s cress 17148 Σ_g cgsu 17351 ↑_s cpws 17357 Mndcmnd 18650 MndHom cmhm 18697 CMndccmn 19700 mulGrpcmgp 20066 1_rcur 20107 Ringcrg 20159 CRingccrg 20160 RingHom crh 20396 SubRingcsubrg 20493 mPoly cmpl 21853 evalSub ces 22018

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-cnex 11073 ax-resscn 11074 ax-1cn 11075 ax-icn 11076 ax-addcl 11077 ax-addrcl 11078 ax-mulcl 11079 ax-mulrcl 11080 ax-mulcom 11081 ax-addass 11082 ax-mulass 11083 ax-distr 11084 ax-i2m1 11085 ax-1ne0 11086 ax-1rid 11087 ax-rnegex 11088 ax-rrecex 11089 ax-cnre 11090 ax-pre-lttri 11091 ax-pre-lttrn 11092 ax-pre-ltadd 11093 ax-pre-mulgt0 11094

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4861 df-int 4900 df-iun 4945 df-iin 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-isom 6498 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-of 7619 df-ofr 7620 df-om 7806 df-1st 7930 df-2nd 7931 df-supp 8100 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-2o 8395 df-er 8631 df-map 8761 df-pm 8762 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9257 df-sup 9337 df-oi 9407 df-card 9843 df-pnf 11159 df-mnf 11160 df-xr 11161 df-ltxr 11162 df-le 11163 df-sub 11357 df-neg 11358 df-nn 12137 df-2 12199 df-3 12200 df-4 12201 df-5 12202 df-6 12203 df-7 12204 df-8 12205 df-9 12206 df-n0 12393 df-z 12480 df-dec 12599 df-uz 12743 df-fz 13415 df-fzo 13562 df-seq 13916 df-hash 14245 df-struct 17065 df-sets 17082 df-slot 17100 df-ndx 17112 df-base 17128 df-ress 17149 df-plusg 17181 df-mulr 17182 df-sca 17184 df-vsca 17185 df-ip 17186 df-tset 17187 df-ple 17188 df-ds 17190 df-hom 17192 df-cco 17193 df-0g 17352 df-gsum 17353 df-prds 17358 df-pws 17360 df-mre 17496 df-mrc 17497 df-acs 17499 df-mgm 18556 df-sgrp 18635 df-mnd 18651 df-mhm 18699 df-submnd 18700 df-grp 18857 df-minusg 18858 df-sbg 18859 df-mulg 18989 df-subg 19044 df-ghm 19133 df-cntz 19237 df-cmn 19702 df-abl 19703 df-mgp 20067 df-rng 20079 df-ur 20108 df-srg 20113 df-ring 20161 df-cring 20162 df-rhm 20399 df-subrng 20470 df-subrg 20494 df-lmod 20804 df-lss 20874 df-lsp 20914 df-assa 21799 df-asp 21800 df-ascl 21801 df-psr 21856 df-mvr 21857 df-mpl 21858 df-evls 22020

This theorem is referenced by: (None)

W3C validator