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Theorem pwsexpg 20281

Description: Value of a group exponentiation in a structure power. Compare pwsmulg 19066. (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 20280	. . 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 20097	. . . 4 ⊢ 𝐵 = (Base‘𝑀)
12		pwsexpg.s	. . . 4 ⊢ ∙ = (._g‘𝑀)
13		pwsexpg.g	. . . 4 ⊢ · = (._g‘𝑇)
14	11, 12, 13	mhmmulg 19062	. . 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 1374	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘(𝑁 ∙ 𝑋)) = (𝑁 · ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘𝑋)))
16	1	pwsring 20276	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑌 ∈ Ring)
17	5, 6, 16	syl2anc 585	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ Ring)
18	3	ringmgp 20191	. . . . 5 ⊢ (𝑌 ∈ Ring → 𝑀 ∈ Mnd)
19	17, 18	syl 17	. . . 4 ⊢ (𝜑 → 𝑀 ∈ Mnd)
20	11, 12, 19, 9, 10	mulgnn0cld 19042	. . 3 ⊢ (𝜑 → (𝑁 ∙ 𝑋) ∈ 𝐵)
21		fveq1 6843	. . . 4 ⊢ (𝑥 = (𝑁 ∙ 𝑋) → (𝑥‘𝐴) = ((𝑁 ∙ 𝑋)‘𝐴))
22		eqid 2737	. . . 4 ⊢ (𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴)) = (𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))
23		fvex 6857	. . . 4 ⊢ ((𝑁 ∙ 𝑋)‘𝐴) ∈ V
24	21, 22, 23	fvmpt 6951	. . 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 6843	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥‘𝐴) = (𝑋‘𝐴))
27		fvex 6857	. . . . 5 ⊢ (𝑋‘𝐴) ∈ V
28	26, 22, 27	fvmpt 6951	. . . 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 7386	. 2 ⊢ (𝜑 → (𝑁 · ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘𝑋)) = (𝑁 · (𝑋‘𝐴)))
31	15, 25, 30	3eqtr3d 2780	1 ⊢ (𝜑 → ((𝑁 ∙ 𝑋)‘𝐴) = (𝑁 · (𝑋‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ↦ cmpt 5181 ‘cfv 6502 (class class class)co 7370 ℕ₀cn0 12415 Basecbs 17150 ↑_s cpws 17380 Mndcmnd 18673 MndHom cmhm 18720 ._gcmg 19014 mulGrpcmgp 20092 Ringcrg 20185

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5314 ax-pr 5381 ax-un 7692 ax-cnex 11096 ax-resscn 11097 ax-1cn 11098 ax-icn 11099 ax-addcl 11100 ax-addrcl 11101 ax-mulcl 11102 ax-mulrcl 11103 ax-mulcom 11104 ax-addass 11105 ax-mulass 11106 ax-distr 11107 ax-i2m1 11108 ax-1ne0 11109 ax-1rid 11110 ax-rnegex 11111 ax-rrecex 11112 ax-cnre 11113 ax-pre-lttri 11114 ax-pre-lttrn 11115 ax-pre-ltadd 11116 ax-pre-mulgt0 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5529 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5587 df-we 5589 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6269 df-ord 6330 df-on 6331 df-lim 6332 df-suc 6333 df-iota 6458 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-riota 7327 df-ov 7373 df-oprab 7374 df-mpo 7375 df-of 7634 df-om 7821 df-1st 7945 df-2nd 7946 df-frecs 8235 df-wrecs 8266 df-recs 8315 df-rdg 8353 df-1o 8409 df-er 8647 df-map 8779 df-ixp 8850 df-en 8898 df-dom 8899 df-sdom 8900 df-fin 8901 df-sup 9359 df-pnf 11182 df-mnf 11183 df-xr 11184 df-ltxr 11185 df-le 11186 df-sub 11380 df-neg 11381 df-nn 12160 df-2 12222 df-3 12223 df-4 12224 df-5 12225 df-6 12226 df-7 12227 df-8 12228 df-9 12229 df-n0 12416 df-z 12503 df-dec 12622 df-uz 12766 df-fz 13438 df-seq 13939 df-struct 17088 df-sets 17105 df-slot 17123 df-ndx 17135 df-base 17151 df-plusg 17204 df-mulr 17205 df-sca 17207 df-vsca 17208 df-ip 17209 df-tset 17210 df-ple 17211 df-ds 17213 df-hom 17215 df-cco 17216 df-0g 17375 df-prds 17381 df-pws 17383 df-mgm 18579 df-sgrp 18658 df-mnd 18674 df-mhm 18722 df-grp 18883 df-minusg 18884 df-mulg 19015 df-cmn 19728 df-abl 19729 df-mgp 20093 df-rng 20105 df-ur 20134 df-ring 20187

This theorem is referenced by: evlsvvval 22065 evls1expd 22328 evlsexpval 42957

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