Theorem gsumzmhm 18337

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 = (0g‘𝐺)
gsumzmhm.w	⊢ (𝜑 → 𝐹 finSupp 0 )

Assertion

Ref	Expression
gsumzmhm	⊢ (𝜑 → (𝐻 Σg (𝐾 ∘ 𝐹)) = (𝐾‘(𝐺 Σg 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 2622	. . . . . . . 8 ⊢ (0g‘𝐻) = (0g‘𝐻)
4	3	gsumz 17374	. . . . . . 7 ⊢ ((𝐻 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐻 Σg (𝑘 ∈ 𝐴 ↦ (0g‘𝐻))) = (0g‘𝐻))
5	1, 2, 4	syl2anc 693	. . . . . 6 ⊢ (𝜑 → (𝐻 Σg (𝑘 ∈ 𝐴 ↦ (0g‘𝐻))) = (0g‘𝐻))
6	5	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐻 Σg (𝑘 ∈ 𝐴 ↦ (0g‘𝐻))) = (0g‘𝐻))
7		gsumzmhm.k	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (𝐺 MndHom 𝐻))
8		gsumzmhm.0	. . . . . . . 8 ⊢ 0 = (0g‘𝐺)
9	8, 3	mhm0 17343	. . . . . . 7 ⊢ (𝐾 ∈ (𝐺 MndHom 𝐻) → (𝐾‘ 0 ) = (0g‘𝐻))
10	7, 9	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐾‘ 0 ) = (0g‘𝐻))
11	10	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐾‘ 0 ) = (0g‘𝐻))
12	6, 11	eqtr4d 2659	. . . 4 ⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐻 Σg (𝑘 ∈ 𝐴 ↦ (0g‘𝐻))) = (𝐾‘ 0 ))
13		gsumzmhm.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ Mnd)
14		gsumzmhm.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐺)
15	14, 8	mndidcl 17308	. . . . . . . . 9 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵)
16	13, 15	syl 17	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ 𝐵)
17	16	ad2antrr 762	. . . . . . 7 ⊢ (((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) ∧ 𝑘 ∈ 𝐴) → 0 ∈ 𝐵)
18		gsumzmhm.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
19		fvex 6201	. . . . . . . . . 10 ⊢ (0g‘𝐺) ∈ V
20	8, 19	eqeltri 2697	. . . . . . . . 9 ⊢ 0 ∈ V
21	20	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ V)
22		fex 6490	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V)
23	18, 2, 22	syl2anc 693	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ V)
24		suppimacnv 7306	. . . . . . . . . 10 ⊢ ((𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 })))
25	23, 21, 24	syl2anc 693	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 })))
26		ssid 3624	. . . . . . . . 9 ⊢ (◡𝐹 “ (V ∖ { 0 })) ⊆ (◡𝐹 “ (V ∖ { 0 }))
27	25, 26	syl6eqss 3655	. . . . . . . 8 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ (◡𝐹 “ (V ∖ { 0 })))
28	18, 2, 21, 27	gsumcllem 18309	. . . . . . 7 ⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) → 𝐹 = (𝑘 ∈ 𝐴 ↦ 0 ))
29		eqid 2622	. . . . . . . . . . 11 ⊢ (Base‘𝐻) = (Base‘𝐻)
30	14, 29	mhmf 17340	. . . . . . . . . 10 ⊢ (𝐾 ∈ (𝐺 MndHom 𝐻) → 𝐾:𝐵⟶(Base‘𝐻))
31	7, 30	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐾:𝐵⟶(Base‘𝐻))
32	31	feqmptd 6249	. . . . . . . 8 ⊢ (𝜑 → 𝐾 = (𝑥 ∈ 𝐵 ↦ (𝐾‘𝑥)))
33	32	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) → 𝐾 = (𝑥 ∈ 𝐵 ↦ (𝐾‘𝑥)))
34		fveq2 6191	. . . . . . 7 ⊢ (𝑥 = 0 → (𝐾‘𝑥) = (𝐾‘ 0 ))
35	17, 28, 33, 34	fmptco 6396	. . . . . 6 ⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐾 ∘ 𝐹) = (𝑘 ∈ 𝐴 ↦ (𝐾‘ 0 )))
36	10	mpteq2dv 4745	. . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ (𝐾‘ 0 )) = (𝑘 ∈ 𝐴 ↦ (0g‘𝐻)))
37	36	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝑘 ∈ 𝐴 ↦ (𝐾‘ 0 )) = (𝑘 ∈ 𝐴 ↦ (0g‘𝐻)))
38	35, 37	eqtrd 2656	. . . . 5 ⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐾 ∘ 𝐹) = (𝑘 ∈ 𝐴 ↦ (0g‘𝐻)))
39	38	oveq2d 6666	. . . 4 ⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐻 Σg (𝐾 ∘ 𝐹)) = (𝐻 Σg (𝑘 ∈ 𝐴 ↦ (0g‘𝐻))))
40	28	oveq2d 6666	. . . . . 6 ⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )))
41	8	gsumz 17374	. . . . . . . 8 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )
42	13, 2, 41	syl2anc 693	. . . . . . 7 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )
43	42	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )
44	40, 43	eqtrd 2656	. . . . 5 ⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐺 Σg 𝐹) = 0 )
45	44	fveq2d 6195	. . . 4 ⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐾‘(𝐺 Σg 𝐹)) = (𝐾‘ 0 ))
46	12, 39, 45	3eqtr4d 2666	. . 3 ⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐻 Σg (𝐾 ∘ 𝐹)) = (𝐾‘(𝐺 Σg 𝐹)))
47	46	ex 450	. 2 ⊢ (𝜑 → ((◡𝐹 “ (V ∖ { 0 })) = ∅ → (𝐻 Σg (𝐾 ∘ 𝐹)) = (𝐾‘(𝐺 Σg 𝐹))))
48	13	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝐺 ∈ Mnd)
49		eqid 2622	. . . . . . . . . 10 ⊢ (+g‘𝐺) = (+g‘𝐺)
50	14, 49	mndcl 17301	. . . . . . . . 9 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+g‘𝐺)𝑦) ∈ 𝐵)
51	50	3expb 1266	. . . . . . . 8 ⊢ ((𝐺 ∈ Mnd ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐺)𝑦) ∈ 𝐵)
52	48, 51	sylan 488	. . . . . . 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)
53	18	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝐹:𝐴⟶𝐵)
54		f1of1 6136	. . . . . . . . . . . 12 ⊢ (𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) → 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1→(◡𝐹 “ (V ∖ { 0 })))
55	54	ad2antll 765	. . . . . . . . . . 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 })))
56		cnvimass 5485	. . . . . . . . . . . 12 ⊢ (◡𝐹 “ (V ∖ { 0 })) ⊆ dom 𝐹
57		fdm 6051	. . . . . . . . . . . . 13 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
58	53, 57	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → dom 𝐹 = 𝐴)
59	56, 58	syl5sseq 3653	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (◡𝐹 “ (V ∖ { 0 })) ⊆ 𝐴)
60		f1ss 6106	. . . . . . . . . . 11 ⊢ ((𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1→(◡𝐹 “ (V ∖ { 0 })) ∧ (◡𝐹 “ (V ∖ { 0 })) ⊆ 𝐴) → 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1→𝐴)
61	55, 59, 60	syl2anc 693	. . . . . . . . . 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)
62		f1f 6101	. . . . . . . . . 10 ⊢ (𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1→𝐴 → 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))⟶𝐴)
63	61, 62	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)
64		fco 6058	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))⟶𝐴) → (𝐹 ∘ 𝑓):(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))⟶𝐵)
65	53, 63, 64	syl2anc 693	. . . . . . . 8 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝐹 ∘ 𝑓):(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))⟶𝐵)
66	65	ffvelrnda 6359	. . . . . . 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)
67		simprl 794	. . . . . . . 8 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ)
68		nnuz 11723	. . . . . . . 8 ⊢ ℕ = (ℤ≥‘1)
69	67, 68	syl6eleq 2711	. . . . . . 7 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ (ℤ≥‘1))
70	7	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝐾 ∈ (𝐺 MndHom 𝐻))
71		eqid 2622	. . . . . . . . . 10 ⊢ (+g‘𝐻) = (+g‘𝐻)
72	14, 49, 71	mhmlin 17342	. . . . . . . . 9 ⊢ ((𝐾 ∈ (𝐺 MndHom 𝐻) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝐾‘(𝑥(+g‘𝐺)𝑦)) = ((𝐾‘𝑥)(+g‘𝐻)(𝐾‘𝑦)))
73	72	3expb 1266	. . . . . . . 8 ⊢ ((𝐾 ∈ (𝐺 MndHom 𝐻) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐾‘(𝑥(+g‘𝐺)𝑦)) = ((𝐾‘𝑥)(+g‘𝐻)(𝐾‘𝑦)))
74	70, 73	sylan 488	. . . . . . 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)))
75		coass 5654	. . . . . . . . 9 ⊢ ((𝐾 ∘ 𝐹) ∘ 𝑓) = (𝐾 ∘ (𝐹 ∘ 𝑓))
76	75	fveq1i 6192	. . . . . . . 8 ⊢ (((𝐾 ∘ 𝐹) ∘ 𝑓)‘𝑥) = ((𝐾 ∘ (𝐹 ∘ 𝑓))‘𝑥)
77		fvco3 6275	. . . . . . . . 9 ⊢ (((𝐹 ∘ 𝑓):(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))⟶𝐵 ∧ 𝑥 ∈ (1...(#‘(◡𝐹 “ (V ∖ { 0 }))))) → ((𝐾 ∘ (𝐹 ∘ 𝑓))‘𝑥) = (𝐾‘((𝐹 ∘ 𝑓)‘𝑥)))
78	65, 77	sylan 488	. . . . . . . 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)))
79	76, 78	syl5req 2669	. . . . . . 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))
80	52, 66, 69, 74, 79	seqhomo 12848	. . . . . 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 })))))
81		gsumzmhm.z	. . . . . . . 8 ⊢ 𝑍 = (Cntz‘𝐺)
82	2	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝐴 ∈ 𝑉)
83		gsumzmhm.c	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
84	83	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
85	27	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝐹 supp 0 ) ⊆ (◡𝐹 “ (V ∖ { 0 })))
86		f1ofo 6144	. . . . . . . . . . 11 ⊢ (𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) → 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–onto→(◡𝐹 “ (V ∖ { 0 })))
87		forn 6118	. . . . . . . . . . 11 ⊢ (𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–onto→(◡𝐹 “ (V ∖ { 0 })) → ran 𝑓 = (◡𝐹 “ (V ∖ { 0 })))
88	86, 87	syl 17	. . . . . . . . . 10 ⊢ (𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) → ran 𝑓 = (◡𝐹 “ (V ∖ { 0 })))
89	88	ad2antll 765	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → ran 𝑓 = (◡𝐹 “ (V ∖ { 0 })))
90	85, 89	sseqtr4d 3642	. . . . . . . 8 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝐹 supp 0 ) ⊆ ran 𝑓)
91		eqid 2622	. . . . . . . 8 ⊢ ((𝐹 ∘ 𝑓) supp 0 ) = ((𝐹 ∘ 𝑓) supp 0 )
92	14, 8, 49, 81, 48, 82, 53, 84, 67, 61, 90, 91	gsumval3 18308	. . . . . . 7 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝐺 Σg 𝐹) = (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(#‘(◡𝐹 “ (V ∖ { 0 })))))
93	92	fveq2d 6195	. . . . . 6 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝐾‘(𝐺 Σg 𝐹)) = (𝐾‘(seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(#‘(◡𝐹 “ (V ∖ { 0 }))))))
94		eqid 2622	. . . . . . 7 ⊢ (Cntz‘𝐻) = (Cntz‘𝐻)
95	1	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝐻 ∈ Mnd)
96	31	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝐾:𝐵⟶(Base‘𝐻))
97		fco 6058	. . . . . . . 8 ⊢ ((𝐾:𝐵⟶(Base‘𝐻) ∧ 𝐹:𝐴⟶𝐵) → (𝐾 ∘ 𝐹):𝐴⟶(Base‘𝐻))
98	96, 53, 97	syl2anc 693	. . . . . . 7 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝐾 ∘ 𝐹):𝐴⟶(Base‘𝐻))
99	81, 94	cntzmhm2 17772	. . . . . . . . 9 ⊢ ((𝐾 ∈ (𝐺 MndHom 𝐻) ∧ ran 𝐹 ⊆ (𝑍‘ran 𝐹)) → (𝐾 “ ran 𝐹) ⊆ ((Cntz‘𝐻)‘(𝐾 “ ran 𝐹)))
100	70, 84, 99	syl2anc 693	. . . . . . . 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)))
101		rnco2 5642	. . . . . . . 8 ⊢ ran (𝐾 ∘ 𝐹) = (𝐾 “ ran 𝐹)
102	101	fveq2i 6194	. . . . . . . 8 ⊢ ((Cntz‘𝐻)‘ran (𝐾 ∘ 𝐹)) = ((Cntz‘𝐻)‘(𝐾 “ ran 𝐹))
103	100, 101, 102	3sstr4g 3646	. . . . . . 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)))
104		eldifi 3732	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 ∖ (◡𝐹 “ (V ∖ { 0 }))) → 𝑥 ∈ 𝐴)
105		fvco3 6275	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐾 ∘ 𝐹)‘𝑥) = (𝐾‘(𝐹‘𝑥)))
106	53, 104, 105	syl2an 494	. . . . . . . . . 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)))
107	20	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 0 ∈ V)
108	53, 85, 82, 107	suppssr 7326	. . . . . . . . . . 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 )
109	108	fveq2d 6195	. . . . . . . . . 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 ))
110	10	ad2antrr 762	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) ∧ 𝑥 ∈ (𝐴 ∖ (◡𝐹 “ (V ∖ { 0 })))) → (𝐾‘ 0 ) = (0g‘𝐻))
111	106, 109, 110	3eqtrd 2660	. . . . . . . . 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) = (0g‘𝐻))
112	98, 111	suppss 7325	. . . . . . . 8 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → ((𝐾 ∘ 𝐹) supp (0g‘𝐻)) ⊆ (◡𝐹 “ (V ∖ { 0 })))
113	112, 89	sseqtr4d 3642	. . . . . . 7 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → ((𝐾 ∘ 𝐹) supp (0g‘𝐻)) ⊆ ran 𝑓)
114		eqid 2622	. . . . . . 7 ⊢ (((𝐾 ∘ 𝐹) ∘ 𝑓) supp (0g‘𝐻)) = (((𝐾 ∘ 𝐹) ∘ 𝑓) supp (0g‘𝐻))
115	29, 3, 71, 94, 95, 82, 98, 103, 67, 61, 113, 114	gsumval3 18308	. . . . . 6 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝐻 Σg (𝐾 ∘ 𝐹)) = (seq1((+g‘𝐻), ((𝐾 ∘ 𝐹) ∘ 𝑓))‘(#‘(◡𝐹 “ (V ∖ { 0 })))))
116	80, 93, 115	3eqtr4rd 2667	. . . . 5 ⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝐻 Σg (𝐾 ∘ 𝐹)) = (𝐾‘(𝐺 Σg 𝐹)))
117	116	expr 643	. . . 4 ⊢ ((𝜑 ∧ (#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ) → (𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) → (𝐻 Σg (𝐾 ∘ 𝐹)) = (𝐾‘(𝐺 Σg 𝐹))))
118	117	exlimdv 1861	. . 3 ⊢ ((𝜑 ∧ (#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ) → (∃𝑓 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) → (𝐻 Σg (𝐾 ∘ 𝐹)) = (𝐾‘(𝐺 Σg 𝐹))))
119	118	expimpd 629	. 2 ⊢ (𝜑 → (((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 }))) → (𝐻 Σg (𝐾 ∘ 𝐹)) = (𝐾‘(𝐺 Σg 𝐹))))
120		gsumzmhm.w	. . . . 5 ⊢ (𝜑 → 𝐹 finSupp 0 )
121	120	fsuppimpd 8282	. . . 4 ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin)
122	25, 121	eqeltrrd 2702	. . 3 ⊢ (𝜑 → (◡𝐹 “ (V ∖ { 0 })) ∈ Fin)
123		fz1f1o 14441	. . 3 ⊢ ((◡𝐹 “ (V ∖ { 0 })) ∈ Fin → ((◡𝐹 “ (V ∖ { 0 })) = ∅ ∨ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))))
124	122, 123	syl 17	. 2 ⊢ (𝜑 → ((◡𝐹 “ (V ∖ { 0 })) = ∅ ∨ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))))
125	47, 119, 124	mpjaod 396	1 ⊢ (𝜑 → (𝐻 Σg (𝐾 ∘ 𝐹)) = (𝐾‘(𝐺 Σg 𝐹)))

Syntax hints:   → wi 4   ∨ wo 383   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574  ∅c0 3915  {csn 4177   class class class wbr 4653   ↦ cmpt 4729  ^◡ccnv 5113  dom cdm 5114  ran crn 5115   “ cima 5117   ∘ ccom 5118  ⟶wf 5884  –1-1→wf1 5885  –onto→wfo 5886  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650   supp csupp 7295  Fincfn 7955   finSupp cfsupp 8275  1c1 9937  ℕcn 11020  ℤ_≥cuz 11687  ...cfz 12326  seqcseq 12801  #chash 13117  Basecbs 15857  +_gcplusg 15941  0_gc0g 16100   Σ_g cgsu 16101  Mndcmnd 17294   MndHom cmhm 17333  Cntzccntz 17748

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-0g 16102  df-gsum 16103  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-cntz 17750

This theorem is referenced by:  gsummhm  18338  gsumzinv  18345