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Theorem gsumzmhm 19851

Description: Apply a group homomorphism to a group sum. (Contributed by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 6-Jun-2019.)

**Hypotheses**
Ref	Expression
gsumzmhm.b	⊢ 𝐵 = (Base‘𝐺)
gsumzmhm.z	⊢ 𝑍 = (Cntz‘𝐺)
gsumzmhm.g	⊢ (𝜑 → 𝐺 ∈ Mnd)
gsumzmhm.h	⊢ (𝜑 → 𝐻 ∈ Mnd)
gsumzmhm.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
gsumzmhm.k	⊢ (𝜑 → 𝐾 ∈ (𝐺 MndHom 𝐻))
gsumzmhm.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
gsumzmhm.c	⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
gsumzmhm.0	⊢ 0 = (0_g‘𝐺)
gsumzmhm.w	⊢ (𝜑 → 𝐹 finSupp 0 )

**Assertion**
Ref	Expression
gsumzmhm	⊢ (𝜑 → (𝐻 Σ_g (𝐾 ∘ 𝐹)) = (𝐾‘(𝐺 Σ_g 𝐹)))

Proof of Theorem gsumzmhm Dummy variables 𝑘 𝑥 𝑦 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumzmhm.h	. . . . . . 7 ⊢ (𝜑 → 𝐻 ∈ Mnd)
2		gsumzmhm.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		eqid 2729	. . . . . . . 8 ⊢ (0_g‘𝐻) = (0_g‘𝐻)
4	3	gsumz 18745	. . . . . . 7 ⊢ ((𝐻 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐻 Σ_g (𝑘 ∈ 𝐴 ↦ (0_g‘𝐻))) = (0_g‘𝐻))
5	1, 2, 4	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝐻 Σ_g (𝑘 ∈ 𝐴 ↦ (0_g‘𝐻))) = (0_g‘𝐻))
6	5	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (^◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐻 Σ_g (𝑘 ∈ 𝐴 ↦ (0_g‘𝐻))) = (0_g‘𝐻))
7		gsumzmhm.k	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (𝐺 MndHom 𝐻))
8		gsumzmhm.0	. . . . . . . 8 ⊢ 0 = (0_g‘𝐺)
9	8, 3	mhm0 18703	. . . . . . 7 ⊢ (𝐾 ∈ (𝐺 MndHom 𝐻) → (𝐾‘ 0 ) = (0_g‘𝐻))
10	7, 9	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐾‘ 0 ) = (0_g‘𝐻))
11	10	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (^◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐾‘ 0 ) = (0_g‘𝐻))
12	6, 11	eqtr4d 2767	. . . 4 ⊢ ((𝜑 ∧ (^◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐻 Σ_g (𝑘 ∈ 𝐴 ↦ (0_g‘𝐻))) = (𝐾‘ 0 ))
13		gsumzmhm.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ Mnd)
14		gsumzmhm.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐺)
15	14, 8	mndidcl 18658	. . . . . . . . 9 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵)
16	13, 15	syl 17	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ 𝐵)
17	16	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ (^◡𝐹 “ (V ∖ { 0 })) = ∅) ∧ 𝑘 ∈ 𝐴) → 0 ∈ 𝐵)
18		gsumzmhm.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
19	8	fvexi 6854	. . . . . . . . 9 ⊢ 0 ∈ V
20	19	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ V)
21	18, 2	fexd 7183	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ V)
22		suppimacnv 8130	. . . . . . . . . 10 ⊢ ((𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 supp 0 ) = (^◡𝐹 “ (V ∖ { 0 })))
23	21, 20, 22	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 supp 0 ) = (^◡𝐹 “ (V ∖ { 0 })))
24		ssid 3966	. . . . . . . . 9 ⊢ (^◡𝐹 “ (V ∖ { 0 })) ⊆ (^◡𝐹 “ (V ∖ { 0 }))
25	23, 24	eqsstrdi 3988	. . . . . . . 8 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ (^◡𝐹 “ (V ∖ { 0 })))
26	18, 2, 20, 25	gsumcllem 19822	. . . . . . 7 ⊢ ((𝜑 ∧ (^◡𝐹 “ (V ∖ { 0 })) = ∅) → 𝐹 = (𝑘 ∈ 𝐴 ↦ 0 ))
27		eqid 2729	. . . . . . . . . . 11 ⊢ (Base‘𝐻) = (Base‘𝐻)
28	14, 27	mhmf 18698	. . . . . . . . . 10 ⊢ (𝐾 ∈ (𝐺 MndHom 𝐻) → 𝐾:𝐵⟶(Base‘𝐻))
29	7, 28	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐾:𝐵⟶(Base‘𝐻))
30	29	feqmptd 6911	. . . . . . . 8 ⊢ (𝜑 → 𝐾 = (𝑥 ∈ 𝐵 ↦ (𝐾‘𝑥)))
31	30	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (^◡𝐹 “ (V ∖ { 0 })) = ∅) → 𝐾 = (𝑥 ∈ 𝐵 ↦ (𝐾‘𝑥)))
32		fveq2 6840	. . . . . . 7 ⊢ (𝑥 = 0 → (𝐾‘𝑥) = (𝐾‘ 0 ))
33	17, 26, 31, 32	fmptco 7083	. . . . . 6 ⊢ ((𝜑 ∧ (^◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐾 ∘ 𝐹) = (𝑘 ∈ 𝐴 ↦ (𝐾‘ 0 )))
34	10	mpteq2dv 5196	. . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ (𝐾‘ 0 )) = (𝑘 ∈ 𝐴 ↦ (0_g‘𝐻)))
35	34	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (^◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝑘 ∈ 𝐴 ↦ (𝐾‘ 0 )) = (𝑘 ∈ 𝐴 ↦ (0_g‘𝐻)))
36	33, 35	eqtrd 2764	. . . . 5 ⊢ ((𝜑 ∧ (^◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐾 ∘ 𝐹) = (𝑘 ∈ 𝐴 ↦ (0_g‘𝐻)))
37	36	oveq2d 7385	. . . 4 ⊢ ((𝜑 ∧ (^◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐻 Σ_g (𝐾 ∘ 𝐹)) = (𝐻 Σ_g (𝑘 ∈ 𝐴 ↦ (0_g‘𝐻))))
38	26	oveq2d 7385	. . . . . 6 ⊢ ((𝜑 ∧ (^◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐺 Σ_g 𝐹) = (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 0 )))
39	8	gsumz 18745	. . . . . . . 8 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )
40	13, 2, 39	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )
41	40	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (^◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )
42	38, 41	eqtrd 2764	. . . . 5 ⊢ ((𝜑 ∧ (^◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐺 Σ_g 𝐹) = 0 )
43	42	fveq2d 6844	. . . 4 ⊢ ((𝜑 ∧ (^◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐾‘(𝐺 Σ_g 𝐹)) = (𝐾‘ 0 ))
44	12, 37, 43	3eqtr4d 2774	. . 3 ⊢ ((𝜑 ∧ (^◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐻 Σ_g (𝐾 ∘ 𝐹)) = (𝐾‘(𝐺 Σ_g 𝐹)))
45	44	ex 412	. 2 ⊢ (𝜑 → ((^◡𝐹 “ (V ∖ { 0 })) = ∅ → (𝐻 Σ_g (𝐾 ∘ 𝐹)) = (𝐾‘(𝐺 Σ_g 𝐹))))
46	13	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) → 𝐺 ∈ Mnd)
47		eqid 2729	. . . . . . . . . 10 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
48	14, 47	mndcl 18651	. . . . . . . . 9 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+_g‘𝐺)𝑦) ∈ 𝐵)
49	48	3expb 1120	. . . . . . . 8 ⊢ ((𝐺 ∈ Mnd ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+_g‘𝐺)𝑦) ∈ 𝐵)
50	46, 49	sylan 580	. . . . . . 7 ⊢ (((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+_g‘𝐺)𝑦) ∈ 𝐵)
51		f1of1 6781	. . . . . . . . . . . 12 ⊢ (𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })) → 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1→(^◡𝐹 “ (V ∖ { 0 })))
52	51	ad2antll 729	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) → 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1→(^◡𝐹 “ (V ∖ { 0 })))
53		cnvimass 6042	. . . . . . . . . . . 12 ⊢ (^◡𝐹 “ (V ∖ { 0 })) ⊆ dom 𝐹
54	18	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) → 𝐹:𝐴⟶𝐵)
55	53, 54	fssdm 6689	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) → (^◡𝐹 “ (V ∖ { 0 })) ⊆ 𝐴)
56		f1ss 6743	. . . . . . . . . . 11 ⊢ ((𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1→(^◡𝐹 “ (V ∖ { 0 })) ∧ (^◡𝐹 “ (V ∖ { 0 })) ⊆ 𝐴) → 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1→𝐴)
57	52, 55, 56	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) → 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1→𝐴)
58		f1f 6738	. . . . . . . . . 10 ⊢ (𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1→𝐴 → 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))⟶𝐴)
59	57, 58	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) → 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))⟶𝐴)
60		fco 6694	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))⟶𝐴) → (𝐹 ∘ 𝑓):(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))⟶𝐵)
61	18, 59, 60	syl2an2r 685	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) → (𝐹 ∘ 𝑓):(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))⟶𝐵)
62	61	ffvelcdmda 7038	. . . . . . 7 ⊢ (((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) ∧ 𝑥 ∈ (1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))) → ((𝐹 ∘ 𝑓)‘𝑥) ∈ 𝐵)
63		simprl 770	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) → (♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ)
64		nnuz 12812	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘1)
65	63, 64	eleqtrdi 2838	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) → (♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ (ℤ_≥‘1))
66	7	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) → 𝐾 ∈ (𝐺 MndHom 𝐻))
67		eqid 2729	. . . . . . . . . 10 ⊢ (+_g‘𝐻) = (+_g‘𝐻)
68	14, 47, 67	mhmlin 18702	. . . . . . . . 9 ⊢ ((𝐾 ∈ (𝐺 MndHom 𝐻) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝐾‘(𝑥(+_g‘𝐺)𝑦)) = ((𝐾‘𝑥)(+_g‘𝐻)(𝐾‘𝑦)))
69	68	3expb 1120	. . . . . . . 8 ⊢ ((𝐾 ∈ (𝐺 MndHom 𝐻) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐾‘(𝑥(+_g‘𝐺)𝑦)) = ((𝐾‘𝑥)(+_g‘𝐻)(𝐾‘𝑦)))
70	66, 69	sylan 580	. . . . . . 7 ⊢ (((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐾‘(𝑥(+_g‘𝐺)𝑦)) = ((𝐾‘𝑥)(+_g‘𝐻)(𝐾‘𝑦)))
71		coass 6226	. . . . . . . . 9 ⊢ ((𝐾 ∘ 𝐹) ∘ 𝑓) = (𝐾 ∘ (𝐹 ∘ 𝑓))
72	71	fveq1i 6841	. . . . . . . 8 ⊢ (((𝐾 ∘ 𝐹) ∘ 𝑓)‘𝑥) = ((𝐾 ∘ (𝐹 ∘ 𝑓))‘𝑥)
73		fvco3 6942	. . . . . . . . 9 ⊢ (((𝐹 ∘ 𝑓):(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))⟶𝐵 ∧ 𝑥 ∈ (1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))) → ((𝐾 ∘ (𝐹 ∘ 𝑓))‘𝑥) = (𝐾‘((𝐹 ∘ 𝑓)‘𝑥)))
74	61, 73	sylan 580	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) ∧ 𝑥 ∈ (1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))) → ((𝐾 ∘ (𝐹 ∘ 𝑓))‘𝑥) = (𝐾‘((𝐹 ∘ 𝑓)‘𝑥)))
75	72, 74	eqtr2id 2777	. . . . . . 7 ⊢ (((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) ∧ 𝑥 ∈ (1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))) → (𝐾‘((𝐹 ∘ 𝑓)‘𝑥)) = (((𝐾 ∘ 𝐹) ∘ 𝑓)‘𝑥))
76	50, 62, 65, 70, 75	seqhomo 13990	. . . . . 6 ⊢ ((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) → (𝐾‘(seq1((+_g‘𝐺), (𝐹 ∘ 𝑓))‘(♯‘(^◡𝐹 “ (V ∖ { 0 }))))) = (seq1((+_g‘𝐻), ((𝐾 ∘ 𝐹) ∘ 𝑓))‘(♯‘(^◡𝐹 “ (V ∖ { 0 })))))
77		gsumzmhm.z	. . . . . . . 8 ⊢ 𝑍 = (Cntz‘𝐺)
78	2	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) → 𝐴 ∈ 𝑉)
79		gsumzmhm.c	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
80	79	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
81	25	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) → (𝐹 supp 0 ) ⊆ (^◡𝐹 “ (V ∖ { 0 })))
82		f1ofo 6789	. . . . . . . . . . 11 ⊢ (𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })) → 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–onto→(^◡𝐹 “ (V ∖ { 0 })))
83		forn 6757	. . . . . . . . . . 11 ⊢ (𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–onto→(^◡𝐹 “ (V ∖ { 0 })) → ran 𝑓 = (^◡𝐹 “ (V ∖ { 0 })))
84	82, 83	syl 17	. . . . . . . . . 10 ⊢ (𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })) → ran 𝑓 = (^◡𝐹 “ (V ∖ { 0 })))
85	84	ad2antll 729	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) → ran 𝑓 = (^◡𝐹 “ (V ∖ { 0 })))
86	81, 85	sseqtrrd 3981	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) → (𝐹 supp 0 ) ⊆ ran 𝑓)
87		eqid 2729	. . . . . . . 8 ⊢ ((𝐹 ∘ 𝑓) supp 0 ) = ((𝐹 ∘ 𝑓) supp 0 )
88	14, 8, 47, 77, 46, 78, 54, 80, 63, 57, 86, 87	gsumval3 19821	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) → (𝐺 Σ_g 𝐹) = (seq1((+_g‘𝐺), (𝐹 ∘ 𝑓))‘(♯‘(^◡𝐹 “ (V ∖ { 0 })))))
89	88	fveq2d 6844	. . . . . 6 ⊢ ((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) → (𝐾‘(𝐺 Σ_g 𝐹)) = (𝐾‘(seq1((+_g‘𝐺), (𝐹 ∘ 𝑓))‘(♯‘(^◡𝐹 “ (V ∖ { 0 }))))))
90		eqid 2729	. . . . . . 7 ⊢ (Cntz‘𝐻) = (Cntz‘𝐻)
91	1	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) → 𝐻 ∈ Mnd)
92		fco 6694	. . . . . . . 8 ⊢ ((𝐾:𝐵⟶(Base‘𝐻) ∧ 𝐹:𝐴⟶𝐵) → (𝐾 ∘ 𝐹):𝐴⟶(Base‘𝐻))
93	29, 54, 92	syl2an2r 685	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) → (𝐾 ∘ 𝐹):𝐴⟶(Base‘𝐻))
94	77, 90	cntzmhm2 19256	. . . . . . . . 9 ⊢ ((𝐾 ∈ (𝐺 MndHom 𝐻) ∧ ran 𝐹 ⊆ (𝑍‘ran 𝐹)) → (𝐾 “ ran 𝐹) ⊆ ((Cntz‘𝐻)‘(𝐾 “ ran 𝐹)))
95	7, 80, 94	syl2an2r 685	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) → (𝐾 “ ran 𝐹) ⊆ ((Cntz‘𝐻)‘(𝐾 “ ran 𝐹)))
96		rnco2 6214	. . . . . . . 8 ⊢ ran (𝐾 ∘ 𝐹) = (𝐾 “ ran 𝐹)
97	96	fveq2i 6843	. . . . . . . 8 ⊢ ((Cntz‘𝐻)‘ran (𝐾 ∘ 𝐹)) = ((Cntz‘𝐻)‘(𝐾 “ ran 𝐹))
98	95, 96, 97	3sstr4g 3997	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) → ran (𝐾 ∘ 𝐹) ⊆ ((Cntz‘𝐻)‘ran (𝐾 ∘ 𝐹)))
99		eldifi 4090	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 ∖ (^◡𝐹 “ (V ∖ { 0 }))) → 𝑥 ∈ 𝐴)
100		fvco3 6942	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐾 ∘ 𝐹)‘𝑥) = (𝐾‘(𝐹‘𝑥)))
101	54, 99, 100	syl2an 596	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡𝐹 “ (V ∖ { 0 })))) → ((𝐾 ∘ 𝐹)‘𝑥) = (𝐾‘(𝐹‘𝑥)))
102	19	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) → 0 ∈ V)
103	54, 81, 78, 102	suppssr 8151	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡𝐹 “ (V ∖ { 0 })))) → (𝐹‘𝑥) = 0 )
104	103	fveq2d 6844	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡𝐹 “ (V ∖ { 0 })))) → (𝐾‘(𝐹‘𝑥)) = (𝐾‘ 0 ))
105	10	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡𝐹 “ (V ∖ { 0 })))) → (𝐾‘ 0 ) = (0_g‘𝐻))
106	101, 104, 105	3eqtrd 2768	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡𝐹 “ (V ∖ { 0 })))) → ((𝐾 ∘ 𝐹)‘𝑥) = (0_g‘𝐻))
107	93, 106	suppss 8150	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) → ((𝐾 ∘ 𝐹) supp (0_g‘𝐻)) ⊆ (^◡𝐹 “ (V ∖ { 0 })))
108	107, 85	sseqtrrd 3981	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) → ((𝐾 ∘ 𝐹) supp (0_g‘𝐻)) ⊆ ran 𝑓)
109		eqid 2729	. . . . . . 7 ⊢ (((𝐾 ∘ 𝐹) ∘ 𝑓) supp (0_g‘𝐻)) = (((𝐾 ∘ 𝐹) ∘ 𝑓) supp (0_g‘𝐻))
110	27, 3, 67, 90, 91, 78, 93, 98, 63, 57, 108, 109	gsumval3 19821	. . . . . 6 ⊢ ((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) → (𝐻 Σ_g (𝐾 ∘ 𝐹)) = (seq1((+_g‘𝐻), ((𝐾 ∘ 𝐹) ∘ 𝑓))‘(♯‘(^◡𝐹 “ (V ∖ { 0 })))))
111	76, 89, 110	3eqtr4rd 2775	. . . . 5 ⊢ ((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) → (𝐻 Σ_g (𝐾 ∘ 𝐹)) = (𝐾‘(𝐺 Σ_g 𝐹)))
112	111	expr 456	. . . 4 ⊢ ((𝜑 ∧ (♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ) → (𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })) → (𝐻 Σ_g (𝐾 ∘ 𝐹)) = (𝐾‘(𝐺 Σ_g 𝐹))))
113	112	exlimdv 1933	. . 3 ⊢ ((𝜑 ∧ (♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })) → (𝐻 Σ_g (𝐾 ∘ 𝐹)) = (𝐾‘(𝐺 Σ_g 𝐹))))
114	113	expimpd 453	. 2 ⊢ (𝜑 → (((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 }))) → (𝐻 Σ_g (𝐾 ∘ 𝐹)) = (𝐾‘(𝐺 Σ_g 𝐹))))
115		gsumzmhm.w	. . . . 5 ⊢ (𝜑 → 𝐹 finSupp 0 )
116	115	fsuppimpd 9296	. . . 4 ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin)
117	23, 116	eqeltrrd 2829	. . 3 ⊢ (𝜑 → (^◡𝐹 “ (V ∖ { 0 })) ∈ Fin)
118		fz1f1o 15652	. . 3 ⊢ ((^◡𝐹 “ (V ∖ { 0 })) ∈ Fin → ((^◡𝐹 “ (V ∖ { 0 })) = ∅ ∨ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))))
119	117, 118	syl 17	. 2 ⊢ (𝜑 → ((^◡𝐹 “ (V ∖ { 0 })) = ∅ ∨ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))))
120	45, 114, 119	mpjaod 860	1 ⊢ (𝜑 → (𝐻 Σ_g (𝐾 ∘ 𝐹)) = (𝐾‘(𝐺 Σ_g 𝐹)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1540 ∃wex 1779 ∈ wcel 2109 Vcvv 3444 ∖ cdif 3908 ⊆ wss 3911 ∅c0 4292 {csn 4585 class class class wbr 5102 ↦ cmpt 5183 ^◡ccnv 5630 ran crn 5632 “ cima 5634 ∘ ccom 5635 ⟶wf 6495 –1-1→wf1 6496 –onto→wfo 6497 –1-1-onto→wf1o 6498 ‘cfv 6499 (class class class)co 7369 supp csupp 8116 Fincfn 8895 finSupp cfsupp 9288 1c1 11045 ℕcn 12162 ℤ_≥cuz 12769 ...cfz 13444 seqcseq 13942 ♯chash 14271 Basecbs 17155 +_gcplusg 17196 0_gc0g 17378 Σ_g cgsu 17379 Mndcmnd 18643 MndHom cmhm 18690 Cntzccntz 19229

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-er 8648 df-map 8778 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-oi 9439 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-nn 12163 df-n0 12419 df-z 12506 df-uz 12770 df-fz 13445 df-fzo 13592 df-seq 13943 df-hash 14272 df-0g 17380 df-gsum 17381 df-mgm 18549 df-sgrp 18628 df-mnd 18644 df-mhm 18692 df-cntz 19231

This theorem is referenced by: gsummhm 19852 gsumzinv 19859

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