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Theorem pwsexpg 20132

Description: Value of a group exponentiation in a structure power. Compare pwsmulg 18993. (Contributed by SN, 30-Jul-2024.)

**Hypotheses**
Ref	Expression
pwsexpg.y	⊢ 𝑌 = (𝑅 ↑_s 𝐼)
pwsexpg.b	⊢ 𝐵 = (Base‘𝑌)
pwsexpg.m	⊢ 𝑀 = (mulGrp‘𝑌)
pwsexpg.t	⊢ 𝑇 = (mulGrp‘𝑅)
pwsexpg.s	⊢ ∙ = (._g‘𝑀)
pwsexpg.g	⊢ · = (._g‘𝑇)
pwsexpg.r	⊢ (𝜑 → 𝑅 ∈ Ring)
pwsexpg.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
pwsexpg.n	⊢ (𝜑 → 𝑁 ∈ ℕ₀)
pwsexpg.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
pwsexpg.a	⊢ (𝜑 → 𝐴 ∈ 𝐼)

**Assertion**
Ref	Expression
pwsexpg	⊢ (𝜑 → ((𝑁 ∙ 𝑋)‘𝐴) = (𝑁 · (𝑋‘𝐴)))

Proof of Theorem pwsexpg Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pwsexpg.y	. . . 4 ⊢ 𝑌 = (𝑅 ↑_s 𝐼)
2		pwsexpg.b	. . . 4 ⊢ 𝐵 = (Base‘𝑌)
3		pwsexpg.m	. . . 4 ⊢ 𝑀 = (mulGrp‘𝑌)
4		pwsexpg.t	. . . 4 ⊢ 𝑇 = (mulGrp‘𝑅)
5		pwsexpg.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring)
6		pwsexpg.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
7		pwsexpg.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐼)
8	1, 2, 3, 4, 5, 6, 7	pwspjmhmmgpd 20131	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴)) ∈ (𝑀 MndHom 𝑇))
9		pwsexpg.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
10		pwsexpg.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
11	3, 2	mgpbas 19985	. . . 4 ⊢ 𝐵 = (Base‘𝑀)
12		pwsexpg.s	. . . 4 ⊢ ∙ = (._g‘𝑀)
13		pwsexpg.g	. . . 4 ⊢ · = (._g‘𝑇)
14	11, 12, 13	mhmmulg 18989	. . 3 ⊢ (((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴)) ∈ (𝑀 MndHom 𝑇) ∧ 𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ 𝐵) → ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘(𝑁 ∙ 𝑋)) = (𝑁 · ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘𝑋)))
15	8, 9, 10, 14	syl3anc 1372	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘(𝑁 ∙ 𝑋)) = (𝑁 · ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘𝑋)))
16	1	pwsring 20127	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑌 ∈ Ring)
17	5, 6, 16	syl2anc 585	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ Ring)
18	3	ringmgp 20053	. . . . 5 ⊢ (𝑌 ∈ Ring → 𝑀 ∈ Mnd)
19	17, 18	syl 17	. . . 4 ⊢ (𝜑 → 𝑀 ∈ Mnd)
20	11, 12, 19, 9, 10	mulgnn0cld 18969	. . 3 ⊢ (𝜑 → (𝑁 ∙ 𝑋) ∈ 𝐵)
21		fveq1 6887	. . . 4 ⊢ (𝑥 = (𝑁 ∙ 𝑋) → (𝑥‘𝐴) = ((𝑁 ∙ 𝑋)‘𝐴))
22		eqid 2733	. . . 4 ⊢ (𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴)) = (𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))
23		fvex 6901	. . . 4 ⊢ ((𝑁 ∙ 𝑋)‘𝐴) ∈ V
24	21, 22, 23	fvmpt 6994	. . 3 ⊢ ((𝑁 ∙ 𝑋) ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘(𝑁 ∙ 𝑋)) = ((𝑁 ∙ 𝑋)‘𝐴))
25	20, 24	syl 17	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘(𝑁 ∙ 𝑋)) = ((𝑁 ∙ 𝑋)‘𝐴))
26		fveq1 6887	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥‘𝐴) = (𝑋‘𝐴))
27		fvex 6901	. . . . 5 ⊢ (𝑋‘𝐴) ∈ V
28	26, 22, 27	fvmpt 6994	. . . 4 ⊢ (𝑋 ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘𝑋) = (𝑋‘𝐴))
29	10, 28	syl 17	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘𝑋) = (𝑋‘𝐴))
30	29	oveq2d 7420	. 2 ⊢ (𝜑 → (𝑁 · ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘𝑋)) = (𝑁 · (𝑋‘𝐴)))
31	15, 25, 30	3eqtr3d 2781	1 ⊢ (𝜑 → ((𝑁 ∙ 𝑋)‘𝐴) = (𝑁 · (𝑋‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ↦ cmpt 5230 ‘cfv 6540 (class class class)co 7404 ℕ₀cn0 12468 Basecbs 17140 ↑_s cpws 17388 Mndcmnd 18621 MndHom cmhm 18665 ._gcmg 18944 mulGrpcmgp 19979 Ringcrg 20047

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7665 df-om 7851 df-1st 7970 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-map 8818 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-seq 13963 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-plusg 17206 df-mulr 17207 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-hom 17217 df-cco 17218 df-0g 17383 df-prds 17389 df-pws 17391 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-mhm 18667 df-grp 18818 df-minusg 18819 df-mulg 18945 df-mgp 19980 df-ur 19997 df-ring 20049

This theorem is referenced by: evls1expd 32594 evlsvvval 41085 evlsexpval 41089

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