evlsexpval - Mathbox for Steven Nguyen

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Theorem evlsexpval 39875

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 2738	. . . . . 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 20902	. . . . 5 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑃 RingHom (𝑆 ↑_s (𝐾 ↑_m 𝐼))))
10	1, 2, 3, 9	syl3anc 1372	. . . 4 ⊢ (𝜑 → 𝑄 ∈ (𝑃 RingHom (𝑆 ↑_s (𝐾 ↑_m 𝐼))))
11		rhmrcl1 19593	. . . 4 ⊢ (𝑄 ∈ (𝑃 RingHom (𝑆 ↑_s (𝐾 ↑_m 𝐼))) → 𝑃 ∈ Ring)
12		eqid 2738	. . . . 5 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
13	12	ringmgp 19422	. . . 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 498	. . 3 ⊢ (𝜑 → 𝑀 ∈ 𝐵)
18		evlsaddval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑃)
19	12, 18	mgpbas 19364	. . . 4 ⊢ 𝐵 = (Base‘(mulGrp‘𝑃))
20		evlsexpval.g	. . . 4 ⊢ ∙ = (._g‘(mulGrp‘𝑃))
21	19, 20	mulgnn0cl 18362	. . 3 ⊢ (((mulGrp‘𝑃) ∈ Mnd ∧ 𝑁 ∈ ℕ₀ ∧ 𝑀 ∈ 𝐵) → (𝑁 ∙ 𝑀) ∈ 𝐵)
22	14, 15, 17, 21	syl3anc 1372	. 2 ⊢ (𝜑 → (𝑁 ∙ 𝑀) ∈ 𝐵)
23		eqid 2738	. . . . 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 20907	. . . 4 ⊢ (𝜑 → (𝑄‘(𝑁 ∙ 𝑀)) = (𝑁(._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑄‘𝑀)))
25	24	fveq1d 6676	. . 3 ⊢ (𝜑 → ((𝑄‘(𝑁 ∙ 𝑀))‘𝐴) = ((𝑁(._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑄‘𝑀))‘𝐴))
26		eqid 2738	. . . 4 ⊢ (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) = (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))
27		eqid 2738	. . . 4 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆)
28		eqid 2738	. . . 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 19429	. . . 4 ⊢ (𝜑 → 𝑆 ∈ Ring)
31		ovexd 7205	. . . 4 ⊢ (𝜑 → (𝐾 ↑_m 𝐼) ∈ V)
32	18, 26	rhmf 19600	. . . . . 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	ffvelrnd 6862	. . . 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 39869	. . 3 ⊢ (𝜑 → ((𝑁(._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑄‘𝑀))‘𝐴) = (𝑁 ↑ ((𝑄‘𝑀)‘𝐴)))
37	16	simprd 499	. . . 4 ⊢ (𝜑 → ((𝑄‘𝑀)‘𝐴) = 𝑉)
38	37	oveq2d 7186	. . 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 515	1 ⊢ (𝜑 → ((𝑁 ∙ 𝑀) ∈ 𝐵 ∧ ((𝑄‘(𝑁 ∙ 𝑀))‘𝐴) = (𝑁 ↑ 𝑉)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 Vcvv 3398 ⟶wf 6335 ‘cfv 6339 (class class class)co 7170 ↑_m cmap 8437 ℕ₀cn0 11976 Basecbs 16586 ↾_s cress 16587 ↑_s cpws 16823 Mndcmnd 18027 ._gcmg 18342 mulGrpcmgp 19358 Ringcrg 19416 CRingccrg 19417 RingHom crh 19586 SubRingcsubrg 19650 mPoly cmpl 20719 evalSub ces 20884

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-se 5484 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-isom 6348 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-of 7425 df-ofr 7426 df-om 7600 df-1st 7714 df-2nd 7715 df-supp 7857 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-er 8320 df-map 8439 df-pm 8440 df-ixp 8508 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-fsupp 8907 df-sup 8979 df-oi 9047 df-card 9441 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-7 11784 df-8 11785 df-9 11786 df-n0 11977 df-z 12063 df-dec 12180 df-uz 12325 df-fz 12982 df-fzo 13125 df-seq 13461 df-hash 13783 df-struct 16588 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-ress 16594 df-plusg 16681 df-mulr 16682 df-sca 16684 df-vsca 16685 df-ip 16686 df-tset 16687 df-ple 16688 df-ds 16690 df-hom 16692 df-cco 16693 df-0g 16818 df-gsum 16819 df-prds 16824 df-pws 16826 df-mre 16960 df-mrc 16961 df-acs 16963 df-mgm 17968 df-sgrp 18017 df-mnd 18028 df-mhm 18072 df-submnd 18073 df-grp 18222 df-minusg 18223 df-sbg 18224 df-mulg 18343 df-subg 18394 df-ghm 18474 df-cntz 18565 df-cmn 19026 df-abl 19027 df-mgp 19359 df-ur 19371 df-srg 19375 df-ring 19418 df-cring 19419 df-rnghom 19589 df-subrg 19652 df-lmod 19755 df-lss 19823 df-lsp 19863 df-assa 20669 df-asp 20670 df-ascl 20671 df-psr 20722 df-mvr 20723 df-mpl 20724 df-evls 20886

This theorem is referenced by: (None)

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