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Theorem pwsexpg 20352

Description: Value of a group exponentiation in a structure power. Compare pwsmulg 19159. (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 20351	. . 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 20167	. . . 4 ⊢ 𝐵 = (Base‘𝑀)
12		pwsexpg.s	. . . 4 ⊢ ∙ = (._g‘𝑀)
13		pwsexpg.g	. . . 4 ⊢ · = (._g‘𝑇)
14	11, 12, 13	mhmmulg 19155	. . 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 1371	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘(𝑁 ∙ 𝑋)) = (𝑁 · ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘𝑋)))
16	1	pwsring 20347	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑌 ∈ Ring)
17	5, 6, 16	syl2anc 583	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ Ring)
18	3	ringmgp 20266	. . . . 5 ⊢ (𝑌 ∈ Ring → 𝑀 ∈ Mnd)
19	17, 18	syl 17	. . . 4 ⊢ (𝜑 → 𝑀 ∈ Mnd)
20	11, 12, 19, 9, 10	mulgnn0cld 19135	. . 3 ⊢ (𝜑 → (𝑁 ∙ 𝑋) ∈ 𝐵)
21		fveq1 6919	. . . 4 ⊢ (𝑥 = (𝑁 ∙ 𝑋) → (𝑥‘𝐴) = ((𝑁 ∙ 𝑋)‘𝐴))
22		eqid 2740	. . . 4 ⊢ (𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴)) = (𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))
23		fvex 6933	. . . 4 ⊢ ((𝑁 ∙ 𝑋)‘𝐴) ∈ V
24	21, 22, 23	fvmpt 7029	. . 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 6919	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥‘𝐴) = (𝑋‘𝐴))
27		fvex 6933	. . . . 5 ⊢ (𝑋‘𝐴) ∈ V
28	26, 22, 27	fvmpt 7029	. . . 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 7464	. 2 ⊢ (𝜑 → (𝑁 · ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘𝑋)) = (𝑁 · (𝑋‘𝐴)))
31	15, 25, 30	3eqtr3d 2788	1 ⊢ (𝜑 → ((𝑁 ∙ 𝑋)‘𝐴) = (𝑁 · (𝑋‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2108 ↦ cmpt 5249 ‘cfv 6573 (class class class)co 7448 ℕ₀cn0 12553 Basecbs 17258 ↑_s cpws 17506 Mndcmnd 18772 MndHom cmhm 18816 ._gcmg 19107 mulGrpcmgp 20161 Ringcrg 20260

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 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-er 8763 df-map 8886 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-sup 9511 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-fz 13568 df-seq 14053 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-plusg 17324 df-mulr 17325 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-hom 17335 df-cco 17336 df-0g 17501 df-prds 17507 df-pws 17509 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-mhm 18818 df-grp 18976 df-minusg 18977 df-mulg 19108 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-ring 20262

This theorem is referenced by: evls1expd 22392 evlsvvval 42518 evlsexpval 42522

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