evlsexpval - Metamath Proof Explorer

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Theorem evlsexpval 22258

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 2761	. . . 4 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
2		evlsaddval.b	. . . 4 ⊢ 𝐵 = (Base‘𝑃)
3	1, 2	mgpbas 20220	. . 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 2761	. . . . . 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 22218	. . . . 5 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑃 RingHom (𝑆 ↑_s (𝐾 ↑_m 𝐼))))
14	5, 6, 7, 13	syl3anc 1396	. . . 4 ⊢ (𝜑 → 𝑄 ∈ (𝑃 RingHom (𝑆 ↑_s (𝐾 ↑_m 𝐼))))
15		rhmrcl1 20557	. . . 4 ⊢ (𝑄 ∈ (𝑃 RingHom (𝑆 ↑_s (𝐾 ↑_m 𝐼))) → 𝑃 ∈ Ring)
16	1	ringmgp 20320	. . . 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 499	. . 3 ⊢ (𝜑 → 𝑀 ∈ 𝐵)
21	3, 4, 17, 18, 20	mulgnn0cld 19160	. 2 ⊢ (𝜑 → (𝑁 ∙ 𝑀) ∈ 𝐵)
22		eqid 2761	. . . . 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 22228	. . . 4 ⊢ (𝜑 → (𝑄‘(𝑁 ∙ 𝑀)) = (𝑁(._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑄‘𝑀)))
24	23	fveq1d 6883	. . 3 ⊢ (𝜑 → ((𝑄‘(𝑁 ∙ 𝑀))‘𝐴) = ((𝑁(._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑄‘𝑀))‘𝐴))
25		eqid 2761	. . . 4 ⊢ (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) = (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))
26		eqid 2761	. . . 4 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆)
27		eqid 2761	. . . 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 20327	. . . 4 ⊢ (𝜑 → 𝑆 ∈ Ring)
30		ovexd 7445	. . . 4 ⊢ (𝜑 → (𝐾 ↑_m 𝐼) ∈ V)
31	2, 25	rhmf 20565	. . . . . 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 7080	. . . 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 20409	. . 3 ⊢ (𝜑 → ((𝑁(._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑄‘𝑀))‘𝐴) = (𝑁 ↑ ((𝑄‘𝑀)‘𝐴)))
36	19	simprd 500	. . . 4 ⊢ (𝜑 → ((𝑄‘𝑀)‘𝐴) = 𝑉)
37	36	oveq2d 7426	. . 3 ⊢ (𝜑 → (𝑁 ↑ ((𝑄‘𝑀)‘𝐴)) = (𝑁 ↑ 𝑉))
38	24, 35, 37	3eqtrd 2800	. 2 ⊢ (𝜑 → ((𝑄‘(𝑁 ∙ 𝑀))‘𝐴) = (𝑁 ↑ 𝑉))
39	21, 38	jca 520	1 ⊢ (𝜑 → ((𝑁 ∙ 𝑀) ∈ 𝐵 ∧ ((𝑄‘(𝑁 ∙ 𝑀))‘𝐴) = (𝑁 ↑ 𝑉)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1568 ∈ wcel 2141 Vcvv 3453 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 ↑_m cmap 8823 ℕ₀cn0 12503 Basecbs 17268 ↾_s cress 17289 ↑_s cpws 17498 Mndcmnd 18791 ._gcmg 19132 mulGrpcmgp 20215 Ringcrg 20314 CRingccrg 20315 RingHom crh 20550 SubRingcsubrg 20653 mPoly cmpl 22035 evalSub ces 22202

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2095 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 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-iin 4958 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-ofr 7675 df-om 7862 df-1st 7985 df-2nd 7986 df-supp 8156 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8452 df-2o 8453 df-er 8693 df-map 8825 df-pm 8826 df-ixp 8895 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-fsupp 9321 df-sup 9401 df-oi 9471 df-card 9924 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-nn 12233 df-2 12302 df-3 12303 df-4 12304 df-5 12305 df-6 12306 df-7 12307 df-8 12308 df-9 12309 df-n0 12504 df-z 12591 df-dec 12711 df-uz 12862 df-fz 13535 df-fzo 13682 df-seq 14037 df-hash 14366 df-struct 17206 df-sets 17223 df-slot 17241 df-ndx 17253 df-base 17269 df-ress 17290 df-plusg 17322 df-mulr 17323 df-sca 17325 df-vsca 17326 df-ip 17327 df-tset 17328 df-ple 17329 df-ds 17331 df-hom 17333 df-cco 17334 df-0g 17493 df-gsum 17494 df-prds 17499 df-pws 17501 df-mre 17637 df-mrc 17638 df-acs 17640 df-mgm 18697 df-sgrp 18776 df-mnd 18792 df-mhm 18840 df-submnd 18841 df-grp 19002 df-minusg 19003 df-sbg 19004 df-mulg 19133 df-subg 19188 df-ghm 19283 df-cntz 19386 df-cmn 19851 df-abl 19852 df-mgp 20216 df-rng 20230 df-ur 20263 df-srg 20268 df-ring 20316 df-cring 20317 df-rhm 20553 df-subrng 20630 df-subrg 20654 df-lmod 20962 df-lss 21032 df-lsp 21072 df-assa 21982 df-asp 21983 df-ascl 21984 df-psr 22038 df-mvr 22039 df-mpl 22040 df-evls 22204

This theorem is referenced by: (None)

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