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Theorem gsummulglem 19959

Description: Lemma for gsummulg 19960 and gsummulgz 19961. (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 19815	. . 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 19846	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝑁 ∈ ℤ) → (𝑥 ∈ 𝐵 ↦ (𝑁 · 𝑥)) ∈ (𝐺 GrpHom 𝐺))
10		ghmmhm 19244	. . . . . 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 19845	. . . . 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 861	. 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 7439	. 2 ⊢ (𝑥 = 𝑋 → (𝑁 · 𝑥) = (𝑁 · 𝑋))
22		oveq2 7439	. 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 19957	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 848 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 finSupp cfsupp 9401 ℕ₀cn0 12526 ℤcz 12613 Basecbs 17247 0_gc0g 17484 Σ_g cgsu 17485 Mndcmnd 18747 MndHom cmhm 18794 ._gcmg 19085 GrpHom cghm 19230 CMndccmn 19798 Abelcabl 19799

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-0g 17486 df-gsum 17487 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-grp 18954 df-minusg 18955 df-mulg 19086 df-ghm 19231 df-cntz 19335 df-cmn 19800 df-abl 19801

This theorem is referenced by: gsummulg 19960 gsummulgz 19961

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