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Theorem pwsmulg 18046

Description: Value of a group multiple in a structure power. (Contributed by Mario Carneiro, 15-Jun-2015.)

**Hypotheses**
Ref	Expression
pwsmulg.y	⊢ 𝑌 = (𝑅 ↑_s 𝐼)
pwsmulg.b	⊢ 𝐵 = (Base‘𝑌)
pwsmulg.s	⊢ ∙ = (._g‘𝑌)
pwsmulg.t	⊢ · = (._g‘𝑅)

**Assertion**
Ref	Expression
pwsmulg	⊢ (((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) ∧ (𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ 𝐵 ∧ 𝐴 ∈ 𝐼)) → ((𝑁 ∙ 𝑋)‘𝐴) = (𝑁 · (𝑋‘𝐴)))

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

Step	Hyp	Ref	Expression
1		simpll 754	. . . 4 ⊢ (((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) ∧ (𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ 𝐵 ∧ 𝐴 ∈ 𝐼)) → 𝑅 ∈ Mnd)
2		simplr 756	. . . 4 ⊢ (((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) ∧ (𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ 𝐵 ∧ 𝐴 ∈ 𝐼)) → 𝐼 ∈ 𝑉)
3		simpr3 1176	. . . 4 ⊢ (((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) ∧ (𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ 𝐵 ∧ 𝐴 ∈ 𝐼)) → 𝐴 ∈ 𝐼)
4		pwsmulg.y	. . . . 5 ⊢ 𝑌 = (𝑅 ↑_s 𝐼)
5		pwsmulg.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑌)
6	4, 5	pwspjmhm 17826	. . . 4 ⊢ ((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴)) ∈ (𝑌 MndHom 𝑅))
7	1, 2, 3, 6	syl3anc 1351	. . 3 ⊢ (((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) ∧ (𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ 𝐵 ∧ 𝐴 ∈ 𝐼)) → (𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴)) ∈ (𝑌 MndHom 𝑅))
8		simpr1 1174	. . 3 ⊢ (((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) ∧ (𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ 𝐵 ∧ 𝐴 ∈ 𝐼)) → 𝑁 ∈ ℕ₀)
9		simpr2 1175	. . 3 ⊢ (((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) ∧ (𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ 𝐵 ∧ 𝐴 ∈ 𝐼)) → 𝑋 ∈ 𝐵)
10		pwsmulg.s	. . . 4 ⊢ ∙ = (._g‘𝑌)
11		pwsmulg.t	. . . 4 ⊢ · = (._g‘𝑅)
12	5, 10, 11	mhmmulg 18042	. . 3 ⊢ (((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴)) ∈ (𝑌 MndHom 𝑅) ∧ 𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ 𝐵) → ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘(𝑁 ∙ 𝑋)) = (𝑁 · ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘𝑋)))
13	7, 8, 9, 12	syl3anc 1351	. 2 ⊢ (((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) ∧ (𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ 𝐵 ∧ 𝐴 ∈ 𝐼)) → ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘(𝑁 ∙ 𝑋)) = (𝑁 · ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘𝑋)))
14	4	pwsmnd 17783	. . . . 5 ⊢ ((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) → 𝑌 ∈ Mnd)
15	14	adantr 473	. . . 4 ⊢ (((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) ∧ (𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ 𝐵 ∧ 𝐴 ∈ 𝐼)) → 𝑌 ∈ Mnd)
16	5, 10	mulgnn0cl 18019	. . . 4 ⊢ ((𝑌 ∈ Mnd ∧ 𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ 𝐵) → (𝑁 ∙ 𝑋) ∈ 𝐵)
17	15, 8, 9, 16	syl3anc 1351	. . 3 ⊢ (((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) ∧ (𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ 𝐵 ∧ 𝐴 ∈ 𝐼)) → (𝑁 ∙ 𝑋) ∈ 𝐵)
18		fveq1 6492	. . . 4 ⊢ (𝑥 = (𝑁 ∙ 𝑋) → (𝑥‘𝐴) = ((𝑁 ∙ 𝑋)‘𝐴))
19		eqid 2772	. . . 4 ⊢ (𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴)) = (𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))
20		fvex 6506	. . . 4 ⊢ ((𝑁 ∙ 𝑋)‘𝐴) ∈ V
21	18, 19, 20	fvmpt 6589	. . 3 ⊢ ((𝑁 ∙ 𝑋) ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘(𝑁 ∙ 𝑋)) = ((𝑁 ∙ 𝑋)‘𝐴))
22	17, 21	syl 17	. 2 ⊢ (((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) ∧ (𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ 𝐵 ∧ 𝐴 ∈ 𝐼)) → ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘(𝑁 ∙ 𝑋)) = ((𝑁 ∙ 𝑋)‘𝐴))
23		fveq1 6492	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥‘𝐴) = (𝑋‘𝐴))
24		fvex 6506	. . . . 5 ⊢ (𝑋‘𝐴) ∈ V
25	23, 19, 24	fvmpt 6589	. . . 4 ⊢ (𝑋 ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘𝑋) = (𝑋‘𝐴))
26	9, 25	syl 17	. . 3 ⊢ (((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) ∧ (𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ 𝐵 ∧ 𝐴 ∈ 𝐼)) → ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘𝑋) = (𝑋‘𝐴))
27	26	oveq2d 6986	. 2 ⊢ (((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) ∧ (𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ 𝐵 ∧ 𝐴 ∈ 𝐼)) → (𝑁 · ((𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴))‘𝑋)) = (𝑁 · (𝑋‘𝐴)))
28	13, 22, 27	3eqtr3d 2816	1 ⊢ (((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) ∧ (𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ 𝐵 ∧ 𝐴 ∈ 𝐼)) → ((𝑁 ∙ 𝑋)‘𝐴) = (𝑁 · (𝑋‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2048 ↦ cmpt 5002 ‘cfv 6182 (class class class)co 6970 ℕ₀cn0 11700 Basecbs 16329 ↑_s cpws 16566 Mndcmnd 17752 MndHom cmhm 17791 ._gcmg 18001

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7494 df-2nd 7495 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-1o 7897 df-oadd 7901 df-er 8081 df-map 8200 df-ixp 8252 df-en 8299 df-dom 8300 df-sdom 8301 df-fin 8302 df-sup 8693 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-nn 11432 df-2 11496 df-3 11497 df-4 11498 df-5 11499 df-6 11500 df-7 11501 df-8 11502 df-9 11503 df-n0 11701 df-z 11787 df-dec 11905 df-uz 12052 df-fz 12702 df-seq 13178 df-struct 16331 df-ndx 16332 df-slot 16333 df-base 16335 df-plusg 16424 df-mulr 16425 df-sca 16427 df-vsca 16428 df-ip 16429 df-tset 16430 df-ple 16431 df-ds 16433 df-hom 16435 df-cco 16436 df-0g 16561 df-prds 16567 df-pws 16569 df-mgm 17700 df-sgrp 17742 df-mnd 17753 df-mhm 17793 df-mulg 18002

This theorem is referenced by: evl1expd 20200

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