evlsexpval - Mathbox for Steven Nguyen

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Theorem evlsexpval 40299

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		evlsaddval.i	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑍)
2		evlsaddval.s	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ CRing)
3		evlsaddval.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))
4		evlsaddval.q	. . . . . 6 ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)
5		evlsaddval.p	. . . . . 6 ⊢ 𝑃 = (𝐼 mPoly 𝑈)
6		evlsaddval.u	. . . . . 6 ⊢ 𝑈 = (𝑆 ↾_s 𝑅)
7		eqid 2733	. . . . . 6 ⊢ (𝑆 ↑_s (𝐾 ↑_m 𝐼)) = (𝑆 ↑_s (𝐾 ↑_m 𝐼))
8		evlsaddval.k	. . . . . 6 ⊢ 𝐾 = (Base‘𝑆)
9	4, 5, 6, 7, 8	evlsrhm 21326	. . . . 5 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑃 RingHom (𝑆 ↑_s (𝐾 ↑_m 𝐼))))
10	1, 2, 3, 9	syl3anc 1369	. . . 4 ⊢ (𝜑 → 𝑄 ∈ (𝑃 RingHom (𝑆 ↑_s (𝐾 ↑_m 𝐼))))
11		rhmrcl1 19991	. . . 4 ⊢ (𝑄 ∈ (𝑃 RingHom (𝑆 ↑_s (𝐾 ↑_m 𝐼))) → 𝑃 ∈ Ring)
12		eqid 2733	. . . . 5 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
13	12	ringmgp 19817	. . . 4 ⊢ (𝑃 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
14	10, 11, 13	3syl 18	. . 3 ⊢ (𝜑 → (mulGrp‘𝑃) ∈ Mnd)
15		evlsexpval.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
16		evlsaddval.m	. . . 4 ⊢ (𝜑 → (𝑀 ∈ 𝐵 ∧ ((𝑄‘𝑀)‘𝐴) = 𝑉))
17	16	simpld 494	. . 3 ⊢ (𝜑 → 𝑀 ∈ 𝐵)
18		evlsaddval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑃)
19	12, 18	mgpbas 19754	. . . 4 ⊢ 𝐵 = (Base‘(mulGrp‘𝑃))
20		evlsexpval.g	. . . 4 ⊢ ∙ = (._g‘(mulGrp‘𝑃))
21	19, 20	mulgnn0cl 18748	. . 3 ⊢ (((mulGrp‘𝑃) ∈ Mnd ∧ 𝑁 ∈ ℕ₀ ∧ 𝑀 ∈ 𝐵) → (𝑁 ∙ 𝑀) ∈ 𝐵)
22	14, 15, 17, 21	syl3anc 1369	. 2 ⊢ (𝜑 → (𝑁 ∙ 𝑀) ∈ 𝐵)
23		eqid 2733	. . . . 5 ⊢ (mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) = (mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))
24	4, 5, 12, 20, 6, 7, 23, 8, 18, 1, 2, 3, 15, 17	evlspw 21331	. . . 4 ⊢ (𝜑 → (𝑄‘(𝑁 ∙ 𝑀)) = (𝑁(._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑄‘𝑀)))
25	24	fveq1d 6794	. . 3 ⊢ (𝜑 → ((𝑄‘(𝑁 ∙ 𝑀))‘𝐴) = ((𝑁(._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑄‘𝑀))‘𝐴))
26		eqid 2733	. . . 4 ⊢ (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) = (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))
27		eqid 2733	. . . 4 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆)
28		eqid 2733	. . . 4 ⊢ (._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))) = (._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
29		evlsexpval.f	. . . 4 ⊢ ↑ = (._g‘(mulGrp‘𝑆))
30	2	crngringd 19824	. . . 4 ⊢ (𝜑 → 𝑆 ∈ Ring)
31		ovexd 7330	. . . 4 ⊢ (𝜑 → (𝐾 ↑_m 𝐼) ∈ V)
32	18, 26	rhmf 19998	. . . . . 6 ⊢ (𝑄 ∈ (𝑃 RingHom (𝑆 ↑_s (𝐾 ↑_m 𝐼))) → 𝑄:𝐵⟶(Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
33	10, 32	syl 17	. . . . 5 ⊢ (𝜑 → 𝑄:𝐵⟶(Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
34	33, 17	ffvelcdmd 6982	. . . 4 ⊢ (𝜑 → (𝑄‘𝑀) ∈ (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
35		evlsaddval.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑_m 𝐼))
36	7, 26, 23, 27, 28, 29, 30, 31, 15, 34, 35	pwsexpg 40291	. . 3 ⊢ (𝜑 → ((𝑁(._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑄‘𝑀))‘𝐴) = (𝑁 ↑ ((𝑄‘𝑀)‘𝐴)))
37	16	simprd 495	. . . 4 ⊢ (𝜑 → ((𝑄‘𝑀)‘𝐴) = 𝑉)
38	37	oveq2d 7311	. . 3 ⊢ (𝜑 → (𝑁 ↑ ((𝑄‘𝑀)‘𝐴)) = (𝑁 ↑ 𝑉))
39	25, 36, 38	3eqtrd 2777	. 2 ⊢ (𝜑 → ((𝑄‘(𝑁 ∙ 𝑀))‘𝐴) = (𝑁 ↑ 𝑉))
40	22, 39	jca 511	1 ⊢ (𝜑 → ((𝑁 ∙ 𝑀) ∈ 𝐵 ∧ ((𝑄‘(𝑁 ∙ 𝑀))‘𝐴) = (𝑁 ↑ 𝑉)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2101 Vcvv 3434 ⟶wf 6443 ‘cfv 6447 (class class class)co 7295 ↑_m cmap 8635 ℕ₀cn0 12261 Basecbs 16940 ↾_s cress 16969 ↑_s cpws 17185 Mndcmnd 18413 ._gcmg 18728 mulGrpcmgp 19748 Ringcrg 19811 CRingccrg 19812 RingHom crh 19984 SubRingcsubrg 20048 mPoly cmpl 21137 evalSub ces 21308

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-rep 5212 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7608 ax-cnex 10955 ax-resscn 10956 ax-1cn 10957 ax-icn 10958 ax-addcl 10959 ax-addrcl 10960 ax-mulcl 10961 ax-mulrcl 10962 ax-mulcom 10963 ax-addass 10964 ax-mulass 10965 ax-distr 10966 ax-i2m1 10967 ax-1ne0 10968 ax-1rid 10969 ax-rnegex 10970 ax-rrecex 10971 ax-cnre 10972 ax-pre-lttri 10973 ax-pre-lttrn 10974 ax-pre-ltadd 10975 ax-pre-mulgt0 10976

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3222 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3908 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-uni 4842 df-int 4883 df-iun 4929 df-iin 4930 df-br 5078 df-opab 5140 df-mpt 5161 df-tr 5195 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-se 5547 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-isom 6456 df-riota 7252 df-ov 7298 df-oprab 7299 df-mpo 7300 df-of 7553 df-ofr 7554 df-om 7733 df-1st 7851 df-2nd 7852 df-supp 7998 df-frecs 8117 df-wrecs 8148 df-recs 8222 df-rdg 8261 df-1o 8317 df-er 8518 df-map 8637 df-pm 8638 df-ixp 8706 df-en 8754 df-dom 8755 df-sdom 8756 df-fin 8757 df-fsupp 9157 df-sup 9229 df-oi 9297 df-card 9725 df-pnf 11039 df-mnf 11040 df-xr 11041 df-ltxr 11042 df-le 11043 df-sub 11235 df-neg 11236 df-nn 12002 df-2 12064 df-3 12065 df-4 12066 df-5 12067 df-6 12068 df-7 12069 df-8 12070 df-9 12071 df-n0 12262 df-z 12348 df-dec 12466 df-uz 12611 df-fz 13268 df-fzo 13411 df-seq 13750 df-hash 14073 df-struct 16876 df-sets 16893 df-slot 16911 df-ndx 16923 df-base 16941 df-ress 16970 df-plusg 17003 df-mulr 17004 df-sca 17006 df-vsca 17007 df-ip 17008 df-tset 17009 df-ple 17010 df-ds 17012 df-hom 17014 df-cco 17015 df-0g 17180 df-gsum 17181 df-prds 17186 df-pws 17188 df-mre 17323 df-mrc 17324 df-acs 17326 df-mgm 18354 df-sgrp 18403 df-mnd 18414 df-mhm 18458 df-submnd 18459 df-grp 18608 df-minusg 18609 df-sbg 18610 df-mulg 18729 df-subg 18780 df-ghm 18860 df-cntz 18951 df-cmn 19416 df-abl 19417 df-mgp 19749 df-ur 19766 df-srg 19770 df-ring 19813 df-cring 19814 df-rnghom 19987 df-subrg 20050 df-lmod 20153 df-lss 20222 df-lsp 20262 df-assa 21088 df-asp 21089 df-ascl 21090 df-psr 21140 df-mvr 21141 df-mpl 21142 df-evls 21310

This theorem is referenced by: (None)

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