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Theorem gsummulglem 19457

Description: Lemma for gsummulg 19458 and gsummulgz 19459. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by AV, 6-Jun-2019.)

**Hypotheses**
Ref	Expression
gsummulg.b	⊢ 𝐵 = (Base‘𝐺)
gsummulg.z	⊢ 0 = (0_g‘𝐺)
gsummulg.t	⊢ · = (._g‘𝐺)
gsummulg.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
gsummulg.f	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵)
gsummulg.w	⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp 0 )
gsummulglem.g	⊢ (𝜑 → 𝐺 ∈ CMnd)
gsummulglem.n	⊢ (𝜑 → 𝑁 ∈ ℤ)
gsummulglem.o	⊢ (𝜑 → (𝐺 ∈ Abel ∨ 𝑁 ∈ ℕ₀))

**Assertion**
Ref	Expression
gsummulglem	⊢ (𝜑 → (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ (𝑁 · 𝑋))) = (𝑁 · (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 𝑋))))

Distinct variable groups: 𝐴,𝑘 𝐵,𝑘 𝑘,𝑁 𝜑,𝑘 · ,𝑘 Allowed substitution hints: 𝐺(𝑘) 𝑉(𝑘) 𝑋(𝑘) 0 (𝑘)

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

Step	Hyp	Ref	Expression
1		gsummulg.b	. 2 ⊢ 𝐵 = (Base‘𝐺)
2		gsummulg.z	. 2 ⊢ 0 = (0_g‘𝐺)
3		gsummulglem.g	. 2 ⊢ (𝜑 → 𝐺 ∈ CMnd)
4		cmnmnd 19317	. . 3 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
5	3, 4	syl 17	. 2 ⊢ (𝜑 → 𝐺 ∈ Mnd)
6		gsummulg.a	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
7		gsummulglem.n	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ)
8		gsummulg.t	. . . . . . 7 ⊢ · = (._g‘𝐺)
9	1, 8	mulgghm 19345	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝑁 ∈ ℤ) → (𝑥 ∈ 𝐵 ↦ (𝑁 · 𝑥)) ∈ (𝐺 GrpHom 𝐺))
10		ghmmhm 18759	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ↦ (𝑁 · 𝑥)) ∈ (𝐺 GrpHom 𝐺) → (𝑥 ∈ 𝐵 ↦ (𝑁 · 𝑥)) ∈ (𝐺 MndHom 𝐺))
11	9, 10	syl 17	. . . . 5 ⊢ ((𝐺 ∈ Abel ∧ 𝑁 ∈ ℤ) → (𝑥 ∈ 𝐵 ↦ (𝑁 · 𝑥)) ∈ (𝐺 MndHom 𝐺))
12	11	expcom 413	. . . 4 ⊢ (𝑁 ∈ ℤ → (𝐺 ∈ Abel → (𝑥 ∈ 𝐵 ↦ (𝑁 · 𝑥)) ∈ (𝐺 MndHom 𝐺)))
13	7, 12	syl 17	. . 3 ⊢ (𝜑 → (𝐺 ∈ Abel → (𝑥 ∈ 𝐵 ↦ (𝑁 · 𝑥)) ∈ (𝐺 MndHom 𝐺)))
14	1, 8	mulgmhm 19344	. . . . 5 ⊢ ((𝐺 ∈ CMnd ∧ 𝑁 ∈ ℕ₀) → (𝑥 ∈ 𝐵 ↦ (𝑁 · 𝑥)) ∈ (𝐺 MndHom 𝐺))
15	14	ex 412	. . . 4 ⊢ (𝐺 ∈ CMnd → (𝑁 ∈ ℕ₀ → (𝑥 ∈ 𝐵 ↦ (𝑁 · 𝑥)) ∈ (𝐺 MndHom 𝐺)))
16	3, 15	syl 17	. . 3 ⊢ (𝜑 → (𝑁 ∈ ℕ₀ → (𝑥 ∈ 𝐵 ↦ (𝑁 · 𝑥)) ∈ (𝐺 MndHom 𝐺)))
17		gsummulglem.o	. . 3 ⊢ (𝜑 → (𝐺 ∈ Abel ∨ 𝑁 ∈ ℕ₀))
18	13, 16, 17	mpjaod 856	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ (𝑁 · 𝑥)) ∈ (𝐺 MndHom 𝐺))
19		gsummulg.f	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵)
20		gsummulg.w	. 2 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp 0 )
21		oveq2 7263	. 2 ⊢ (𝑥 = 𝑋 → (𝑁 · 𝑥) = (𝑁 · 𝑋))
22		oveq2 7263	. 2 ⊢ (𝑥 = (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 𝑋)) → (𝑁 · 𝑥) = (𝑁 · (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 𝑋))))
23	1, 2, 3, 5, 6, 18, 19, 20, 21, 22	gsummhm2 19455	1 ⊢ (𝜑 → (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ (𝑁 · 𝑋))) = (𝑁 · (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 𝑋))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 843 = wceq 1539 ∈ wcel 2108 class class class wbr 5070 ↦ cmpt 5153 ‘cfv 6418 (class class class)co 7255 finSupp cfsupp 9058 ℕ₀cn0 12163 ℤcz 12249 Basecbs 16840 0_gc0g 17067 Σ_g cgsu 17068 Mndcmnd 18300 MndHom cmhm 18343 ._gcmg 18615 GrpHom cghm 18746 CMndccmn 19301 Abelcabl 19302

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-n0 12164 df-z 12250 df-uz 12512 df-fz 13169 df-fzo 13312 df-seq 13650 df-hash 13973 df-0g 17069 df-gsum 17070 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-mhm 18345 df-grp 18495 df-minusg 18496 df-mulg 18616 df-ghm 18747 df-cntz 18838 df-cmn 19303 df-abl 19304

This theorem is referenced by: gsummulg 19458 gsummulgz 19459

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