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Theorem gsumzf1o 18136

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 = (0_g‘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 Σ_g 𝐹) = (𝐺 Σ_g (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 = (0_g‘𝐺)
4	3	gsumz 17197	. . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )
5	1, 2, 4	syl2anc 691	. . . . . 6 ⊢ (𝜑 → (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )
6		gsumzf1o.h	. . . . . . . . 9 ⊢ (𝜑 → 𝐻:𝐶–1-1-onto→𝐴)
7		f1of1 6049	. . . . . . . . 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 7029	. . . . . . . 8 ⊢ ((𝐻:𝐶–1-1→𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐶 ∈ V)
10	8, 2, 9	syl2anc 691	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ V)
11	3	gsumz 17197	. . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ 𝐶 ∈ V) → (𝐺 Σ_g (𝑥 ∈ 𝐶 ↦ 0 )) = 0 )
12	1, 10, 11	syl2anc 691	. . . . . 6 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ 𝐶 ↦ 0 )) = 0 )
13	5, 12	eqtr4d 2647	. . . . 5 ⊢ (𝜑 → (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 0 )) = (𝐺 Σ_g (𝑥 ∈ 𝐶 ↦ 0 )))
14	13	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 0 )) = (𝐺 Σ_g (𝑥 ∈ 𝐶 ↦ 0 )))
15		gsumzcl.f	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
16		fvex 6113	. . . . . . . 8 ⊢ (0_g‘𝐺) ∈ V
17	3, 16	eqeltri 2684	. . . . . . 7 ⊢ 0 ∈ V
18	17	a1i 11	. . . . . 6 ⊢ (𝜑 → 0 ∈ V)
19		ssid 3587	. . . . . . 7 ⊢ (𝐹 supp 0 ) ⊆ (𝐹 supp 0 )
20	19	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 supp 0 ))
21	15, 2, 18, 20	gsumcllem 18132	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → 𝐹 = (𝑘 ∈ 𝐴 ↦ 0 ))
22	21	oveq2d 6565	. . . 4 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σ_g 𝐹) = (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 0 )))
23		f1of 6050	. . . . . . . . 9 ⊢ (𝐻:𝐶–1-1-onto→𝐴 → 𝐻:𝐶⟶𝐴)
24	6, 23	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐻:𝐶⟶𝐴)
25	24	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → 𝐻:𝐶⟶𝐴)
26	25	ffvelrnda 6267	. . . . . 6 ⊢ (((𝜑 ∧ (𝐹 supp 0 ) = ∅) ∧ 𝑥 ∈ 𝐶) → (𝐻‘𝑥) ∈ 𝐴)
27	25	feqmptd 6159	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → 𝐻 = (𝑥 ∈ 𝐶 ↦ (𝐻‘𝑥)))
28		eqidd 2611	. . . . . 6 ⊢ (𝑘 = (𝐻‘𝑥) → 0 = 0 )
29	26, 27, 21, 28	fmptco 6303	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐹 ∘ 𝐻) = (𝑥 ∈ 𝐶 ↦ 0 ))
30	29	oveq2d 6565	. . . 4 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σ_g (𝐹 ∘ 𝐻)) = (𝐺 Σ_g (𝑥 ∈ 𝐶 ↦ 0 )))
31	14, 22, 30	3eqtr4d 2654	. . 3 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σ_g 𝐹) = (𝐺 Σ_g (𝐹 ∘ 𝐻)))
32	31	ex 449	. 2 ⊢ (𝜑 → ((𝐹 supp 0 ) = ∅ → (𝐺 Σ_g 𝐹) = (𝐺 Σ_g (𝐹 ∘ 𝐻))))
33		coass 5571	. . . . . . . . . . 11 ⊢ ((𝐻 ∘ ^◡𝐻) ∘ 𝑓) = (𝐻 ∘ (^◡𝐻 ∘ 𝑓))
34	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)
35		f1ococnv2 6076	. . . . . . . . . . . . . 14 ⊢ (𝐻:𝐶–1-1-onto→𝐴 → (𝐻 ∘ ^◡𝐻) = ( I ↾ 𝐴))
36	34, 35	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐻 ∘ ^◡𝐻) = ( I ↾ 𝐴))
37	36	coeq1d 5205	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ((𝐻 ∘ ^◡𝐻) ∘ 𝑓) = (( I ↾ 𝐴) ∘ 𝑓))
38		f1of1 6049	. . . . . . . . . . . . . . 15 ⊢ (𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1→(𝐹 supp 0 ))
39	38	ad2antll 761	. . . . . . . . . . . . . 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 ))
40		suppssdm 7195	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 supp 0 ) ⊆ dom 𝐹
41		fdm 5964	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
42	15, 41	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom 𝐹 = 𝐴)
43	40, 42	syl5sseq 3616	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ 𝐴)
44	43	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ 𝐴)
45		f1ss 6019	. . . . . . . . . . . . . 14 ⊢ ((𝑓:(1...(#‘(𝐹 supp 0 )))–1-1→(𝐹 supp 0 ) ∧ (𝐹 supp 0 ) ⊆ 𝐴) → 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1→𝐴)
46	39, 44, 45	syl2anc 691	. . . . . . . . . . . . 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)
47		f1f 6014	. . . . . . . . . . . . 13 ⊢ (𝑓:(1...(#‘(𝐹 supp 0 )))–1-1→𝐴 → 𝑓:(1...(#‘(𝐹 supp 0 )))⟶𝐴)
48		fcoi2 5992	. . . . . . . . . . . . 13 ⊢ (𝑓:(1...(#‘(𝐹 supp 0 )))⟶𝐴 → (( I ↾ 𝐴) ∘ 𝑓) = 𝑓)
49	46, 47, 48	3syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (( I ↾ 𝐴) ∘ 𝑓) = 𝑓)
50	37, 49	eqtrd 2644	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ((𝐻 ∘ ^◡𝐻) ∘ 𝑓) = 𝑓)
51	33, 50	syl5reqr 2659	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝑓 = (𝐻 ∘ (^◡𝐻 ∘ 𝑓)))
52	51	coeq2d 5206	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 ∘ 𝑓) = (𝐹 ∘ (𝐻 ∘ (^◡𝐻 ∘ 𝑓))))
53		coass 5571	. . . . . . . . 9 ⊢ ((𝐹 ∘ 𝐻) ∘ (^◡𝐻 ∘ 𝑓)) = (𝐹 ∘ (𝐻 ∘ (^◡𝐻 ∘ 𝑓)))
54	52, 53	syl6eqr 2662	. . . . . . . 8 ⊢ ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 ∘ 𝑓) = ((𝐹 ∘ 𝐻) ∘ (^◡𝐻 ∘ 𝑓)))
55	54	seqeq3d 12671	. . . . . . 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))))
56	55	fveq1d 6105	. . . . . 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 ))))
57		gsumzcl.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐺)
58		eqid 2610	. . . . . . 7 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
59		gsumzcl.z	. . . . . . 7 ⊢ 𝑍 = (Cntz‘𝐺)
60	1	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐺 ∈ Mnd)
61	2	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐴 ∈ 𝑉)
62	15	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐹:𝐴⟶𝐵)
63		gsumzcl.c	. . . . . . . 8 ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
64	63	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
65		simprl 790	. . . . . . 7 ⊢ ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (#‘(𝐹 supp 0 )) ∈ ℕ)
66		f1ofo 6057	. . . . . . . . . 10 ⊢ (𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → 𝑓:(1...(#‘(𝐹 supp 0 )))–onto→(𝐹 supp 0 ))
67		forn 6031	. . . . . . . . . 10 ⊢ (𝑓:(1...(#‘(𝐹 supp 0 )))–onto→(𝐹 supp 0 ) → ran 𝑓 = (𝐹 supp 0 ))
68	66, 67	syl 17	. . . . . . . . 9 ⊢ (𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → ran 𝑓 = (𝐹 supp 0 ))
69	68	ad2antll 761	. . . . . . . 8 ⊢ ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ran 𝑓 = (𝐹 supp 0 ))
70	19, 69	syl5sseqr 3617	. . . . . . 7 ⊢ ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ ran 𝑓)
71		eqid 2610	. . . . . . 7 ⊢ ((𝐹 ∘ 𝑓) supp 0 ) = ((𝐹 ∘ 𝑓) supp 0 )
72	57, 3, 58, 59, 60, 61, 62, 64, 65, 46, 70, 71	gsumval3 18131	. . . . . 6 ⊢ ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐺 Σ_g 𝐹) = (seq1((+_g‘𝐺), (𝐹 ∘ 𝑓))‘(#‘(𝐹 supp 0 ))))
73	10	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐶 ∈ V)
74		fco 5971	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐻:𝐶⟶𝐴) → (𝐹 ∘ 𝐻):𝐶⟶𝐵)
75	15, 24, 74	syl2anc 691	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∘ 𝐻):𝐶⟶𝐵)
76	75	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 ∘ 𝐻):𝐶⟶𝐵)
77		rncoss 5307	. . . . . . . . 9 ⊢ ran (𝐹 ∘ 𝐻) ⊆ ran 𝐹
78	59	cntzidss 17593	. . . . . . . . 9 ⊢ ((ran 𝐹 ⊆ (𝑍‘ran 𝐹) ∧ ran (𝐹 ∘ 𝐻) ⊆ ran 𝐹) → ran (𝐹 ∘ 𝐻) ⊆ (𝑍‘ran (𝐹 ∘ 𝐻)))
79	63, 77, 78	sylancl 693	. . . . . . . 8 ⊢ (𝜑 → ran (𝐹 ∘ 𝐻) ⊆ (𝑍‘ran (𝐹 ∘ 𝐻)))
80	79	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)))
81		f1ocnv 6062	. . . . . . . . . 10 ⊢ (𝐻:𝐶–1-1-onto→𝐴 → ^◡𝐻:𝐴–1-1-onto→𝐶)
82		f1of1 6049	. . . . . . . . . 10 ⊢ (^◡𝐻:𝐴–1-1-onto→𝐶 → ^◡𝐻:𝐴–1-1→𝐶)
83	6, 81, 82	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → ^◡𝐻:𝐴–1-1→𝐶)
84	83	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ^◡𝐻:𝐴–1-1→𝐶)
85		f1co 6023	. . . . . . . 8 ⊢ ((^◡𝐻:𝐴–1-1→𝐶 ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1→𝐴) → (^◡𝐻 ∘ 𝑓):(1...(#‘(𝐹 supp 0 )))–1-1→𝐶)
86	84, 46, 85	syl2anc 691	. . . . . . 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)
87	19	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ (𝐹 supp 0 ))
88		fex 6394	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V)
89	15, 2, 88	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 ∈ V)
90		suppimacnv 7193	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 supp 0 ) = (^◡𝐹 “ (V ∖ { 0 })))
91	89, 17, 90	sylancl 693	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹 supp 0 ) = (^◡𝐹 “ (V ∖ { 0 })))
92	91	eqcomd 2616	. . . . . . . . . . . 12 ⊢ (𝜑 → (^◡𝐹 “ (V ∖ { 0 })) = (𝐹 supp 0 ))
93	92	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 ))
94	87, 93, 69	3sstr4d 3611	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (^◡𝐹 “ (V ∖ { 0 })) ⊆ ran 𝑓)
95		imass2 5420	. . . . . . . . . 10 ⊢ ((^◡𝐹 “ (V ∖ { 0 })) ⊆ ran 𝑓 → (^◡𝐻 “ (^◡𝐹 “ (V ∖ { 0 }))) ⊆ (^◡𝐻 “ ran 𝑓))
96	94, 95	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))
97		cnvco 5230	. . . . . . . . . . 11 ⊢ ^◡(𝐹 ∘ 𝐻) = (^◡𝐻 ∘ ^◡𝐹)
98	97	imaeq1i 5382	. . . . . . . . . 10 ⊢ (^◡(𝐹 ∘ 𝐻) “ (V ∖ { 0 })) = ((^◡𝐻 ∘ ^◡𝐹) “ (V ∖ { 0 }))
99		imaco 5557	. . . . . . . . . 10 ⊢ ((^◡𝐻 ∘ ^◡𝐹) “ (V ∖ { 0 })) = (^◡𝐻 “ (^◡𝐹 “ (V ∖ { 0 })))
100	98, 99	eqtri 2632	. . . . . . . . 9 ⊢ (^◡(𝐹 ∘ 𝐻) “ (V ∖ { 0 })) = (^◡𝐻 “ (^◡𝐹 “ (V ∖ { 0 })))
101		rnco2 5559	. . . . . . . . 9 ⊢ ran (^◡𝐻 ∘ 𝑓) = (^◡𝐻 “ ran 𝑓)
102	96, 100, 101	3sstr4g 3609	. . . . . . . 8 ⊢ ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (^◡(𝐹 ∘ 𝐻) “ (V ∖ { 0 })) ⊆ ran (^◡𝐻 ∘ 𝑓))
103		f1oexrnex 7008	. . . . . . . . . . . . 13 ⊢ ((𝐻:𝐶–1-1-onto→𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐻 ∈ V)
104	6, 2, 103	syl2anc 691	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐻 ∈ V)
105		coexg 7010	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ V ∧ 𝐻 ∈ V) → (𝐹 ∘ 𝐻) ∈ V)
106	89, 104, 105	syl2anc 691	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 ∘ 𝐻) ∈ V)
107		suppimacnv 7193	. . . . . . . . . . 11 ⊢ (((𝐹 ∘ 𝐻) ∈ V ∧ 0 ∈ V) → ((𝐹 ∘ 𝐻) supp 0 ) = (^◡(𝐹 ∘ 𝐻) “ (V ∖ { 0 })))
108	106, 17, 107	sylancl 693	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐹 ∘ 𝐻) supp 0 ) = (^◡(𝐹 ∘ 𝐻) “ (V ∖ { 0 })))
109	108	sseq1d 3595	. . . . . . . . 9 ⊢ (𝜑 → (((𝐹 ∘ 𝐻) supp 0 ) ⊆ ran (^◡𝐻 ∘ 𝑓) ↔ (^◡(𝐹 ∘ 𝐻) “ (V ∖ { 0 })) ⊆ ran (^◡𝐻 ∘ 𝑓)))
110	109	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)))
111	102, 110	mpbird 246	. . . . . . 7 ⊢ ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ((𝐹 ∘ 𝐻) supp 0 ) ⊆ ran (^◡𝐻 ∘ 𝑓))
112		eqid 2610	. . . . . . 7 ⊢ (((𝐹 ∘ 𝐻) ∘ (^◡𝐻 ∘ 𝑓)) supp 0 ) = (((𝐹 ∘ 𝐻) ∘ (^◡𝐻 ∘ 𝑓)) supp 0 )
113	57, 3, 58, 59, 60, 73, 76, 80, 65, 86, 111, 112	gsumval3 18131	. . . . . 6 ⊢ ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐺 Σ_g (𝐹 ∘ 𝐻)) = (seq1((+_g‘𝐺), ((𝐹 ∘ 𝐻) ∘ (^◡𝐻 ∘ 𝑓)))‘(#‘(𝐹 supp 0 ))))
114	56, 72, 113	3eqtr4d 2654	. . . . 5 ⊢ ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐺 Σ_g 𝐹) = (𝐺 Σ_g (𝐹 ∘ 𝐻)))
115	114	expr 641	. . . 4 ⊢ ((𝜑 ∧ (#‘(𝐹 supp 0 )) ∈ ℕ) → (𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → (𝐺 Σ_g 𝐹) = (𝐺 Σ_g (𝐹 ∘ 𝐻))))
116	115	exlimdv 1848	. . 3 ⊢ ((𝜑 ∧ (#‘(𝐹 supp 0 )) ∈ ℕ) → (∃𝑓 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → (𝐺 Σ_g 𝐹) = (𝐺 Σ_g (𝐹 ∘ 𝐻))))
117	116	expimpd 627	. 2 ⊢ (𝜑 → (((#‘(𝐹 supp 0 )) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 )) → (𝐺 Σ_g 𝐹) = (𝐺 Σ_g (𝐹 ∘ 𝐻))))
118		gsumzcl.w	. . 3 ⊢ (𝜑 → 𝐹 finSupp 0 )
119		fsuppimp 8164	. . . 4 ⊢ (𝐹 finSupp 0 → (Fun 𝐹 ∧ (𝐹 supp 0 ) ∈ Fin))
120	119	simprd 478	. . 3 ⊢ (𝐹 finSupp 0 → (𝐹 supp 0 ) ∈ Fin)
121		fz1f1o 14288	. . 3 ⊢ ((𝐹 supp 0 ) ∈ Fin → ((𝐹 supp 0 ) = ∅ ∨ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))))
122	118, 120, 121	3syl 18	. 2 ⊢ (𝜑 → ((𝐹 supp 0 ) = ∅ ∨ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))))
123	32, 117, 122	mpjaod 395	1 ⊢ (𝜑 → (𝐺 Σ_g 𝐹) = (𝐺 Σ_g (𝐹 ∘ 𝐻)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 Vcvv 3173 ∖ cdif 3537 ⊆ wss 3540 ∅c0 3874 {csn 4125 class class class wbr 4583 ↦ cmpt 4643 I cid 4948 ^◡ccnv 5037 dom cdm 5038 ran crn 5039 ↾ cres 5040 “ cima 5041 ∘ ccom 5042 Fun wfun 5798 ⟶wf 5800 –1-1→wf1 5801 –onto→wfo 5802 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 supp csupp 7182 Fincfn 7841 finSupp cfsupp 8158 1c1 9816 ℕcn 10897 ...cfz 12197 seqcseq 12663 #chash 12979 Basecbs 15695 +_gcplusg 15768 0_gc0g 15923 Σ_g cgsu 15924 Mndcmnd 17117 Cntzccntz 17571

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-0g 15925 df-gsum 15926 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-cntz 17573

This theorem is referenced by: gsumf1o 18140 smadiadetlem3 20293

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