Theorem gsumzf1o 19930

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 18849	. . . . . . 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 6847	. . . . . . . . 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 7981	. . . . . . . 8 ⊢ ((𝐻:𝐶–1-1→𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐶 ∈ V)
10	8, 2, 9	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ V)
11	3	gsumz 18849	. . . . . . 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 2780	. . . . 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 6920	. . . . . . 7 ⊢ 0 ∈ V
17	16	a1i 11	. . . . . 6 ⊢ (𝜑 → 0 ∈ V)
18		ssidd 4007	. . . . . 6 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 supp 0 ))
19	15, 2, 17, 18	gsumcllem 19926	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → 𝐹 = (𝑘 ∈ 𝐴 ↦ 0 ))
20	19	oveq2d 7447	. . . 4 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )))
21		f1of 6848	. . . . . . . . 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 7104	. . . . . 6 ⊢ (((𝜑 ∧ (𝐹 supp 0 ) = ∅) ∧ 𝑥 ∈ 𝐶) → (𝐻‘𝑥) ∈ 𝐴)
25	23	feqmptd 6977	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → 𝐻 = (𝑥 ∈ 𝐶 ↦ (𝐻‘𝑥)))
26		eqidd 2738	. . . . . 6 ⊢ (𝑘 = (𝐻‘𝑥) → 0 = 0 )
27	24, 25, 19, 26	fmptco 7149	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐹 ∘ 𝐻) = (𝑥 ∈ 𝐶 ↦ 0 ))
28	27	oveq2d 7447	. . . 4 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg (𝐹 ∘ 𝐻)) = (𝐺 Σg (𝑥 ∈ 𝐶 ↦ 0 )))
29	14, 20, 28	3eqtr4d 2787	. . 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 6875	. . . . . . . . . . . . . 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 5872	. . . . . . . . . . . 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 6847	. . . . . . . . . . . . . . 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 8202	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 supp 0 ) ⊆ dom 𝐹
38	37, 15	fssdm 6755	. . . . . . . . . . . . . . 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 6809	. . . . . . . . . . . . . 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 6804	. . . . . . . . . . . . 13 ⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1→𝐴 → 𝑓:(1...(♯‘(𝐹 supp 0 )))⟶𝐴)
43		fcoi2 6783	. . . . . . . . . . . . 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 2777	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ((𝐻 ∘ ◡𝐻) ∘ 𝑓) = 𝑓)
46		coass 6285	. . . . . . . . . . 11 ⊢ ((𝐻 ∘ ◡𝐻) ∘ 𝑓) = (𝐻 ∘ (◡𝐻 ∘ 𝑓))
47	45, 46	eqtr3di 2792	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝑓 = (𝐻 ∘ (◡𝐻 ∘ 𝑓)))
48	47	coeq2d 5873	. . . . . . . . 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 6285	. . . . . . . . 9 ⊢ ((𝐹 ∘ 𝐻) ∘ (◡𝐻 ∘ 𝑓)) = (𝐹 ∘ (𝐻 ∘ (◡𝐻 ∘ 𝑓)))
50	48, 49	eqtr4di 2795	. . . . . . . 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 14050	. . . . . . 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 6908	. . . . . 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 2737	. . . . . . 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 771	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (♯‘(𝐹 supp 0 )) ∈ ℕ)
62		ssid 4006	. . . . . . . 8 ⊢ (𝐹 supp 0 ) ⊆ (𝐹 supp 0 )
63		f1ofo 6855	. . . . . . . . . 10 ⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → 𝑓:(1...(♯‘(𝐹 supp 0 )))–onto→(𝐹 supp 0 ))
64		forn 6823	. . . . . . . . . 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 4027	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ ran 𝑓)
68		eqid 2737	. . . . . . 7 ⊢ ((𝐹 ∘ 𝑓) supp 0 ) = ((𝐹 ∘ 𝑓) supp 0 )
69	53, 3, 54, 55, 56, 57, 58, 60, 61, 41, 67, 68	gsumval3 19925	. . . . . 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 6760	. . . . . . . . 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 5986	. . . . . . . . 9 ⊢ ran (𝐹 ∘ 𝐻) ⊆ ran 𝐹
75	55	cntzidss 19358	. . . . . . . . 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 6860	. . . . . . . . . 10 ⊢ (𝐻:𝐶–1-1-onto→𝐴 → ◡𝐻:𝐴–1-1-onto→𝐶)
79		f1of1 6847	. . . . . . . . . 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 6815	. . . . . . . 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 4007	. . . . . . . . . . 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 7247	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 ∈ V)
86		suppimacnv 8199	. . . . . . . . . . . . . 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 2743	. . . . . . . . . . . 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 4039	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (◡𝐹 “ (V ∖ { 0 })) ⊆ ran 𝑓)
91		imass2 6120	. . . . . . . . . 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 5896	. . . . . . . . . . 11 ⊢ ◡(𝐹 ∘ 𝐻) = (◡𝐻 ∘ ◡𝐹)
94	93	imaeq1i 6075	. . . . . . . . . 10 ⊢ (◡(𝐹 ∘ 𝐻) “ (V ∖ { 0 })) = ((◡𝐻 ∘ ◡𝐹) “ (V ∖ { 0 }))
95		imaco 6271	. . . . . . . . . 10 ⊢ ((◡𝐻 ∘ ◡𝐹) “ (V ∖ { 0 })) = (◡𝐻 “ (◡𝐹 “ (V ∖ { 0 })))
96	94, 95	eqtri 2765	. . . . . . . . 9 ⊢ (◡(𝐹 ∘ 𝐻) “ (V ∖ { 0 })) = (◡𝐻 “ (◡𝐹 “ (V ∖ { 0 })))
97		rnco2 6273	. . . . . . . . 9 ⊢ ran (◡𝐻 ∘ 𝑓) = (◡𝐻 “ ran 𝑓)
98	92, 96, 97	3sstr4g 4037	. . . . . . . 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 7949	. . . . . . . . . . . . 13 ⊢ ((𝐻:𝐶–1-1-onto→𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐻 ∈ V)
100	6, 2, 99	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐻 ∈ V)
101		coexg 7951	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ V ∧ 𝐻 ∈ V) → (𝐹 ∘ 𝐻) ∈ V)
102	85, 100, 101	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 ∘ 𝐻) ∈ V)
103		suppimacnv 8199	. . . . . . . . . . 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 4015	. . . . . . . . 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 2737	. . . . . . 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 19925	. . . . . 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 2787	. . . . 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 9408	. . . 4 ⊢ (𝐹 finSupp 0 → (Fun 𝐹 ∧ (𝐹 supp 0 ) ∈ Fin))
116	115	simprd 495	. . 3 ⊢ (𝐹 finSupp 0 → (𝐹 supp 0 ) ∈ Fin)
117		fz1f1o 15746	. . 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 861	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ∘ 𝐻)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1540  ∃wex 1779   ∈ wcel 2108  Vcvv 3480   ∖ cdif 3948   ⊆ wss 3951  ∅c0 4333  {csn 4626   class class class wbr 5143   ↦ cmpt 5225   I cid 5577  ^◡ccnv 5684  ran crn 5686   ↾ cres 5687   “ cima 5688   ∘ ccom 5689  Fun wfun 6555  ⟶wf 6557  –1-1→wf1 6558  –onto→wfo 6559  –1-1-onto→wf1o 6560  ‘cfv 6561  (class class class)co 7431   supp csupp 8185  Fincfn 8985   finSupp cfsupp 9401  1c1 11156  ℕcn 12266  ...cfz 13547  seqcseq 14042  ♯chash 14369  Basecbs 17247  +_gcplusg 17297  0_gc0g 17484   Σ_g cgsu 17485  Mndcmnd 18747  Cntzccntz 19333

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-seq 14043  df-hash 14370  df-0g 17486  df-gsum 17487  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-cntz 19335

This theorem is referenced by:  gsumf1o  19934  smadiadetlem3  22674