pwsexpg - Mathbox for Steven Nguyen

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Theorem pwsexpg 39810

Description: Value of a group exponentiation in a structure power. Compare pwsmulg 18352. (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 39809	. . 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 19326	. . . 4 ⊢ 𝐵 = (Base‘𝑀)
12		pwsexpg.s	. . . 4 ⊢ ∙ = (._g‘𝑀)
13		pwsexpg.g	. . . 4 ⊢ · = (._g‘𝑇)
14	11, 12, 13	mhmmulg 18348	. . 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 1368	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘(𝑁 ∙ 𝑋)) = (𝑁 · ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘𝑋)))
16	1	pwsring 19449	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑌 ∈ Ring)
17	5, 6, 16	syl2anc 587	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ Ring)
18	3	ringmgp 19384	. . . . 5 ⊢ (𝑌 ∈ Ring → 𝑀 ∈ Mnd)
19	17, 18	syl 17	. . . 4 ⊢ (𝜑 → 𝑀 ∈ Mnd)
20	11, 12	mulgnn0cl 18324	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ 𝐵) → (𝑁 ∙ 𝑋) ∈ 𝐵)
21	19, 9, 10, 20	syl3anc 1368	. . 3 ⊢ (𝜑 → (𝑁 ∙ 𝑋) ∈ 𝐵)
22		fveq1 6662	. . . 4 ⊢ (𝑥 = (𝑁 ∙ 𝑋) → (𝑥‘𝐴) = ((𝑁 ∙ 𝑋)‘𝐴))
23		eqid 2758	. . . 4 ⊢ (𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴)) = (𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))
24		fvex 6676	. . . 4 ⊢ ((𝑁 ∙ 𝑋)‘𝐴) ∈ V
25	22, 23, 24	fvmpt 6764	. . 3 ⊢ ((𝑁 ∙ 𝑋) ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘(𝑁 ∙ 𝑋)) = ((𝑁 ∙ 𝑋)‘𝐴))
26	21, 25	syl 17	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘(𝑁 ∙ 𝑋)) = ((𝑁 ∙ 𝑋)‘𝐴))
27		fveq1 6662	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥‘𝐴) = (𝑋‘𝐴))
28		fvex 6676	. . . . 5 ⊢ (𝑋‘𝐴) ∈ V
29	27, 23, 28	fvmpt 6764	. . . 4 ⊢ (𝑋 ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘𝑋) = (𝑋‘𝐴))
30	10, 29	syl 17	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘𝑋) = (𝑋‘𝐴))
31	30	oveq2d 7172	. 2 ⊢ (𝜑 → (𝑁 · ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘𝑋)) = (𝑁 · (𝑋‘𝐴)))
32	15, 26, 31	3eqtr3d 2801	1 ⊢ (𝜑 → ((𝑁 ∙ 𝑋)‘𝐴) = (𝑁 · (𝑋‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ↦ cmpt 5116 ‘cfv 6340 (class class class)co 7156 ℕ₀cn0 11947 Basecbs 16554 ↑_s cpws 16791 Mndcmnd 17990 MndHom cmhm 18033 ._gcmg 18304 mulGrpcmgp 19320 Ringcrg 19378

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5160 ax-sep 5173 ax-nul 5180 ax-pow 5238 ax-pr 5302 ax-un 7465 ax-cnex 10644 ax-resscn 10645 ax-1cn 10646 ax-icn 10647 ax-addcl 10648 ax-addrcl 10649 ax-mulcl 10650 ax-mulrcl 10651 ax-mulcom 10652 ax-addass 10653 ax-mulass 10654 ax-distr 10655 ax-i2m1 10656 ax-1ne0 10657 ax-1rid 10658 ax-rnegex 10659 ax-rrecex 10660 ax-cnre 10661 ax-pre-lttri 10662 ax-pre-lttrn 10663 ax-pre-ltadd 10664 ax-pre-mulgt0 10665

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-pss 3879 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4802 df-iun 4888 df-br 5037 df-opab 5099 df-mpt 5117 df-tr 5143 df-id 5434 df-eprel 5439 df-po 5447 df-so 5448 df-fr 5487 df-we 5489 df-xp 5534 df-rel 5535 df-cnv 5536 df-co 5537 df-dm 5538 df-rn 5539 df-res 5540 df-ima 5541 df-pred 6131 df-ord 6177 df-on 6178 df-lim 6179 df-suc 6180 df-iota 6299 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7411 df-om 7586 df-1st 7699 df-2nd 7700 df-wrecs 7963 df-recs 8024 df-rdg 8062 df-1o 8118 df-er 8305 df-map 8424 df-ixp 8493 df-en 8541 df-dom 8542 df-sdom 8543 df-fin 8544 df-sup 8952 df-pnf 10728 df-mnf 10729 df-xr 10730 df-ltxr 10731 df-le 10732 df-sub 10923 df-neg 10924 df-nn 11688 df-2 11750 df-3 11751 df-4 11752 df-5 11753 df-6 11754 df-7 11755 df-8 11756 df-9 11757 df-n0 11948 df-z 12034 df-dec 12151 df-uz 12296 df-fz 12953 df-seq 13432 df-struct 16556 df-ndx 16557 df-slot 16558 df-base 16560 df-sets 16561 df-plusg 16649 df-mulr 16650 df-sca 16652 df-vsca 16653 df-ip 16654 df-tset 16655 df-ple 16656 df-ds 16658 df-hom 16660 df-cco 16661 df-0g 16786 df-prds 16792 df-pws 16794 df-mgm 17931 df-sgrp 17980 df-mnd 17991 df-mhm 18035 df-grp 18185 df-minusg 18186 df-mulg 18305 df-mgp 19321 df-ur 19333 df-ring 19380

This theorem is referenced by: evlsbagval 39815 evlsexpval 39816

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