Theorem gsumzf1o 19791

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 18710	. . . . . . 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 6763	. . . . . . . . 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 7892	. . . . . . . 8 ⊢ ((𝐻:𝐶–1-1→𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐶 ∈ V)
10	8, 2, 9	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ V)
11	3	gsumz 18710	. . . . . . 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 2767	. . . . 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 6836	. . . . . . 7 ⊢ 0 ∈ V
17	16	a1i 11	. . . . . 6 ⊢ (𝜑 → 0 ∈ V)
18		ssidd 3959	. . . . . 6 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 supp 0 ))
19	15, 2, 17, 18	gsumcllem 19787	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → 𝐹 = (𝑘 ∈ 𝐴 ↦ 0 ))
20	19	oveq2d 7365	. . . 4 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )))
21		f1of 6764	. . . . . . . . 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 7018	. . . . . 6 ⊢ (((𝜑 ∧ (𝐹 supp 0 ) = ∅) ∧ 𝑥 ∈ 𝐶) → (𝐻‘𝑥) ∈ 𝐴)
25	23	feqmptd 6891	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → 𝐻 = (𝑥 ∈ 𝐶 ↦ (𝐻‘𝑥)))
26		eqidd 2730	. . . . . 6 ⊢ (𝑘 = (𝐻‘𝑥) → 0 = 0 )
27	24, 25, 19, 26	fmptco 7063	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐹 ∘ 𝐻) = (𝑥 ∈ 𝐶 ↦ 0 ))
28	27	oveq2d 7365	. . . 4 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg (𝐹 ∘ 𝐻)) = (𝐺 Σg (𝑥 ∈ 𝐶 ↦ 0 )))
29	14, 20, 28	3eqtr4d 2774	. . 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 6791	. . . . . . . . . . . . . 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 5804	. . . . . . . . . . . 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 6763	. . . . . . . . . . . . . . 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 8110	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 supp 0 ) ⊆ dom 𝐹
38	37, 15	fssdm 6671	. . . . . . . . . . . . . . 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 6725	. . . . . . . . . . . . . 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 6720	. . . . . . . . . . . . 13 ⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1→𝐴 → 𝑓:(1...(♯‘(𝐹 supp 0 )))⟶𝐴)
43		fcoi2 6699	. . . . . . . . . . . . 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 2764	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ((𝐻 ∘ ◡𝐻) ∘ 𝑓) = 𝑓)
46		coass 6214	. . . . . . . . . . 11 ⊢ ((𝐻 ∘ ◡𝐻) ∘ 𝑓) = (𝐻 ∘ (◡𝐻 ∘ 𝑓))
47	45, 46	eqtr3di 2779	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝑓 = (𝐻 ∘ (◡𝐻 ∘ 𝑓)))
48	47	coeq2d 5805	. . . . . . . . 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 6214	. . . . . . . . 9 ⊢ ((𝐹 ∘ 𝐻) ∘ (◡𝐻 ∘ 𝑓)) = (𝐹 ∘ (𝐻 ∘ (◡𝐻 ∘ 𝑓)))
50	48, 49	eqtr4di 2782	. . . . . . . 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 13916	. . . . . . 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 6824	. . . . . 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 2729	. . . . . . 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 3958	. . . . . . . 8 ⊢ (𝐹 supp 0 ) ⊆ (𝐹 supp 0 )
63		f1ofo 6771	. . . . . . . . . 10 ⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → 𝑓:(1...(♯‘(𝐹 supp 0 )))–onto→(𝐹 supp 0 ))
64		forn 6739	. . . . . . . . . 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 3979	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ ran 𝑓)
68		eqid 2729	. . . . . . 7 ⊢ ((𝐹 ∘ 𝑓) supp 0 ) = ((𝐹 ∘ 𝑓) supp 0 )
69	53, 3, 54, 55, 56, 57, 58, 60, 61, 41, 67, 68	gsumval3 19786	. . . . . 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 6676	. . . . . . . . 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 5918	. . . . . . . . 9 ⊢ ran (𝐹 ∘ 𝐻) ⊆ ran 𝐹
75	55	cntzidss 19219	. . . . . . . . 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 6776	. . . . . . . . . 10 ⊢ (𝐻:𝐶–1-1-onto→𝐴 → ◡𝐻:𝐴–1-1-onto→𝐶)
79		f1of1 6763	. . . . . . . . . 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 6731	. . . . . . . 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 3959	. . . . . . . . . . 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 7163	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 ∈ V)
86		suppimacnv 8107	. . . . . . . . . . . . . 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 2735	. . . . . . . . . . . 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 3991	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (◡𝐹 “ (V ∖ { 0 })) ⊆ ran 𝑓)
91		imass2 6053	. . . . . . . . . 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 5828	. . . . . . . . . . 11 ⊢ ◡(𝐹 ∘ 𝐻) = (◡𝐻 ∘ ◡𝐹)
94	93	imaeq1i 6008	. . . . . . . . . 10 ⊢ (◡(𝐹 ∘ 𝐻) “ (V ∖ { 0 })) = ((◡𝐻 ∘ ◡𝐹) “ (V ∖ { 0 }))
95		imaco 6200	. . . . . . . . . 10 ⊢ ((◡𝐻 ∘ ◡𝐹) “ (V ∖ { 0 })) = (◡𝐻 “ (◡𝐹 “ (V ∖ { 0 })))
96	94, 95	eqtri 2752	. . . . . . . . 9 ⊢ (◡(𝐹 ∘ 𝐻) “ (V ∖ { 0 })) = (◡𝐻 “ (◡𝐹 “ (V ∖ { 0 })))
97		rnco2 6202	. . . . . . . . 9 ⊢ ran (◡𝐻 ∘ 𝑓) = (◡𝐻 “ ran 𝑓)
98	92, 96, 97	3sstr4g 3989	. . . . . . . 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 7860	. . . . . . . . . . . . 13 ⊢ ((𝐻:𝐶–1-1-onto→𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐻 ∈ V)
100	6, 2, 99	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐻 ∈ V)
101		coexg 7862	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ V ∧ 𝐻 ∈ V) → (𝐹 ∘ 𝐻) ∈ V)
102	85, 100, 101	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 ∘ 𝐻) ∈ V)
103		suppimacnv 8107	. . . . . . . . . . 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 3967	. . . . . . . . 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 2729	. . . . . . 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 19786	. . . . . 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 2774	. . . . 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 1933	. . 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 9258	. . . 4 ⊢ (𝐹 finSupp 0 → (Fun 𝐹 ∧ (𝐹 supp 0 ) ∈ Fin))
116	115	simprd 495	. . 3 ⊢ (𝐹 finSupp 0 → (𝐹 supp 0 ) ∈ Fin)
117		fz1f1o 15617	. . 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 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3436   ∖ cdif 3900   ⊆ wss 3903  ∅c0 4284  {csn 4577   class class class wbr 5092   ↦ cmpt 5173   I cid 5513  ^◡ccnv 5618  ran crn 5620   ↾ cres 5621   “ cima 5622   ∘ ccom 5623  Fun wfun 6476  ⟶wf 6478  –1-1→wf1 6479  –onto→wfo 6480  –1-1-onto→wf1o 6481  ‘cfv 6482  (class class class)co 7349   supp csupp 8093  Fincfn 8872   finSupp cfsupp 9251  1c1 11010  ℕcn 12128  ...cfz 13410  seqcseq 13908  ♯chash 14237  Basecbs 17120  +_gcplusg 17161  0_gc0g 17343   Σ_g cgsu 17344  Mndcmnd 18608  Cntzccntz 19194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-supp 8094  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-er 8625  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-fsupp 9252  df-oi 9402  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-n0 12385  df-z 12472  df-uz 12736  df-fz 13411  df-fzo 13558  df-seq 13909  df-hash 14238  df-0g 17345  df-gsum 17346  df-mgm 18514  df-sgrp 18593  df-mnd 18609  df-cntz 19196

This theorem is referenced by:  gsumf1o  19795  smadiadetlem3  22553