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Theorem gsumzmhm 20151

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 2761	. . . . . . . 8 ⊢ (0_g‘𝐻) = (0_g‘𝐻)
4	3	gsumz 19032	. . . . . . 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 18989	. . . . . . 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 2799	. . . 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 18939	. . . . . . . . 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 6899	. . . . . . . . 9 ⊢ 0 ∈ V
20	19	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ V)
21	18, 2	fexd 7233	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ V)
22		suppimacnv 8191	. . . . . . . . . 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 20122	. . . . . . 7 ⊢ ((𝜑 ∧ (^◡𝐹 “ (V ∖ { 0 })) = ∅) → 𝐹 = (𝑘 ∈ 𝐴 ↦ 0 ))
27		eqid 2761	. . . . . . . . . . 11 ⊢ (Base‘𝐻) = (Base‘𝐻)
28	14, 27	mhmf 18984	. . . . . . . . . 10 ⊢ (𝐾 ∈ (𝐺 MndHom 𝐻) → 𝐾:𝐵⟶(Base‘𝐻))
29	7, 28	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐾:𝐵⟶(Base‘𝐻))
30	29	feqmptd 6953	. . . . . . . 8 ⊢ (𝜑 → 𝐾 = (𝑥 ∈ 𝐵 ↦ (𝐾‘𝑥)))
31	30	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (^◡𝐹 “ (V ∖ { 0 })) = ∅) → 𝐾 = (𝑥 ∈ 𝐵 ↦ (𝐾‘𝑥)))
32		fveq2 6885	. . . . . . 7 ⊢ (𝑥 = 0 → (𝐾‘𝑥) = (𝐾‘ 0 ))
33	17, 26, 31, 32	fmptco 7130	. . . . . 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 2796	. . . . 5 ⊢ ((𝜑 ∧ (^◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐾 ∘ 𝐹) = (𝑘 ∈ 𝐴 ↦ (0_g‘𝐻)))
37	36	oveq2d 7436	. . . 4 ⊢ ((𝜑 ∧ (^◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐻 Σ_g (𝐾 ∘ 𝐹)) = (𝐻 Σ_g (𝑘 ∈ 𝐴 ↦ (0_g‘𝐻))))
38	26	oveq2d 7436	. . . . . 6 ⊢ ((𝜑 ∧ (^◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐺 Σ_g 𝐹) = (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 0 )))
39	8	gsumz 19032	. . . . . . . 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 2796	. . . . 5 ⊢ ((𝜑 ∧ (^◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐺 Σ_g 𝐹) = 0 )
43	42	fveq2d 6889	. . . 4 ⊢ ((𝜑 ∧ (^◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐾‘(𝐺 Σ_g 𝐹)) = (𝐾‘ 0 ))
44	12, 37, 43	3eqtr4d 2806	. . 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 2761	. . . . . . . . . 10 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
48	14, 47	mndcl 18931	. . . . . . . . 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 6823	. . . . . . . . . . . 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 6198	. . . . . . . . . . . 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 6729	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘(^◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })))) → (^◡𝐹 “ (V ∖ { 0 })) ⊆ 𝐴)
56		f1ss 6785	. . . . . . . . . . 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 6778	. . . . . . . . . 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 6734	. . . . . . . . 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 7084	. . . . . . 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 13004	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘1)
65	63, 64	eleqtrdi 2871	. . . . . . 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 2761	. . . . . . . . . 10 ⊢ (+_g‘𝐻) = (+_g‘𝐻)
68	14, 47, 67	mhmlin 18988	. . . . . . . . 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 6267	. . . . . . . . 9 ⊢ ((𝐾 ∘ 𝐹) ∘ 𝑓) = (𝐾 ∘ (𝐹 ∘ 𝑓))
72	71	fveq1i 6886	. . . . . . . 8 ⊢ (((𝐾 ∘ 𝐹) ∘ 𝑓)‘𝑥) = ((𝐾 ∘ (𝐹 ∘ 𝑓))‘𝑥)
73		fvco3 6985	. . . . . . . . 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 2809	. . . . . . 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 14192	. . . . . 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 6832	. . . . . . . . . . 11 ⊢ (𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(^◡𝐹 “ (V ∖ { 0 })) → 𝑓:(1...(♯‘(^◡𝐹 “ (V ∖ { 0 }))))–onto→(^◡𝐹 “ (V ∖ { 0 })))
83		forn 6799	. . . . . . . . . . 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 2761	. . . . . . . 8 ⊢ ((𝐹 ∘ 𝑓) supp 0 ) = ((𝐹 ∘ 𝑓) supp 0 )
88	14, 8, 47, 77, 46, 78, 54, 80, 63, 57, 86, 87	gsumval3 20121	. . . . . . 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 6889	. . . . . 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 2761	. . . . . . 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 6734	. . . . . . . 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 19556	. . . . . . . . 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 6255	. . . . . . . 8 ⊢ ran (𝐾 ∘ 𝐹) = (𝐾 “ ran 𝐹)
97	96	fveq2i 6888	. . . . . . . 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 6985	. . . . . . . . . . 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 8212	. . . . . . . . . . 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 6889	. . . . . . . . . 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 2800	. . . . . . . . 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 8211	. . . . . . . 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 2761	. . . . . . 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 20121	. . . . . 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 2807	. . . . 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 9361	. . . 4 ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin)
117	23, 116	eqeltrrd 2862	. . 3 ⊢ (𝜑 → (^◡𝐹 “ (V ∖ { 0 })) ∈ Fin)
118		fz1f1o 15876	. . 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 3451 ∖ cdif 3896 ⊆ wss 3899 ∅c0 4279 {csn 4584 class class class wbr 5103 ↦ cmpt 5186 ^◡ccnv 5650 ran crn 5652 “ cima 5654 ∘ ccom 5655 ⟶wf 6534 –1-1→wf1 6535 –onto→wfo 6536 –1-1-onto→wf1o 6537 ‘cfv 6538 (class class class)co 7420 supp csupp 8177 Fincfn 8973 finSupp cfsupp 9353 1c1 11201 ℕcn 12335 ℤ_≥cuz 12965 ...cfz 13639 seqcseq 14144 ♯chash 14474 Basecbs 17387 +_gcplusg 17428 0_gc0g 17610 Σ_g cgsu 17611 Mndcmnd 18923 MndHom cmhm 18976 Cntzccntz 19529

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 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277

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 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 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 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8178 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-er 8717 df-map 8849 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fsupp 9354 df-oi 9504 df-card 10020 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-nn 12336 df-n0 12607 df-z 12694 df-uz 12966 df-fz 13640 df-fzo 13789 df-seq 14145 df-hash 14475 df-0g 17612 df-gsum 17613 df-mgm 18816 df-sgrp 18908 df-mnd 18924 df-mhm 18978 df-cntz 19531

This theorem is used by: gsummhm 20152 gsumzinv 20159

W3C validator