Theorem gsumzf1o 19839

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 18759	. . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )
5	1, 2, 4	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )
6		gsumzf1o.h	. . . . . . . . 9 ⊢ (𝜑 → 𝐻:𝐶–1-1-onto→𝐴)
7		f1of1 6771	. . . . . . . . 9 ⊢ (𝐻:𝐶–1-1-onto→𝐴 → 𝐻:𝐶–1-1→𝐴)
8	6, 7	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐻:𝐶–1-1→𝐴)
9		f1dmex 7899	. . . . . . . 8 ⊢ ((𝐻:𝐶–1-1→𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐶 ∈ V)
10	8, 2, 9	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ V)
11	3	gsumz 18759	. . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ 𝐶 ∈ V) → (𝐺 Σg (𝑥 ∈ 𝐶 ↦ 0 )) = 0 )
12	1, 10, 11	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐶 ↦ 0 )) = 0 )
13	5, 12	eqtr4d 2772	. . . . 5 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = (𝐺 Σg (𝑥 ∈ 𝐶 ↦ 0 )))
14	13	adantr 480	. . . 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 6846	. . . . . . 7 ⊢ 0 ∈ V
17	16	a1i 11	. . . . . 6 ⊢ (𝜑 → 0 ∈ V)
18		ssidd 3955	. . . . . 6 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 supp 0 ))
19	15, 2, 17, 18	gsumcllem 19835	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → 𝐹 = (𝑘 ∈ 𝐴 ↦ 0 ))
20	19	oveq2d 7372	. . . 4 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )))
21		f1of 6772	. . . . . . . . 9 ⊢ (𝐻:𝐶–1-1-onto→𝐴 → 𝐻:𝐶⟶𝐴)
22	6, 21	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐻:𝐶⟶𝐴)
23	22	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → 𝐻:𝐶⟶𝐴)
24	23	ffvelcdmda 7027	. . . . . 6 ⊢ (((𝜑 ∧ (𝐹 supp 0 ) = ∅) ∧ 𝑥 ∈ 𝐶) → (𝐻‘𝑥) ∈ 𝐴)
25	23	feqmptd 6900	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → 𝐻 = (𝑥 ∈ 𝐶 ↦ (𝐻‘𝑥)))
26		eqidd 2735	. . . . . 6 ⊢ (𝑘 = (𝐻‘𝑥) → 0 = 0 )
27	24, 25, 19, 26	fmptco 7072	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐹 ∘ 𝐻) = (𝑥 ∈ 𝐶 ↦ 0 ))
28	27	oveq2d 7372	. . . 4 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg (𝐹 ∘ 𝐻)) = (𝐺 Σg (𝑥 ∈ 𝐶 ↦ 0 )))
29	14, 20, 28	3eqtr4d 2779	. . 3 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ∘ 𝐻)))
30	29	ex 412	. 2 ⊢ (𝜑 → ((𝐹 supp 0 ) = ∅ → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ∘ 𝐻))))
31	6	adantr 480	. . . . . . . . . . . . . 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 6799	. . . . . . . . . . . . . 14 ⊢ (𝐻:𝐶–1-1-onto→𝐴 → (𝐻 ∘ ◡𝐻) = ( I ↾ 𝐴))
33	31, 32	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐻 ∘ ◡𝐻) = ( I ↾ 𝐴))
34	33	coeq1d 5808	. . . . . . . . . . . 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 6771	. . . . . . . . . . . . . . 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 729	. . . . . . . . . . . . . 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 8117	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 supp 0 ) ⊆ dom 𝐹
38	37, 15	fssdm 6679	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ 𝐴)
39	38	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ 𝐴)
40		f1ss 6733	. . . . . . . . . . . . . 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 584	. . . . . . . . . . . . 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 6728	. . . . . . . . . . . . 13 ⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1→𝐴 → 𝑓:(1...(♯‘(𝐹 supp 0 )))⟶𝐴)
43		fcoi2 6707	. . . . . . . . . . . . 13 ⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))⟶𝐴 → (( I ↾ 𝐴) ∘ 𝑓) = 𝑓)
44	41, 42, 43	3syl 18	. . . . . . . . . . . 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 2769	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ((𝐻 ∘ ◡𝐻) ∘ 𝑓) = 𝑓)
46		coass 6222	. . . . . . . . . . 11 ⊢ ((𝐻 ∘ ◡𝐻) ∘ 𝑓) = (𝐻 ∘ (◡𝐻 ∘ 𝑓))
47	45, 46	eqtr3di 2784	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝑓 = (𝐻 ∘ (◡𝐻 ∘ 𝑓)))
48	47	coeq2d 5809	. . . . . . . . 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 6222	. . . . . . . . 9 ⊢ ((𝐹 ∘ 𝐻) ∘ (◡𝐻 ∘ 𝑓)) = (𝐹 ∘ (𝐻 ∘ (◡𝐻 ∘ 𝑓)))
50	48, 49	eqtr4di 2787	. . . . . . . 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 13930	. . . . . . 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 6834	. . . . . 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 2734	. . . . . . 7 ⊢ (+g‘𝐺) = (+g‘𝐺)
55		gsumzcl.z	. . . . . . 7 ⊢ 𝑍 = (Cntz‘𝐺)
56	1	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐺 ∈ Mnd)
57	2	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐴 ∈ 𝑉)
58	15	adantr 480	. . . . . . 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 480	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
61		simprl 770	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (♯‘(𝐹 supp 0 )) ∈ ℕ)
62		ssid 3954	. . . . . . . 8 ⊢ (𝐹 supp 0 ) ⊆ (𝐹 supp 0 )
63		f1ofo 6779	. . . . . . . . . 10 ⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → 𝑓:(1...(♯‘(𝐹 supp 0 )))–onto→(𝐹 supp 0 ))
64		forn 6747	. . . . . . . . . 10 ⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))–onto→(𝐹 supp 0 ) → ran 𝑓 = (𝐹 supp 0 ))
65	63, 64	syl 17	. . . . . . . . 9 ⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → ran 𝑓 = (𝐹 supp 0 ))
66	65	ad2antll 729	. . . . . . . 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 3975	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ ran 𝑓)
68		eqid 2734	. . . . . . 7 ⊢ ((𝐹 ∘ 𝑓) supp 0 ) = ((𝐹 ∘ 𝑓) supp 0 )
69	53, 3, 54, 55, 56, 57, 58, 60, 61, 41, 67, 68	gsumval3 19834	. . . . . 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 480	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐶 ∈ V)
71		fco 6684	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐻:𝐶⟶𝐴) → (𝐹 ∘ 𝐻):𝐶⟶𝐵)
72	15, 22, 71	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∘ 𝐻):𝐶⟶𝐵)
73	72	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 ∘ 𝐻):𝐶⟶𝐵)
74		rncoss 5924	. . . . . . . . 9 ⊢ ran (𝐹 ∘ 𝐻) ⊆ ran 𝐹
75	55	cntzidss 19267	. . . . . . . . 9 ⊢ ((ran 𝐹 ⊆ (𝑍‘ran 𝐹) ∧ ran (𝐹 ∘ 𝐻) ⊆ ran 𝐹) → ran (𝐹 ∘ 𝐻) ⊆ (𝑍‘ran (𝐹 ∘ 𝐻)))
76	59, 74, 75	sylancl 586	. . . . . . . 8 ⊢ (𝜑 → ran (𝐹 ∘ 𝐻) ⊆ (𝑍‘ran (𝐹 ∘ 𝐻)))
77	76	adantr 480	. . . . . . 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 6784	. . . . . . . . . 10 ⊢ (𝐻:𝐶–1-1-onto→𝐴 → ◡𝐻:𝐴–1-1-onto→𝐶)
79		f1of1 6771	. . . . . . . . . 10 ⊢ (◡𝐻:𝐴–1-1-onto→𝐶 → ◡𝐻:𝐴–1-1→𝐶)
80	6, 78, 79	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → ◡𝐻:𝐴–1-1→𝐶)
81	80	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ◡𝐻:𝐴–1-1→𝐶)
82		f1co 6739	. . . . . . . 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 584	. . . . . . 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 3955	. . . . . . . . . . 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 7171	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 ∈ V)
86		suppimacnv 8114	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 })))
87	85, 16, 86	sylancl 586	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 })))
88	87	eqcomd 2740	. . . . . . . . . . . 12 ⊢ (𝜑 → (◡𝐹 “ (V ∖ { 0 })) = (𝐹 supp 0 ))
89	88	adantr 480	. . . . . . . . . . 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 3987	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (◡𝐹 “ (V ∖ { 0 })) ⊆ ran 𝑓)
91		imass2 6059	. . . . . . . . . 10 ⊢ ((◡𝐹 “ (V ∖ { 0 })) ⊆ ran 𝑓 → (◡𝐻 “ (◡𝐹 “ (V ∖ { 0 }))) ⊆ (◡𝐻 “ ran 𝑓))
92	90, 91	syl 17	. . . . . . . . 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 5832	. . . . . . . . . . 11 ⊢ ◡(𝐹 ∘ 𝐻) = (◡𝐻 ∘ ◡𝐹)
94	93	imaeq1i 6014	. . . . . . . . . 10 ⊢ (◡(𝐹 ∘ 𝐻) “ (V ∖ { 0 })) = ((◡𝐻 ∘ ◡𝐹) “ (V ∖ { 0 }))
95		imaco 6207	. . . . . . . . . 10 ⊢ ((◡𝐻 ∘ ◡𝐹) “ (V ∖ { 0 })) = (◡𝐻 “ (◡𝐹 “ (V ∖ { 0 })))
96	94, 95	eqtri 2757	. . . . . . . . 9 ⊢ (◡(𝐹 ∘ 𝐻) “ (V ∖ { 0 })) = (◡𝐻 “ (◡𝐹 “ (V ∖ { 0 })))
97		rnco2 6210	. . . . . . . . 9 ⊢ ran (◡𝐻 ∘ 𝑓) = (◡𝐻 “ ran 𝑓)
98	92, 96, 97	3sstr4g 3985	. . . . . . . 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 7867	. . . . . . . . . . . . 13 ⊢ ((𝐻:𝐶–1-1-onto→𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐻 ∈ V)
100	6, 2, 99	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐻 ∈ V)
101		coexg 7869	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ V ∧ 𝐻 ∈ V) → (𝐹 ∘ 𝐻) ∈ V)
102	85, 100, 101	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 ∘ 𝐻) ∈ V)
103		suppimacnv 8114	. . . . . . . . . . 11 ⊢ (((𝐹 ∘ 𝐻) ∈ V ∧ 0 ∈ V) → ((𝐹 ∘ 𝐻) supp 0 ) = (◡(𝐹 ∘ 𝐻) “ (V ∖ { 0 })))
104	102, 16, 103	sylancl 586	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐹 ∘ 𝐻) supp 0 ) = (◡(𝐹 ∘ 𝐻) “ (V ∖ { 0 })))
105	104	sseq1d 3963	. . . . . . . . 9 ⊢ (𝜑 → (((𝐹 ∘ 𝐻) supp 0 ) ⊆ ran (◡𝐻 ∘ 𝑓) ↔ (◡(𝐹 ∘ 𝐻) “ (V ∖ { 0 })) ⊆ ran (◡𝐻 ∘ 𝑓)))
106	105	adantr 480	. . . . . . . 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 257	. . . . . . 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 2734	. . . . . . 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 19834	. . . . . 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 2779	. . . . 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 456	. . . 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 1934	. . 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 453	. 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 9269	. . . 4 ⊢ (𝐹 finSupp 0 → (Fun 𝐹 ∧ (𝐹 supp 0 ) ∈ Fin))
116	115	simprd 495	. . 3 ⊢ (𝐹 finSupp 0 → (𝐹 supp 0 ) ∈ Fin)
117		fz1f1o 15631	. . 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 18	. 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 860	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ∘ 𝐻)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1541  ∃wex 1780   ∈ wcel 2113  Vcvv 3438   ∖ cdif 3896   ⊆ wss 3899  ∅c0 4283  {csn 4578   class class class wbr 5096   ↦ cmpt 5177   I cid 5516  ^◡ccnv 5621  ran crn 5623   ↾ cres 5624   “ cima 5625   ∘ ccom 5626  Fun wfun 6484  ⟶wf 6486  –1-1→wf1 6487  –onto→wfo 6488  –1-1-onto→wf1o 6489  ‘cfv 6490  (class class class)co 7356   supp csupp 8100  Fincfn 8881   finSupp cfsupp 9262  1c1 11025  ℕcn 12143  ...cfz 13421  seqcseq 13922  ♯chash 14251  Basecbs 17134  +_gcplusg 17175  0_gc0g 17357   Σ_g cgsu 17358  Mndcmnd 18657  Cntzccntz 19242

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 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101

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 2567  df-clab 2713  df-cleq 2726  df-clel 2809  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 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-isom 6499  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-supp 8101  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8633  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885  df-fsupp 9263  df-oi 9413  df-card 9849  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-nn 12144  df-n0 12400  df-z 12487  df-uz 12750  df-fz 13422  df-fzo 13569  df-seq 13923  df-hash 14252  df-0g 17359  df-gsum 17360  df-mgm 18563  df-sgrp 18642  df-mnd 18658  df-cntz 19244

This theorem is referenced by:  gsumf1o  19843  smadiadetlem3  22610