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Theorem gsumzmhm 20067

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 2760	. . . . . . . 8 ⊢ (0_g‘𝐻) = (0_g‘𝐻)
4	3	gsumz 18948	. . . . . . 7 ⊢ ((𝐻 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐻 Σ_g (𝑘 ∈ 𝐴 ↦ (0_g‘𝐻))) = (0_g‘𝐻))
5	1, 2, 4	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (𝐻 Σ_g (𝑘 ∈ 𝐴 ↦ (0_g‘𝐻))) = (0_g‘𝐻))
6	5	adantr 486	. . . . 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 18905	. . . . . . 7 ⊢ (𝐾 ∈ (𝐺 MndHom 𝐻) → (𝐾‘ 0 ) = (0_g‘𝐻))
10	7, 9	syl 18	. . . . . 6 ⊢ (𝜑 → (𝐾‘ 0 ) = (0_g‘𝐻))
11	10	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (^◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐾‘ 0 ) = (0_g‘𝐻))
12	6, 11	eqtr4d 2798	. . . 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 18855	. . . . . . . . 9 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵)
16	13, 15	syl 18	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ 𝐵)
17	16	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ (^◡𝐹 “ (V ∖ { 0 })) = ∅) ∧ 𝑘 ∈ 𝐴) → 0 ∈ 𝐵)
18		gsumzmhm.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
19	8	fvexi 6893	. . . . . . . . 9 ⊢ 0 ∈ V
20	19	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ V)
21	18, 2	fexd 7227	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ V)
22		suppimacnv 8173	. . . . . . . . . 10 ⊢ ((𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 supp 0 ) = (^◡𝐹 “ (V ∖ { 0 })))
23	21, 20, 22	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 supp 0 ) = (^◡𝐹 “ (V ∖ { 0 })))
24		ssid 3953	. . . . . . . . 9 ⊢ (^◡𝐹 “ (V ∖ { 0 })) ⊆ (^◡𝐹 “ (V ∖ { 0 }))
25	23, 24	eqsstrdi 3975	. . . . . . . 8 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ (^◡𝐹 “ (V ∖ { 0 })))
26	18, 2, 20, 25	gsumcllem 20038	. . . . . . 7 ⊢ ((𝜑 ∧ (^◡𝐹 “ (V ∖ { 0 })) = ∅) → 𝐹 = (𝑘 ∈ 𝐴 ↦ 0 ))
27		eqid 2760	. . . . . . . . . . 11 ⊢ (Base‘𝐻) = (Base‘𝐻)
28	14, 27	mhmf 18900	. . . . . . . . . 10 ⊢ (𝐾 ∈ (𝐺 MndHom 𝐻) → 𝐾:𝐵⟶(Base‘𝐻))
29	7, 28	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐾:𝐵⟶(Base‘𝐻))
30	29	feqmptd 6947	. . . . . . . 8 ⊢ (𝜑 → 𝐾 = (𝑥 ∈ 𝐵 ↦ (𝐾‘𝑥)))
31	30	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (^◡𝐹 “ (V ∖ { 0 })) = ∅) → 𝐾 = (𝑥 ∈ 𝐵 ↦ (𝐾‘𝑥)))
32		fveq2 6879	. . . . . . 7 ⊢ (𝑥 = 0 → (𝐾‘𝑥) = (𝐾‘ 0 ))
33	17, 26, 31, 32	fmptco 7124	. . . . . 6 ⊢ ((𝜑 ∧ (^◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐾 ∘ 𝐹) = (𝑘 ∈ 𝐴 ↦ (𝐾‘ 0 )))
34	10	mpteq2dv 5199	. . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ (𝐾‘ 0 )) = (𝑘 ∈ 𝐴 ↦ (0_g‘𝐻)))
35	34	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (^◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝑘 ∈ 𝐴 ↦ (𝐾‘ 0 )) = (𝑘 ∈ 𝐴 ↦ (0_g‘𝐻)))
36	33, 35	eqtrd 2795	. . . . 5 ⊢ ((𝜑 ∧ (^◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐾 ∘ 𝐹) = (𝑘 ∈ 𝐴 ↦ (0_g‘𝐻)))
37	36	oveq2d 7430	. . . 4 ⊢ ((𝜑 ∧ (^◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐻 Σ_g (𝐾 ∘ 𝐹)) = (𝐻 Σ_g (𝑘 ∈ 𝐴 ↦ (0_g‘𝐻))))
38	26	oveq2d 7430	. . . . . 6 ⊢ ((𝜑 ∧ (^◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐺 Σ_g 𝐹) = (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 0 )))
39	8	gsumz 18948	. . . . . . . 8 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )
40	13, 2, 39	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )
41	40	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (^◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )
42	38, 41	eqtrd 2795	. . . . 5 ⊢ ((𝜑 ∧ (^◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐺 Σ_g 𝐹) = 0 )
43	42	fveq2d 6883	. . . 4 ⊢ ((𝜑 ∧ (^◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐾‘(𝐺 Σ_g 𝐹)) = (𝐾‘ 0 ))
44	12, 37, 43	3eqtr4d 2805	. . 3 ⊢ ((𝜑 ∧ (^◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐻 Σ_g (𝐾 ∘ 𝐹)) = (𝐾‘(𝐺 Σ_g 𝐹)))
45	44	ex 418	. 2 ⊢ (𝜑 → ((^◡𝐹 “ (V ∖ { 0 })) = ∅ → (𝐻 Σ_g (𝐾 ∘ 𝐹)) = (𝐾‘(𝐺 Σ_g 𝐹))))
46	13	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) → 𝐺 ∈ Mnd)
47		eqid 2760	. . . . . . . . . 10 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
48	14, 47	mndcl 18847	. . . . . . . . 9 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+_g‘𝐺)𝑦) ∈ 𝐵)
49	48	3expb 1138	. . . . . . . 8 ⊢ ((𝐺 ∈ Mnd ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+_g‘𝐺)𝑦) ∈ 𝐵)
50	46, 49	sylan 592	. . . . . . 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 6817	. . . . . . . . . . . 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 742	. . . . . . . . . . 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 6078	. . . . . . . . . . . 12 ⊢ (^◡𝐹 “ (V ∖ { 0 })) ⊆ dom 𝐹
54	18	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) → 𝐹:𝐴⟶𝐵)
55	53, 54	fssdm 6723	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) → (^◡𝐹 “ (V ∖ { 0 })) ⊆ 𝐴)
56		f1ss 6779	. . . . . . . . . . 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 596	. . . . . . . . . 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 6772	. . . . . . . . . 10 ⊢ (𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1→𝐴 → 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))⟶𝐴)
59	57, 58	syl 18	. . . . . . . . 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 6728	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))⟶𝐴) → (𝐹 ∘ 𝑓):(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))⟶𝐵)
61	18, 59, 60	syl2an2r 698	. . . . . . . 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 7078	. . . . . . 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 783	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) → (♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ)
64		nnuz 12929	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘1)
65	63, 64	eleqtrdi 2870	. . . . . . 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 486	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) → 𝐾 ∈ (𝐺 MndHom 𝐻))
67		eqid 2760	. . . . . . . . . 10 ⊢ (+_g‘𝐻) = (+_g‘𝐻)
68	14, 47, 67	mhmlin 18904	. . . . . . . . 9 ⊢ ((𝐾 ∈ (𝐺 MndHom 𝐻) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝐾‘(𝑥(+_g‘𝐺)𝑦)) = ((𝐾‘𝑥)(+_g‘𝐻)(𝐾‘𝑦)))
69	68	3expb 1138	. . . . . . . 8 ⊢ ((𝐾 ∈ (𝐺 MndHom 𝐻) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐾‘(𝑥(+_g‘𝐺)𝑦)) = ((𝐾‘𝑥)(+_g‘𝐻)(𝐾‘𝑦)))
70	66, 69	sylan 592	. . . . . . 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 6262	. . . . . . . . 9 ⊢ ((𝐾 ∘ 𝐹) ∘ 𝑓) = (𝐾 ∘ (𝐹 ∘ 𝑓))
72	71	fveq1i 6880	. . . . . . . 8 ⊢ (((𝐾 ∘ 𝐹) ∘ 𝑓)‘𝑥) = ((𝐾 ∘ (𝐹 ∘ 𝑓))‘𝑥)
73		fvco3 6979	. . . . . . . . 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 592	. . . . . . . 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 2808	. . . . . . 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 14116	. . . . . 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 486	. . . . . . . 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 486	. . . . . . . 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 486	. . . . . . . . 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 6826	. . . . . . . . . . 11 ⊢ (𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })) → 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–onto→(^◡𝐹 “ (V ∖ { 0 })))
83		forn 6793	. . . . . . . . . . 11 ⊢ (𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–onto→(^◡𝐹 “ (V ∖ { 0 })) → ran 𝑓 = (^◡𝐹 “ (V ∖ { 0 })))
84	82, 83	syl 18	. . . . . . . . . 10 ⊢ (𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })) → ran 𝑓 = (^◡𝐹 “ (V ∖ { 0 })))
85	84	ad2antll 742	. . . . . . . . 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 3968	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) → (𝐹 supp 0 ) ⊆ ran 𝑓)
87		eqid 2760	. . . . . . . 8 ⊢ ((𝐹 ∘ 𝑓) supp 0 ) = ((𝐹 ∘ 𝑓) supp 0 )
88	14, 8, 47, 77, 46, 78, 54, 80, 63, 57, 86, 87	gsumval3 20037	. . . . . . 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 6883	. . . . . 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 2760	. . . . . . 7 ⊢ (Cntz‘𝐻) = (Cntz‘𝐻)
91	1	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) → 𝐻 ∈ Mnd)
92		fco 6728	. . . . . . . 8 ⊢ ((𝐾:𝐵⟶(Base‘𝐻) ∧ 𝐹:𝐴⟶𝐵) → (𝐾 ∘ 𝐹):𝐴⟶(Base‘𝐻))
93	29, 54, 92	syl2an2r 698	. . . . . . 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 19472	. . . . . . . . 9 ⊢ ((𝐾 ∈ (𝐺 MndHom 𝐻) ∧ ran 𝐹 ⊆ (𝑍‘ran 𝐹)) → (𝐾 “ ran 𝐹) ⊆ ((Cntz‘𝐻)‘(𝐾 “ ran 𝐹)))
95	7, 80, 94	syl2an2r 698	. . . . . . . 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 6250	. . . . . . . 8 ⊢ ran (𝐾 ∘ 𝐹) = (𝐾 “ ran 𝐹)
97	96	fveq2i 6882	. . . . . . . 8 ⊢ ((Cntz‘𝐻)‘ran (𝐾 ∘ 𝐹)) = ((Cntz‘𝐻)‘(𝐾 “ ran 𝐹))
98	95, 96, 97	3sstr4g 3984	. . . . . . 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 4078	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 ∖ (^◡𝐹 “ (V ∖ { 0 }))) → 𝑥 ∈ 𝐴)
100		fvco3 6979	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐾 ∘ 𝐹)‘𝑥) = (𝐾‘(𝐹‘𝑥)))
101	54, 99, 100	syl2an 608	. . . . . . . . . 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 8194	. . . . . . . . . . 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 6883	. . . . . . . . . 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 739	. . . . . . . . . 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 2799	. . . . . . . . 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 8193	. . . . . . . 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 3968	. . . . . . 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 2760	. . . . . . 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 20037	. . . . . 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 2806	. . . . 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 462	. . . 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 1966	. . 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 459	. 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 9342	. . . 4 ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin)
117	23, 116	eqeltrrd 2861	. . 3 ⊢ (𝜑 → (^◡𝐹 “ (V ∖ { 0 })) ∈ Fin)
118		fz1f1o 15799	. . 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 18	. 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 874	1 ⊢ (𝜑 → (𝐻 Σ_g (𝐾 ∘ 𝐹)) = (𝐾‘(𝐺 Σ_g 𝐹)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∨ wo 861 = wceq 1570 ∃wex 1812 ∈ wcel 2145 Vcvv 3450 ∖ cdif 3896 ⊆ wss 3899 ∅c0 4279 {csn 4584 class class class wbr 5103 ↦ cmpt 5186 ^◡ccnv 5654 ran crn 5656 “ cima 5658 ∘ ccom 5659 ⟶wf 6529 –1-1→wf1 6530 –onto→wfo 6531 –1-1-onto→wf1o 6532 ‘cfv 6533 (class class class)co 7414 supp csupp 8159 Fincfn 8955 finSupp cfsupp 9334 1c1 11128 ℕcn 12260 ℤ_≥cuz 12890 ...cfz 13564 seqcseq 14068 ♯chash 14397 Basecbs 17304 +_gcplusg 17345 0_gc0g 17527 Σ_g cgsu 17528 Mndcmnd 18839 MndHom cmhm 18892 Cntzccntz 19445

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7864 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8458 df-er 8699 df-map 8831 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9335 df-oi 9485 df-card 9947 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-nn 12261 df-n0 12532 df-z 12619 df-uz 12891 df-fz 13565 df-fzo 13713 df-seq 14069 df-hash 14398 df-0g 17529 df-gsum 17530 df-mgm 18733 df-sgrp 18824 df-mnd 18840 df-mhm 18894 df-cntz 19447

This theorem is used by: gsummhm 20068 gsumzinv 20075

W3C validator