evlsexpval - Metamath Proof Explorer

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Theorem evlsexpval 22384

Description: Polynomial evaluation builder for exponentiation. (Contributed by SN, 27-Jul-2024.)

**Hypotheses**
Ref	Expression
evlsaddval.q	⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)
evlsaddval.p	⊢ 𝑃 = (𝐼 mPoly 𝑈)
evlsaddval.u	⊢ 𝑈 = (𝑆 ↾_s 𝑅)
evlsaddval.k	⊢ 𝐾 = (Base‘𝑆)
evlsaddval.b	⊢ 𝐵 = (Base‘𝑃)
evlsaddval.i	⊢ (𝜑 → 𝐼 ∈ 𝑍)
evlsaddval.s	⊢ (𝜑 → 𝑆 ∈ CRing)
evlsaddval.r	⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))
evlsaddval.a	⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑_m 𝐼))
evlsaddval.m	⊢ (𝜑 → (𝑀 ∈ 𝐵 ∧ ((𝑄‘𝑀)‘𝐴) = 𝑉))
evlsexpval.g	⊢ ∙ = (._g‘(mulGrp‘𝑃))
evlsexpval.f	⊢ ↑ = (._g‘(mulGrp‘𝑆))
evlsexpval.n	⊢ (𝜑 → 𝑁 ∈ ℕ₀)

**Assertion**
Ref	Expression
evlsexpval	⊢ (𝜑 → ((𝑁 ∙ 𝑀) ∈ 𝐵 ∧ ((𝑄‘(𝑁 ∙ 𝑀))‘𝐴) = (𝑁 ↑ 𝑉)))

**Proof of Theorem evlsexpval**
Step	Hyp	Ref	Expression
1		eqid 2760	. . . 4 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
2		evlsaddval.b	. . . 4 ⊢ 𝐵 = (Base‘𝑃)
3	1, 2	mgpbas 20312	. . 3 ⊢ 𝐵 = (Base‘(mulGrp‘𝑃))
4		evlsexpval.g	. . 3 ⊢ ∙ = (._g‘(mulGrp‘𝑃))
5		evlsaddval.i	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑍)
6		evlsaddval.s	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ CRing)
7		evlsaddval.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))
8		evlsaddval.q	. . . . . 6 ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)
9		evlsaddval.p	. . . . . 6 ⊢ 𝑃 = (𝐼 mPoly 𝑈)
10		evlsaddval.u	. . . . . 6 ⊢ 𝑈 = (𝑆 ↾_s 𝑅)
11		eqid 2760	. . . . . 6 ⊢ (𝑆 ↑_s (𝐾 ↑_m 𝐼)) = (𝑆 ↑_s (𝐾 ↑_m 𝐼))
12		evlsaddval.k	. . . . . 6 ⊢ 𝐾 = (Base‘𝑆)
13	8, 9, 10, 11, 12	evlsrhm 22344	. . . . 5 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑃 RingHom (𝑆 ↑_s (𝐾 ↑_m 𝐼))))
14	5, 6, 7, 13	syl3anc 1398	. . . 4 ⊢ (𝜑 → 𝑄 ∈ (𝑃 RingHom (𝑆 ↑_s (𝐾 ↑_m 𝐼))))
15		rhmrcl1 20654	. . . 4 ⊢ (𝑄 ∈ (𝑃 RingHom (𝑆 ↑_s (𝐾 ↑_m 𝐼))) → 𝑃 ∈ Ring)
16	1	ringmgp 20412	. . . 4 ⊢ (𝑃 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
17	14, 15, 16	3syl 19	. . 3 ⊢ (𝜑 → (mulGrp‘𝑃) ∈ Mnd)
18		evlsexpval.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
19		evlsaddval.m	. . . 4 ⊢ (𝜑 → (𝑀 ∈ 𝐵 ∧ ((𝑄‘𝑀)‘𝐴) = 𝑉))
20	19	simpld 500	. . 3 ⊢ (𝜑 → 𝑀 ∈ 𝐵)
21	3, 4, 17, 18, 20	mulgnn0cld 19252	. 2 ⊢ (𝜑 → (𝑁 ∙ 𝑀) ∈ 𝐵)
22		eqid 2760	. . . . 5 ⊢ (mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) = (mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))
23	8, 9, 1, 4, 10, 11, 22, 12, 2, 5, 6, 7, 18, 20	evlspw 22354	. . . 4 ⊢ (𝜑 → (𝑄‘(𝑁 ∙ 𝑀)) = (𝑁(._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑄‘𝑀)))
24	23	fveq1d 6876	. . 3 ⊢ (𝜑 → ((𝑄‘(𝑁 ∙ 𝑀))‘𝐴) = ((𝑁(._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑄‘𝑀))‘𝐴))
25		eqid 2760	. . . 4 ⊢ (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) = (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))
26		eqid 2760	. . . 4 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆)
27		eqid 2760	. . . 4 ⊢ (._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))) = (._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
28		evlsexpval.f	. . . 4 ⊢ ↑ = (._g‘(mulGrp‘𝑆))
29	6	crngringd 20420	. . . 4 ⊢ (𝜑 → 𝑆 ∈ Ring)
30		ovexd 7444	. . . 4 ⊢ (𝜑 → (𝐾 ↑_m 𝐼) ∈ V)
31	2, 25	rhmf 20662	. . . . . 6 ⊢ (𝑄 ∈ (𝑃 RingHom (𝑆 ↑_s (𝐾 ↑_m 𝐼))) → 𝑄:𝐵⟶(Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
32	14, 31	syl 18	. . . . 5 ⊢ (𝜑 → 𝑄:𝐵⟶(Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
33	32, 20	ffvelcdmd 7074	. . . 4 ⊢ (𝜑 → (𝑄‘𝑀) ∈ (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
34		evlsaddval.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑_m 𝐼))
35	11, 25, 22, 26, 27, 28, 29, 30, 18, 33, 34	pwsexpg 20505	. . 3 ⊢ (𝜑 → ((𝑁(._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑄‘𝑀))‘𝐴) = (𝑁 ↑ ((𝑄‘𝑀)‘𝐴)))
36	19	simprd 501	. . . 4 ⊢ (𝜑 → ((𝑄‘𝑀)‘𝐴) = 𝑉)
37	36	oveq2d 7425	. . 3 ⊢ (𝜑 → (𝑁 ↑ ((𝑄‘𝑀)‘𝐴)) = (𝑁 ↑ 𝑉))
38	24, 35, 37	3eqtrd 2799	. 2 ⊢ (𝜑 → ((𝑄‘(𝑁 ∙ 𝑀))‘𝐴) = (𝑁 ↑ 𝑉))
39	21, 38	jca 521	1 ⊢ (𝜑 → ((𝑁 ∙ 𝑀) ∈ 𝐵 ∧ ((𝑄‘(𝑁 ∙ 𝑀))‘𝐴) = (𝑁 ↑ 𝑉)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 Vcvv 3450 ⟶wf 6524 ‘cfv 6528 (class class class)co 7409 ↑_m cmap 8826 ℕ₀cn0 12561 Basecbs 17334 ↾_s cress 17355 ↑_s cpws 17564 Mndcmnd 18870 ._gcmg 19224 mulGrpcmgp 20307 Ringcrg 20406 CRingccrg 20407 RingHom crh 20646 SubRingcsubrg 20768 mPoly cmpl 22161 evalSub ces 22328

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 2732 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7735 ax-cnex 11213 ax-resscn 11214 ax-1cn 11215 ax-icn 11216 ax-addcl 11217 ax-addrcl 11218 ax-mulcl 11219 ax-mulrcl 11220 ax-mulcom 11221 ax-addass 11222 ax-mulass 11223 ax-distr 11224 ax-i2m1 11225 ax-1ne0 11226 ax-1rid 11227 ax-rnegex 11228 ax-rrecex 11229 ax-cnre 11230 ax-pre-lttri 11231 ax-pre-lttrn 11232 ax-pre-ltadd 11233 ax-pre-mulgt0 11234

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 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 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 5543 df-eprel 5548 df-po 5556 df-so 5557 df-fr 5601 df-se 5602 df-we 5603 df-xp 5654 df-rel 5655 df-cnv 5656 df-co 5657 df-dm 5658 df-rn 5659 df-res 5660 df-ima 5661 df-pred 6294 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6530 df-fn 6531 df-f 6532 df-f1 6533 df-fo 6534 df-f1o 6535 df-fv 6536 df-isom 6537 df-riota 7366 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7677 df-ofr 7678 df-om 7862 df-1st 7985 df-2nd 7986 df-supp 8157 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8455 df-2o 8456 df-er 8696 df-map 8828 df-pm 8829 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fsupp 9332 df-sup 9412 df-oi 9482 df-card 9977 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-sub 11500 df-neg 11501 df-nn 12291 df-2 12360 df-3 12361 df-4 12362 df-5 12363 df-6 12364 df-7 12365 df-8 12366 df-9 12367 df-n0 12562 df-z 12649 df-dec 12770 df-uz 12921 df-fz 13595 df-fzo 13743 df-seq 14099 df-hash 14428 df-struct 17272 df-sets 17289 df-slot 17307 df-ndx 17319 df-base 17335 df-ress 17356 df-plusg 17388 df-mulr 17389 df-sca 17391 df-vsca 17392 df-ip 17393 df-tset 17394 df-ple 17395 df-ds 17397 df-hom 17399 df-cco 17400 df-0g 17559 df-gsum 17560 df-prds 17565 df-pws 17567 df-mre 17703 df-mrc 17704 df-acs 17706 df-mgm 18763 df-sgrp 18855 df-mnd 18871 df-mhm 18925 df-submnd 18926 df-grp 19094 df-minusg 19095 df-sbg 19096 df-mulg 19225 df-subg 19280 df-ghm 19375 df-cntz 19478 df-cmn 19943 df-abl 19944 df-mgp 20308 df-rng 20322 df-ur 20355 df-srg 20360 df-ring 20408 df-cring 20409 df-rhm 20649 df-subrng 20745 df-subrg 20769 df-lmod 21084 df-lss 21154 df-lsp 21194 df-assa 22108 df-asp 22109 df-ascl 22110 df-psr 22164 df-mvr 22165 df-mpl 22166 df-evls 22330

This theorem is used by: (None)

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