evlsgsumadd - Metamath Proof Explorer

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

Theorem evlsgsumadd 22405

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 20826	. . . . . 6 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑈 ∈ Ring)
8	5, 7	syl 18	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ Ring)
9	3, 4, 8	mplringd 22330	. . . 4 ⊢ (𝜑 → 𝑊 ∈ Ring)
10		ringcmn 20511	. . . 4 ⊢ (𝑊 ∈ Ring → 𝑊 ∈ CMnd)
11	9, 10	syl 18	. . 3 ⊢ (𝜑 → 𝑊 ∈ CMnd)
12		evlsgsumadd.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ CRing)
13		crngring 20472	. . . . . 6 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring)
14	12, 13	syl 18	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ Ring)
15		ovex 7453	. . . . 5 ⊢ (𝐾 ↑_m 𝐼) ∈ V
16	14, 15	jctir 530	. . . 4 ⊢ (𝜑 → (𝑆 ∈ Ring ∧ (𝐾 ↑_m 𝐼) ∈ V))
17		evlsgsumadd.p	. . . . 5 ⊢ 𝑃 = (𝑆 ↑_s (𝐾 ↑_m 𝐼))
18	17	pwsring 20553	. . . 4 ⊢ ((𝑆 ∈ Ring ∧ (𝐾 ↑_m 𝐼) ∈ V) → 𝑃 ∈ Ring)
19		ringmnd 20470	. . . 4 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ Mnd)
20	16, 18, 19	3syl 19	. . 3 ⊢ (𝜑 → 𝑃 ∈ Mnd)
21		nn0ex 12612	. . . . 5 ⊢ ℕ₀ ∈ V
22	21	a1i 11	. . . 4 ⊢ (𝜑 → ℕ₀ ∈ V)
23		evlsgsumadd.n	. . . 4 ⊢ (𝜑 → 𝑁 ⊆ ℕ₀)
24	22, 23	ssexd 5286	. . 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 22397	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑊 RingHom 𝑃))
28	4, 12, 5, 27	syl3anc 1398	. . . 4 ⊢ (𝜑 → 𝑄 ∈ (𝑊 RingHom 𝑃))
29		rhmghm 20714	. . . 4 ⊢ (𝑄 ∈ (𝑊 RingHom 𝑃) → 𝑄 ∈ (𝑊 GrpHom 𝑃))
30		ghmmhm 19440	. . . 4 ⊢ (𝑄 ∈ (𝑊 GrpHom 𝑃) → 𝑄 ∈ (𝑊 MndHom 𝑃))
31	28, 29, 30	3syl 19	. . 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 20154	. 2 ⊢ (𝜑 → (𝑃 Σ_g (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌))) = (𝑄‘(𝑊 Σ_g (𝑥 ∈ 𝑁 ↦ 𝑌))))
35	34	eqcomd 2767	1 ⊢ (𝜑 → (𝑄‘(𝑊 Σ_g (𝑥 ∈ 𝑁 ↦ 𝑌))) = (𝑃 Σ_g (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 Vcvv 3451 ⊆ wss 3899 class class class wbr 5103 ↦ cmpt 5186 ‘cfv 6538 (class class class)co 7420 ↑_m cmap 8847 finSupp cfsupp 9353 ℕ₀cn0 12606 Basecbs 17387 ↾_s cress 17408 0_gc0g 17610 Σ_g cgsu 17611 ↑_s cpws 17617 Mndcmnd 18923 MndHom cmhm 18976 GrpHom cghm 19427 CMndccmn 19994 Ringcrg 20459 CRingccrg 20460 RingHom crh 20699 SubRingcsubrg 20821 mPoly cmpl 22214 evalSub ces 22381

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-ofr 7694 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8178 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-2o 8477 df-er 8717 df-map 8849 df-pm 8850 df-ixp 8926 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fsupp 9354 df-sup 9434 df-oi 9504 df-card 10020 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-nn 12336 df-2 12405 df-3 12406 df-4 12407 df-5 12408 df-6 12409 df-7 12410 df-8 12411 df-9 12412 df-n0 12607 df-z 12694 df-dec 12815 df-uz 12966 df-fz 13640 df-fzo 13789 df-seq 14145 df-hash 14475 df-struct 17325 df-sets 17342 df-slot 17360 df-ndx 17372 df-base 17388 df-ress 17409 df-plusg 17441 df-mulr 17442 df-sca 17444 df-vsca 17445 df-ip 17446 df-tset 17447 df-ple 17448 df-ds 17450 df-hom 17452 df-cco 17453 df-0g 17612 df-gsum 17613 df-prds 17618 df-pws 17620 df-mre 17756 df-mrc 17757 df-acs 17759 df-mgm 18816 df-sgrp 18908 df-mnd 18924 df-mhm 18978 df-submnd 18979 df-grp 19147 df-minusg 19148 df-sbg 19149 df-mulg 19278 df-subg 19333 df-ghm 19428 df-cntz 19531 df-cmn 19996 df-abl 19997 df-mgp 20361 df-rng 20375 df-ur 20408 df-srg 20413 df-ring 20461 df-cring 20462 df-rhm 20702 df-subrng 20798 df-subrg 20822 df-lmod 21137 df-lss 21207 df-lsp 21247 df-assa 22161 df-asp 22162 df-ascl 22163 df-psr 22217 df-mvr 22218 df-mpl 22219 df-evls 22383

This theorem is used by: (None)

W3C validator