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Theorem gsumzmhm 19859

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 2733	. . . . . . . 8 ⊢ (0_g‘𝐻) = (0_g‘𝐻)
4	3	gsumz 18754	. . . . . . 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 18712	. . . . . . 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 2771	. . . 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 18667	. . . . . . . . 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 6845	. . . . . . . . 9 ⊢ 0 ∈ V
20	19	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ V)
21	18, 2	fexd 7170	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ V)
22		suppimacnv 8113	. . . . . . . . . 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 3954	. . . . . . . . 9 ⊢ (^◡𝐹 “ (V ∖ { 0 })) ⊆ (^◡𝐹 “ (V ∖ { 0 }))
25	23, 24	eqsstrdi 3976	. . . . . . . 8 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ (^◡𝐹 “ (V ∖ { 0 })))
26	18, 2, 20, 25	gsumcllem 19830	. . . . . . 7 ⊢ ((𝜑 ∧ (^◡𝐹 “ (V ∖ { 0 })) = ∅) → 𝐹 = (𝑘 ∈ 𝐴 ↦ 0 ))
27		eqid 2733	. . . . . . . . . . 11 ⊢ (Base‘𝐻) = (Base‘𝐻)
28	14, 27	mhmf 18707	. . . . . . . . . 10 ⊢ (𝐾 ∈ (𝐺 MndHom 𝐻) → 𝐾:𝐵⟶(Base‘𝐻))
29	7, 28	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐾:𝐵⟶(Base‘𝐻))
30	29	feqmptd 6899	. . . . . . . 8 ⊢ (𝜑 → 𝐾 = (𝑥 ∈ 𝐵 ↦ (𝐾‘𝑥)))
31	30	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (^◡𝐹 “ (V ∖ { 0 })) = ∅) → 𝐾 = (𝑥 ∈ 𝐵 ↦ (𝐾‘𝑥)))
32		fveq2 6831	. . . . . . 7 ⊢ (𝑥 = 0 → (𝐾‘𝑥) = (𝐾‘ 0 ))
33	17, 26, 31, 32	fmptco 7071	. . . . . 6 ⊢ ((𝜑 ∧ (^◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐾 ∘ 𝐹) = (𝑘 ∈ 𝐴 ↦ (𝐾‘ 0 )))
34	10	mpteq2dv 5189	. . . . . . 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 2768	. . . . 5 ⊢ ((𝜑 ∧ (^◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐾 ∘ 𝐹) = (𝑘 ∈ 𝐴 ↦ (0_g‘𝐻)))
37	36	oveq2d 7371	. . . 4 ⊢ ((𝜑 ∧ (^◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐻 Σ_g (𝐾 ∘ 𝐹)) = (𝐻 Σ_g (𝑘 ∈ 𝐴 ↦ (0_g‘𝐻))))
38	26	oveq2d 7371	. . . . . 6 ⊢ ((𝜑 ∧ (^◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐺 Σ_g 𝐹) = (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 0 )))
39	8	gsumz 18754	. . . . . . . 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 2768	. . . . 5 ⊢ ((𝜑 ∧ (^◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐺 Σ_g 𝐹) = 0 )
43	42	fveq2d 6835	. . . 4 ⊢ ((𝜑 ∧ (^◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐾‘(𝐺 Σ_g 𝐹)) = (𝐾‘ 0 ))
44	12, 37, 43	3eqtr4d 2778	. . 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 2733	. . . . . . . . . 10 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
48	14, 47	mndcl 18660	. . . . . . . . 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 6770	. . . . . . . . . . . 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 6038	. . . . . . . . . . . 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 6678	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) → (^◡𝐹 “ (V ∖ { 0 })) ⊆ 𝐴)
56		f1ss 6732	. . . . . . . . . . 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 6727	. . . . . . . . . 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 6683	. . . . . . . . 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 7026	. . . . . . 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 12785	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘1)
65	63, 64	eleqtrdi 2843	. . . . . . 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 2733	. . . . . . . . . 10 ⊢ (+_g‘𝐻) = (+_g‘𝐻)
68	14, 47, 67	mhmlin 18711	. . . . . . . . 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 6221	. . . . . . . . 9 ⊢ ((𝐾 ∘ 𝐹) ∘ 𝑓) = (𝐾 ∘ (𝐹 ∘ 𝑓))
72	71	fveq1i 6832	. . . . . . . 8 ⊢ (((𝐾 ∘ 𝐹) ∘ 𝑓)‘𝑥) = ((𝐾 ∘ (𝐹 ∘ 𝑓))‘𝑥)
73		fvco3 6930	. . . . . . . . 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 2781	. . . . . . 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 13966	. . . . . 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 6778	. . . . . . . . . . 11 ⊢ (𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })) → 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–onto→(^◡𝐹 “ (V ∖ { 0 })))
83		forn 6746	. . . . . . . . . . 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 3969	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) → (𝐹 supp 0 ) ⊆ ran 𝑓)
87		eqid 2733	. . . . . . . 8 ⊢ ((𝐹 ∘ 𝑓) supp 0 ) = ((𝐹 ∘ 𝑓) supp 0 )
88	14, 8, 47, 77, 46, 78, 54, 80, 63, 57, 86, 87	gsumval3 19829	. . . . . . 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 6835	. . . . . 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 2733	. . . . . . 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 6683	. . . . . . . 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 19264	. . . . . . . . 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 6209	. . . . . . . 8 ⊢ ran (𝐾 ∘ 𝐹) = (𝐾 “ ran 𝐹)
97	96	fveq2i 6834	. . . . . . . 8 ⊢ ((Cntz‘𝐻)‘ran (𝐾 ∘ 𝐹)) = ((Cntz‘𝐻)‘(𝐾 “ ran 𝐹))
98	95, 96, 97	3sstr4g 3985	. . . . . . 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 4082	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 ∖ (^◡𝐹 “ (V ∖ { 0 }))) → 𝑥 ∈ 𝐴)
100		fvco3 6930	. . . . . . . . . . 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 8134	. . . . . . . . . . 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 6835	. . . . . . . . . 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 2772	. . . . . . . . 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 8133	. . . . . . . 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 3969	. . . . . . 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 2733	. . . . . . 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 19829	. . . . . 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 2779	. . . . 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 1934	. . 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 9263	. . . 4 ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin)
117	23, 116	eqeltrrd 2834	. . 3 ⊢ (𝜑 → (^◡𝐹 “ (V ∖ { 0 })) ∈ Fin)
118		fz1f1o 15627	. . 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 1541 ∃wex 1780 ∈ wcel 2113 Vcvv 3438 ∖ cdif 3896 ⊆ wss 3899 ∅c0 4284 {csn 4577 class class class wbr 5095 ↦ cmpt 5176 ^◡ccnv 5620 ran crn 5622 “ cima 5624 ∘ ccom 5625 ⟶wf 6485 –1-1→wf1 6486 –onto→wfo 6487 –1-1-onto→wf1o 6488 ‘cfv 6489 (class class class)co 7355 supp csupp 8099 Fincfn 8878 finSupp cfsupp 9255 1c1 11017 ℕcn 12135 ℤ_≥cuz 12742 ...cfz 13417 seqcseq 13918 ♯chash 14247 Basecbs 17130 +_gcplusg 17171 0_gc0g 17353 Σ_g cgsu 17354 Mndcmnd 18652 MndHom cmhm 18699 Cntzccntz 19237

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-cnex 11072 ax-resscn 11073 ax-1cn 11074 ax-icn 11075 ax-addcl 11076 ax-addrcl 11077 ax-mulcl 11078 ax-mulrcl 11079 ax-mulcom 11080 ax-addass 11081 ax-mulass 11082 ax-distr 11083 ax-i2m1 11084 ax-1ne0 11085 ax-1rid 11086 ax-rnegex 11087 ax-rrecex 11088 ax-cnre 11089 ax-pre-lttri 11090 ax-pre-lttrn 11091 ax-pre-ltadd 11092 ax-pre-mulgt0 11093

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4861 df-int 4900 df-iun 4945 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-isom 6498 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-om 7806 df-1st 7930 df-2nd 7931 df-supp 8100 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-er 8631 df-map 8761 df-en 8879 df-dom 8880 df-sdom 8881 df-fin 8882 df-fsupp 9256 df-oi 9406 df-card 9842 df-pnf 11158 df-mnf 11159 df-xr 11160 df-ltxr 11161 df-le 11162 df-sub 11356 df-neg 11357 df-nn 12136 df-n0 12392 df-z 12479 df-uz 12743 df-fz 13418 df-fzo 13565 df-seq 13919 df-hash 14248 df-0g 17355 df-gsum 17356 df-mgm 18558 df-sgrp 18637 df-mnd 18653 df-mhm 18701 df-cntz 19239

This theorem is referenced by: gsummhm 19860 gsumzinv 19867

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