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Theorem gsummulglem 19542

Description: Lemma for gsummulg 19543 and gsummulgz 19544. (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 19402	. . 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 19430	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝑁 ∈ ℤ) → (𝑥 ∈ 𝐵 ↦ (𝑁 · 𝑥)) ∈ (𝐺 GrpHom 𝐺))
10		ghmmhm 18844	. . . . . 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 414	. . . 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 19429	. . . . 5 ⊢ ((𝐺 ∈ CMnd ∧ 𝑁 ∈ ℕ₀) → (𝑥 ∈ 𝐵 ↦ (𝑁 · 𝑥)) ∈ (𝐺 MndHom 𝐺))
15	14	ex 413	. . . 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 857	. 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 7283	. 2 ⊢ (𝑥 = 𝑋 → (𝑁 · 𝑥) = (𝑁 · 𝑋))
22		oveq2 7283	. 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 19540	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 396 ∨ wo 844 = wceq 1539 ∈ wcel 2106 class class class wbr 5074 ↦ cmpt 5157 ‘cfv 6433 (class class class)co 7275 finSupp cfsupp 9128 ℕ₀cn0 12233 ℤcz 12319 Basecbs 16912 0_gc0g 17150 Σ_g cgsu 17151 Mndcmnd 18385 MndHom cmhm 18428 ._gcmg 18700 GrpHom cghm 18831 CMndccmn 19386 Abelcabl 19387

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-fzo 13383 df-seq 13722 df-hash 14045 df-0g 17152 df-gsum 17153 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-mhm 18430 df-grp 18580 df-minusg 18581 df-mulg 18701 df-ghm 18832 df-cntz 18923 df-cmn 19388 df-abl 19389

This theorem is referenced by: gsummulg 19543 gsummulgz 19544

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