evlsexpval - Metamath Proof Explorer

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Theorem evlsexpval 22345

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 2762	. . . 4 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
2		evlsaddval.b	. . . 4 ⊢ 𝐵 = (Base‘𝑃)
3	1, 2	mgpbas 20279	. . 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 2762	. . . . . 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 22305	. . . . 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 20619	. . . 4 ⊢ (𝑄 ∈ (𝑃 RingHom (𝑆 ↑_s (𝐾 ↑_m 𝐼))) → 𝑃 ∈ Ring)
16	1	ringmgp 20379	. . . 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 19219	. 2 ⊢ (𝜑 → (𝑁 ∙ 𝑀) ∈ 𝐵)
22		eqid 2762	. . . . 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 22315	. . . 4 ⊢ (𝜑 → (𝑄‘(𝑁 ∙ 𝑀)) = (𝑁(._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑄‘𝑀)))
24	23	fveq1d 6884	. . 3 ⊢ (𝜑 → ((𝑄‘(𝑁 ∙ 𝑀))‘𝐴) = ((𝑁(._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑄‘𝑀))‘𝐴))
25		eqid 2762	. . . 4 ⊢ (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) = (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))
26		eqid 2762	. . . 4 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆)
27		eqid 2762	. . . 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 20386	. . . 4 ⊢ (𝜑 → 𝑆 ∈ Ring)
30		ovexd 7451	. . . 4 ⊢ (𝜑 → (𝐾 ↑_m 𝐼) ∈ V)
31	2, 25	rhmf 20627	. . . . . 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 7081	. . . 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 20470	. . 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 7432	. . 3 ⊢ (𝜑 → (𝑁 ↑ ((𝑄‘𝑀)‘𝐴)) = (𝑁 ↑ 𝑉))
38	24, 35, 37	3eqtrd 2801	. 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 3453 ⟶wf 6533 ‘cfv 6537 (class class class)co 7416 ↑_m cmap 8829 ℕ₀cn0 12531 Basecbs 17305 ↾_s cress 17326 ↑_s cpws 17535 Mndcmnd 18838 ._gcmg 19191 mulGrpcmgp 20274 Ringcrg 20373 CRingccrg 20374 RingHom crh 20611 SubRingcsubrg 20732 mPoly cmpl 22122 evalSub ces 22289

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 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204

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 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7681 df-ofr 7682 df-om 7866 df-1st 7989 df-2nd 7990 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8458 df-2o 8459 df-er 8699 df-map 8831 df-pm 8832 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9335 df-sup 9415 df-oi 9485 df-card 9947 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-nn 12261 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12532 df-z 12619 df-dec 12740 df-uz 12891 df-fz 13564 df-fzo 13712 df-seq 14068 df-hash 14397 df-struct 17243 df-sets 17260 df-slot 17278 df-ndx 17290 df-base 17306 df-ress 17327 df-plusg 17359 df-mulr 17360 df-sca 17362 df-vsca 17363 df-ip 17364 df-tset 17365 df-ple 17366 df-ds 17368 df-hom 17370 df-cco 17371 df-0g 17530 df-gsum 17531 df-prds 17536 df-pws 17538 df-mre 17674 df-mrc 17675 df-acs 17677 df-mgm 18734 df-sgrp 18823 df-mnd 18839 df-mhm 18892 df-submnd 18893 df-grp 19061 df-minusg 19062 df-sbg 19063 df-mulg 19192 df-subg 19247 df-ghm 19342 df-cntz 19445 df-cmn 19910 df-abl 19911 df-mgp 20275 df-rng 20289 df-ur 20322 df-srg 20327 df-ring 20375 df-cring 20376 df-rhm 20614 df-subrng 20709 df-subrg 20733 df-lmod 21047 df-lss 21117 df-lsp 21157 df-assa 22069 df-asp 22070 df-ascl 22071 df-psr 22125 df-mvr 22126 df-mpl 22127 df-evls 22291

This theorem is used by: (None)

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