Theorem gsumzf1o 20126

Description: Re-index a finite group sum using a bijection. (Contributed by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 2-Jun-2019.)

Hypotheses

Ref	Expression
gsumzcl.b	⊢ 𝐵 = (Base‘𝐺)
gsumzcl.0	⊢ 0 = (0g‘𝐺)
gsumzcl.z	⊢ 𝑍 = (Cntz‘𝐺)
gsumzcl.g	⊢ (𝜑 → 𝐺 ∈ Mnd)
gsumzcl.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
gsumzcl.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
gsumzcl.c	⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
gsumzcl.w	⊢ (𝜑 → 𝐹 finSupp 0 )
gsumzf1o.h	⊢ (𝜑 → 𝐻:𝐶–1-1-onto→𝐴)

Assertion

Ref	Expression
gsumzf1o	⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ∘ 𝐻)))

Proof of Theorem gsumzf1o

Dummy variables 𝑓 𝑘 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumzcl.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ Mnd)
2		gsumzcl.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		gsumzcl.0	. . . . . . . 8 ⊢ 0 = (0g‘𝐺)
4	3	gsumz 19032	. . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )
5	1, 2, 4	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )
6		gsumzf1o.h	. . . . . . . . 9 ⊢ (𝜑 → 𝐻:𝐶–1-1-onto→𝐴)
7		f1of1 6823	. . . . . . . . 9 ⊢ (𝐻:𝐶–1-1-onto→𝐴 → 𝐻:𝐶–1-1→𝐴)
8	6, 7	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐻:𝐶–1-1→𝐴)
9		f1dmex 7969	. . . . . . . 8 ⊢ ((𝐻:𝐶–1-1→𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐶 ∈ V)
10	8, 2, 9	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ V)
11	3	gsumz 19032	. . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ 𝐶 ∈ V) → (𝐺 Σg (𝑥 ∈ 𝐶 ↦ 0 )) = 0 )
12	1, 10, 11	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐶 ↦ 0 )) = 0 )
13	5, 12	eqtr4d 2799	. . . . 5 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = (𝐺 Σg (𝑥 ∈ 𝐶 ↦ 0 )))
14	13	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = (𝐺 Σg (𝑥 ∈ 𝐶 ↦ 0 )))
15		gsumzcl.f	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
16	3	fvexi 6899	. . . . . . 7 ⊢ 0 ∈ V
17	16	a1i 11	. . . . . 6 ⊢ (𝜑 → 0 ∈ V)
18		ssidd 3954	. . . . . 6 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 supp 0 ))
19	15, 2, 17, 18	gsumcllem 20122	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → 𝐹 = (𝑘 ∈ 𝐴 ↦ 0 ))
20	19	oveq2d 7436	. . . 4 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )))
21		f1of 6824	. . . . . . . . 9 ⊢ (𝐻:𝐶–1-1-onto→𝐴 → 𝐻:𝐶⟶𝐴)
22	6, 21	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐻:𝐶⟶𝐴)
23	22	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → 𝐻:𝐶⟶𝐴)
24	23	ffvelcdmda 7084	. . . . . 6 ⊢ (((𝜑 ∧ (𝐹 supp 0 ) = ∅) ∧ 𝑥 ∈ 𝐶) → (𝐻‘𝑥) ∈ 𝐴)
25	23	feqmptd 6953	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → 𝐻 = (𝑥 ∈ 𝐶 ↦ (𝐻‘𝑥)))
26		eqidd 2762	. . . . . 6 ⊢ (𝑘 = (𝐻‘𝑥) → 0 = 0 )
27	24, 25, 19, 26	fmptco 7130	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐹 ∘ 𝐻) = (𝑥 ∈ 𝐶 ↦ 0 ))
28	27	oveq2d 7436	. . . 4 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg (𝐹 ∘ 𝐻)) = (𝐺 Σg (𝑥 ∈ 𝐶 ↦ 0 )))
29	14, 20, 28	3eqtr4d 2806	. . 3 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ∘ 𝐻)))
30	29	ex 418	. 2 ⊢ (𝜑 → ((𝐹 supp 0 ) = ∅ → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ∘ 𝐻))))
31	6	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐻:𝐶–1-1-onto→𝐴)
32		f1ococnv2 6852	. . . . . . . . . . . . . 14 ⊢ (𝐻:𝐶–1-1-onto→𝐴 → (𝐻 ∘ ◡𝐻) = ( I ↾ 𝐴))
33	31, 32	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐻 ∘ ◡𝐻) = ( I ↾ 𝐴))
34	33	coeq1d 5839	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ((𝐻 ∘ ◡𝐻) ∘ 𝑓) = (( I ↾ 𝐴) ∘ 𝑓))
35		f1of1 6823	. . . . . . . . . . . . . . 15 ⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1→(𝐹 supp 0 ))
36	35	ad2antll 742	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1→(𝐹 supp 0 ))
37		suppssdm 8194	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 supp 0 ) ⊆ dom 𝐹
38	37, 15	fssdm 6729	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ 𝐴)
39	38	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ 𝐴)
40		f1ss 6785	. . . . . . . . . . . . . 14 ⊢ ((𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1→(𝐹 supp 0 ) ∧ (𝐹 supp 0 ) ⊆ 𝐴) → 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1→𝐴)
41	36, 39, 40	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1→𝐴)
42		f1f 6778	. . . . . . . . . . . . 13 ⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1→𝐴 → 𝑓:(1...(♯‘(𝐹 supp 0 )))⟶𝐴)
43		fcoi2 6757	. . . . . . . . . . . . 13 ⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))⟶𝐴 → (( I ↾ 𝐴) ∘ 𝑓) = 𝑓)
44	41, 42, 43	3syl 19	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (( I ↾ 𝐴) ∘ 𝑓) = 𝑓)
45	34, 44	eqtrd 2796	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ((𝐻 ∘ ◡𝐻) ∘ 𝑓) = 𝑓)
46		coass 6267	. . . . . . . . . . 11 ⊢ ((𝐻 ∘ ◡𝐻) ∘ 𝑓) = (𝐻 ∘ (◡𝐻 ∘ 𝑓))
47	45, 46	eqtr3di 2811	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝑓 = (𝐻 ∘ (◡𝐻 ∘ 𝑓)))
48	47	coeq2d 5840	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 ∘ 𝑓) = (𝐹 ∘ (𝐻 ∘ (◡𝐻 ∘ 𝑓))))
49		coass 6267	. . . . . . . . 9 ⊢ ((𝐹 ∘ 𝐻) ∘ (◡𝐻 ∘ 𝑓)) = (𝐹 ∘ (𝐻 ∘ (◡𝐻 ∘ 𝑓)))
50	48, 49	eqtr4di 2814	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 ∘ 𝑓) = ((𝐹 ∘ 𝐻) ∘ (◡𝐻 ∘ 𝑓)))
51	50	seqeq3d 14152	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → seq1((+g‘𝐺), (𝐹 ∘ 𝑓)) = seq1((+g‘𝐺), ((𝐹 ∘ 𝐻) ∘ (◡𝐻 ∘ 𝑓))))
52	51	fveq1d 6887	. . . . . 6 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(♯‘(𝐹 supp 0 ))) = (seq1((+g‘𝐺), ((𝐹 ∘ 𝐻) ∘ (◡𝐻 ∘ 𝑓)))‘(♯‘(𝐹 supp 0 ))))
53		gsumzcl.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐺)
54		eqid 2761	. . . . . . 7 ⊢ (+g‘𝐺) = (+g‘𝐺)
55		gsumzcl.z	. . . . . . 7 ⊢ 𝑍 = (Cntz‘𝐺)
56	1	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐺 ∈ Mnd)
57	2	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐴 ∈ 𝑉)
58	15	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐹:𝐴⟶𝐵)
59		gsumzcl.c	. . . . . . . 8 ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
60	59	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
61		simprl 783	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (♯‘(𝐹 supp 0 )) ∈ ℕ)
62		ssid 3953	. . . . . . . 8 ⊢ (𝐹 supp 0 ) ⊆ (𝐹 supp 0 )
63		f1ofo 6832	. . . . . . . . . 10 ⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → 𝑓:(1...(♯‘(𝐹 supp 0 )))–onto→(𝐹 supp 0 ))
64		forn 6799	. . . . . . . . . 10 ⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))–onto→(𝐹 supp 0 ) → ran 𝑓 = (𝐹 supp 0 ))
65	63, 64	syl 18	. . . . . . . . 9 ⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → ran 𝑓 = (𝐹 supp 0 ))
66	65	ad2antll 742	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ran 𝑓 = (𝐹 supp 0 ))
67	62, 66	sseqtrrid 3974	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ ran 𝑓)
68		eqid 2761	. . . . . . 7 ⊢ ((𝐹 ∘ 𝑓) supp 0 ) = ((𝐹 ∘ 𝑓) supp 0 )
69	53, 3, 54, 55, 56, 57, 58, 60, 61, 41, 67, 68	gsumval3 20121	. . . . . 6 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐺 Σg 𝐹) = (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(♯‘(𝐹 supp 0 ))))
70	10	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐶 ∈ V)
71		fco 6734	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐻:𝐶⟶𝐴) → (𝐹 ∘ 𝐻):𝐶⟶𝐵)
72	15, 22, 71	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∘ 𝐻):𝐶⟶𝐵)
73	72	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 ∘ 𝐻):𝐶⟶𝐵)
74		rncoss 5959	. . . . . . . . 9 ⊢ ran (𝐹 ∘ 𝐻) ⊆ ran 𝐹
75	55	cntzidss 19554	. . . . . . . . 9 ⊢ ((ran 𝐹 ⊆ (𝑍‘ran 𝐹) ∧ ran (𝐹 ∘ 𝐻) ⊆ ran 𝐹) → ran (𝐹 ∘ 𝐻) ⊆ (𝑍‘ran (𝐹 ∘ 𝐻)))
76	59, 74, 75	sylancl 598	. . . . . . . 8 ⊢ (𝜑 → ran (𝐹 ∘ 𝐻) ⊆ (𝑍‘ran (𝐹 ∘ 𝐻)))
77	76	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ran (𝐹 ∘ 𝐻) ⊆ (𝑍‘ran (𝐹 ∘ 𝐻)))
78		f1ocnv 6837	. . . . . . . . . 10 ⊢ (𝐻:𝐶–1-1-onto→𝐴 → ◡𝐻:𝐴–1-1-onto→𝐶)
79		f1of1 6823	. . . . . . . . . 10 ⊢ (◡𝐻:𝐴–1-1-onto→𝐶 → ◡𝐻:𝐴–1-1→𝐶)
80	6, 78, 79	3syl 19	. . . . . . . . 9 ⊢ (𝜑 → ◡𝐻:𝐴–1-1→𝐶)
81	80	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ◡𝐻:𝐴–1-1→𝐶)
82		f1co 6791	. . . . . . . 8 ⊢ ((◡𝐻:𝐴–1-1→𝐶 ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1→𝐴) → (◡𝐻 ∘ 𝑓):(1...(♯‘(𝐹 supp 0 )))–1-1→𝐶)
83	81, 41, 82	syl2anc 596	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (◡𝐻 ∘ 𝑓):(1...(♯‘(𝐹 supp 0 )))–1-1→𝐶)
84		ssidd 3954	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ (𝐹 supp 0 ))
85	15, 2	fexd 7233	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 ∈ V)
86		suppimacnv 8191	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 })))
87	85, 16, 86	sylancl 598	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 })))
88	87	eqcomd 2767	. . . . . . . . . . . 12 ⊢ (𝜑 → (◡𝐹 “ (V ∖ { 0 })) = (𝐹 supp 0 ))
89	88	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (◡𝐹 “ (V ∖ { 0 })) = (𝐹 supp 0 ))
90	84, 89, 66	3sstr4d 3986	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (◡𝐹 “ (V ∖ { 0 })) ⊆ ran 𝑓)
91		imass2 6055	. . . . . . . . . 10 ⊢ ((◡𝐹 “ (V ∖ { 0 })) ⊆ ran 𝑓 → (◡𝐻 “ (◡𝐹 “ (V ∖ { 0 }))) ⊆ (◡𝐻 “ ran 𝑓))
92	90, 91	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (◡𝐻 “ (◡𝐹 “ (V ∖ { 0 }))) ⊆ (◡𝐻 “ ran 𝑓))
93		cnvco 5867	. . . . . . . . . . 11 ⊢ ◡(𝐹 ∘ 𝐻) = (◡𝐻 ∘ ◡𝐹)
94	93	imaeq1i 6049	. . . . . . . . . 10 ⊢ (◡(𝐹 ∘ 𝐻) “ (V ∖ { 0 })) = ((◡𝐻 ∘ ◡𝐹) “ (V ∖ { 0 }))
95		imaco 6252	. . . . . . . . . 10 ⊢ ((◡𝐻 ∘ ◡𝐹) “ (V ∖ { 0 })) = (◡𝐻 “ (◡𝐹 “ (V ∖ { 0 })))
96	94, 95	eqtri 2784	. . . . . . . . 9 ⊢ (◡(𝐹 ∘ 𝐻) “ (V ∖ { 0 })) = (◡𝐻 “ (◡𝐹 “ (V ∖ { 0 })))
97		rnco2 6255	. . . . . . . . 9 ⊢ ran (◡𝐻 ∘ 𝑓) = (◡𝐻 “ ran 𝑓)
98	92, 96, 97	3sstr4g 3984	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (◡(𝐹 ∘ 𝐻) “ (V ∖ { 0 })) ⊆ ran (◡𝐻 ∘ 𝑓))
99		f1oexrnex 7939	. . . . . . . . . . . . 13 ⊢ ((𝐻:𝐶–1-1-onto→𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐻 ∈ V)
100	6, 2, 99	syl2anc 596	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐻 ∈ V)
101		coexg 7941	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ V ∧ 𝐻 ∈ V) → (𝐹 ∘ 𝐻) ∈ V)
102	85, 100, 101	syl2anc 596	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 ∘ 𝐻) ∈ V)
103		suppimacnv 8191	. . . . . . . . . . 11 ⊢ (((𝐹 ∘ 𝐻) ∈ V ∧ 0 ∈ V) → ((𝐹 ∘ 𝐻) supp 0 ) = (◡(𝐹 ∘ 𝐻) “ (V ∖ { 0 })))
104	102, 16, 103	sylancl 598	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐹 ∘ 𝐻) supp 0 ) = (◡(𝐹 ∘ 𝐻) “ (V ∖ { 0 })))
105	104	sseq1d 3962	. . . . . . . . 9 ⊢ (𝜑 → (((𝐹 ∘ 𝐻) supp 0 ) ⊆ ran (◡𝐻 ∘ 𝑓) ↔ (◡(𝐹 ∘ 𝐻) “ (V ∖ { 0 })) ⊆ ran (◡𝐻 ∘ 𝑓)))
106	105	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (((𝐹 ∘ 𝐻) supp 0 ) ⊆ ran (◡𝐻 ∘ 𝑓) ↔ (◡(𝐹 ∘ 𝐻) “ (V ∖ { 0 })) ⊆ ran (◡𝐻 ∘ 𝑓)))
107	98, 106	mpbird 260	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ((𝐹 ∘ 𝐻) supp 0 ) ⊆ ran (◡𝐻 ∘ 𝑓))
108		eqid 2761	. . . . . . 7 ⊢ (((𝐹 ∘ 𝐻) ∘ (◡𝐻 ∘ 𝑓)) supp 0 ) = (((𝐹 ∘ 𝐻) ∘ (◡𝐻 ∘ 𝑓)) supp 0 )
109	53, 3, 54, 55, 56, 70, 73, 77, 61, 83, 107, 108	gsumval3 20121	. . . . . 6 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐺 Σg (𝐹 ∘ 𝐻)) = (seq1((+g‘𝐺), ((𝐹 ∘ 𝐻) ∘ (◡𝐻 ∘ 𝑓)))‘(♯‘(𝐹 supp 0 ))))
110	52, 69, 109	3eqtr4d 2806	. . . . 5 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ∘ 𝐻)))
111	110	expr 462	. . . 4 ⊢ ((𝜑 ∧ (♯‘(𝐹 supp 0 )) ∈ ℕ) → (𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ∘ 𝐻))))
112	111	exlimdv 1966	. . 3 ⊢ ((𝜑 ∧ (♯‘(𝐹 supp 0 )) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ∘ 𝐻))))
113	112	expimpd 459	. 2 ⊢ (𝜑 → (((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 )) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ∘ 𝐻))))
114		gsumzcl.w	. . 3 ⊢ (𝜑 → 𝐹 finSupp 0 )
115		fsuppimp 9360	. . . 4 ⊢ (𝐹 finSupp 0 → (Fun 𝐹 ∧ (𝐹 supp 0 ) ∈ Fin))
116	115	simprd 501	. . 3 ⊢ (𝐹 finSupp 0 → (𝐹 supp 0 ) ∈ Fin)
117		fz1f1o 15876	. . 3 ⊢ ((𝐹 supp 0 ) ∈ Fin → ((𝐹 supp 0 ) = ∅ ∨ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))))
118	114, 116, 117	3syl 19	. 2 ⊢ (𝜑 → ((𝐹 supp 0 ) = ∅ ∨ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))))
119	30, 113, 118	mpjaod 874	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ∘ 𝐻)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ 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   I cid 5545  ^◡ccnv 5650  ran crn 5652   ↾ cres 5653   “ cima 5654   ∘ ccom 5655  Fun wfun 6532  ⟶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  ...cfz 13639  seqcseq 14144  ♯chash 14474  Basecbs 17387  +_gcplusg 17428  0_gc0g 17610   Σ_g cgsu 17611  Mndcmnd 18923  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-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-cntz 19531

This theorem is used by:  gsumf1o  20130  smadiadetlem3  22983