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Theorem gsumzf1o 18313

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 17374	. . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )
5	1, 2, 4	syl2anc 693	. . . . . 6 ⊢ (𝜑 → (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )
6		gsumzf1o.h	. . . . . . . . 9 ⊢ (𝜑 → 𝐻:𝐶–1-1-onto→𝐴)
7		f1of1 6136	. . . . . . . . 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 7136	. . . . . . . 8 ⊢ ((𝐻:𝐶–1-1→𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐶 ∈ V)
10	8, 2, 9	syl2anc 693	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ V)
11	3	gsumz 17374	. . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ 𝐶 ∈ V) → (𝐺 Σ_g (𝑥 ∈ 𝐶 ↦ 0 )) = 0 )
12	1, 10, 11	syl2anc 693	. . . . . 6 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ 𝐶 ↦ 0 )) = 0 )
13	5, 12	eqtr4d 2659	. . . . 5 ⊢ (𝜑 → (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 0 )) = (𝐺 Σ_g (𝑥 ∈ 𝐶 ↦ 0 )))
14	13	adantr 481	. . . 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 6201	. . . . . . . 8 ⊢ (0_g‘𝐺) ∈ V
17	3, 16	eqeltri 2697	. . . . . . 7 ⊢ 0 ∈ V
18	17	a1i 11	. . . . . 6 ⊢ (𝜑 → 0 ∈ V)
19		ssid 3624	. . . . . . 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 18309	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → 𝐹 = (𝑘 ∈ 𝐴 ↦ 0 ))
22	21	oveq2d 6666	. . . 4 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σ_g 𝐹) = (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 0 )))
23		f1of 6137	. . . . . . . . 9 ⊢ (𝐻:𝐶–1-1-onto→𝐴 → 𝐻:𝐶⟶𝐴)
24	6, 23	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐻:𝐶⟶𝐴)
25	24	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → 𝐻:𝐶⟶𝐴)
26	25	ffvelrnda 6359	. . . . . 6 ⊢ (((𝜑 ∧ (𝐹 supp 0 ) = ∅) ∧ 𝑥 ∈ 𝐶) → (𝐻‘𝑥) ∈ 𝐴)
27	25	feqmptd 6249	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → 𝐻 = (𝑥 ∈ 𝐶 ↦ (𝐻‘𝑥)))
28		eqidd 2623	. . . . . 6 ⊢ (𝑘 = (𝐻‘𝑥) → 0 = 0 )
29	26, 27, 21, 28	fmptco 6396	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐹 ∘ 𝐻) = (𝑥 ∈ 𝐶 ↦ 0 ))
30	29	oveq2d 6666	. . . 4 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σ_g (𝐹 ∘ 𝐻)) = (𝐺 Σ_g (𝑥 ∈ 𝐶 ↦ 0 )))
31	14, 22, 30	3eqtr4d 2666	. . 3 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σ_g 𝐹) = (𝐺 Σ_g (𝐹 ∘ 𝐻)))
32	31	ex 450	. 2 ⊢ (𝜑 → ((𝐹 supp 0 ) = ∅ → (𝐺 Σ_g 𝐹) = (𝐺 Σ_g (𝐹 ∘ 𝐻))))
33		coass 5654	. . . . . . . . . . 11 ⊢ ((𝐻 ∘ ^◡𝐻) ∘ 𝑓) = (𝐻 ∘ (^◡𝐻 ∘ 𝑓))
34	6	adantr 481	. . . . . . . . . . . . . 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 6163	. . . . . . . . . . . . . 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 5283	. . . . . . . . . . . 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 6136	. . . . . . . . . . . . . . 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 765	. . . . . . . . . . . . . 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 7308	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 supp 0 ) ⊆ dom 𝐹
41		fdm 6051	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
42	15, 41	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom 𝐹 = 𝐴)
43	40, 42	syl5sseq 3653	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ 𝐴)
44	43	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ 𝐴)
45		f1ss 6106	. . . . . . . . . . . . . 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 693	. . . . . . . . . . . . 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 6101	. . . . . . . . . . . . 13 ⊢ (𝑓:(1...(#‘(𝐹 supp 0 )))–1-1→𝐴 → 𝑓:(1...(#‘(𝐹 supp 0 )))⟶𝐴)
48		fcoi2 6079	. . . . . . . . . . . . 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 2656	. . . . . . . . . . 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 2671	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝑓 = (𝐻 ∘ (^◡𝐻 ∘ 𝑓)))
52	51	coeq2d 5284	. . . . . . . . 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 5654	. . . . . . . . 9 ⊢ ((𝐹 ∘ 𝐻) ∘ (^◡𝐻 ∘ 𝑓)) = (𝐹 ∘ (𝐻 ∘ (^◡𝐻 ∘ 𝑓)))
54	52, 53	syl6eqr 2674	. . . . . . . 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 12809	. . . . . . 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 6193	. . . . . 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 2622	. . . . . . 7 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
59		gsumzcl.z	. . . . . . 7 ⊢ 𝑍 = (Cntz‘𝐺)
60	1	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐺 ∈ Mnd)
61	2	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐴 ∈ 𝑉)
62	15	adantr 481	. . . . . . 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 481	. . . . . . 7 ⊢ ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
65		simprl 794	. . . . . . 7 ⊢ ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (#‘(𝐹 supp 0 )) ∈ ℕ)
66		f1ofo 6144	. . . . . . . . . 10 ⊢ (𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → 𝑓:(1...(#‘(𝐹 supp 0 )))–onto→(𝐹 supp 0 ))
67		forn 6118	. . . . . . . . . 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 765	. . . . . . . 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 3654	. . . . . . 7 ⊢ ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ ran 𝑓)
71		eqid 2622	. . . . . . 7 ⊢ ((𝐹 ∘ 𝑓) supp 0 ) = ((𝐹 ∘ 𝑓) supp 0 )
72	57, 3, 58, 59, 60, 61, 62, 64, 65, 46, 70, 71	gsumval3 18308	. . . . . 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 481	. . . . . . 7 ⊢ ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐶 ∈ V)
74		fco 6058	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐻:𝐶⟶𝐴) → (𝐹 ∘ 𝐻):𝐶⟶𝐵)
75	15, 24, 74	syl2anc 693	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∘ 𝐻):𝐶⟶𝐵)
76	75	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 ∘ 𝐻):𝐶⟶𝐵)
77		rncoss 5386	. . . . . . . . 9 ⊢ ran (𝐹 ∘ 𝐻) ⊆ ran 𝐹
78	59	cntzidss 17770	. . . . . . . . 9 ⊢ ((ran 𝐹 ⊆ (𝑍‘ran 𝐹) ∧ ran (𝐹 ∘ 𝐻) ⊆ ran 𝐹) → ran (𝐹 ∘ 𝐻) ⊆ (𝑍‘ran (𝐹 ∘ 𝐻)))
79	63, 77, 78	sylancl 694	. . . . . . . 8 ⊢ (𝜑 → ran (𝐹 ∘ 𝐻) ⊆ (𝑍‘ran (𝐹 ∘ 𝐻)))
80	79	adantr 481	. . . . . . 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 6149	. . . . . . . . . 10 ⊢ (𝐻:𝐶–1-1-onto→𝐴 → ^◡𝐻:𝐴–1-1-onto→𝐶)
82		f1of1 6136	. . . . . . . . . 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 481	. . . . . . . 8 ⊢ ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ^◡𝐻:𝐴–1-1→𝐶)
85		f1co 6110	. . . . . . . 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 693	. . . . . . 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 6490	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V)
89	15, 2, 88	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 ∈ V)
90		suppimacnv 7306	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 supp 0 ) = (^◡𝐹 “ (V ∖ { 0 })))
91	89, 17, 90	sylancl 694	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹 supp 0 ) = (^◡𝐹 “ (V ∖ { 0 })))
92	91	eqcomd 2628	. . . . . . . . . . . 12 ⊢ (𝜑 → (^◡𝐹 “ (V ∖ { 0 })) = (𝐹 supp 0 ))
93	92	adantr 481	. . . . . . . . . . 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 3648	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (^◡𝐹 “ (V ∖ { 0 })) ⊆ ran 𝑓)
95		imass2 5501	. . . . . . . . . 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 5308	. . . . . . . . . . 11 ⊢ ^◡(𝐹 ∘ 𝐻) = (^◡𝐻 ∘ ^◡𝐹)
98	97	imaeq1i 5463	. . . . . . . . . 10 ⊢ (^◡(𝐹 ∘ 𝐻) “ (V ∖ { 0 })) = ((^◡𝐻 ∘ ^◡𝐹) “ (V ∖ { 0 }))
99		imaco 5640	. . . . . . . . . 10 ⊢ ((^◡𝐻 ∘ ^◡𝐹) “ (V ∖ { 0 })) = (^◡𝐻 “ (^◡𝐹 “ (V ∖ { 0 })))
100	98, 99	eqtri 2644	. . . . . . . . 9 ⊢ (^◡(𝐹 ∘ 𝐻) “ (V ∖ { 0 })) = (^◡𝐻 “ (^◡𝐹 “ (V ∖ { 0 })))
101		rnco2 5642	. . . . . . . . 9 ⊢ ran (^◡𝐻 ∘ 𝑓) = (^◡𝐻 “ ran 𝑓)
102	96, 100, 101	3sstr4g 3646	. . . . . . . 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 7115	. . . . . . . . . . . . 13 ⊢ ((𝐻:𝐶–1-1-onto→𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐻 ∈ V)
104	6, 2, 103	syl2anc 693	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐻 ∈ V)
105		coexg 7117	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ V ∧ 𝐻 ∈ V) → (𝐹 ∘ 𝐻) ∈ V)
106	89, 104, 105	syl2anc 693	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 ∘ 𝐻) ∈ V)
107		suppimacnv 7306	. . . . . . . . . . 11 ⊢ (((𝐹 ∘ 𝐻) ∈ V ∧ 0 ∈ V) → ((𝐹 ∘ 𝐻) supp 0 ) = (^◡(𝐹 ∘ 𝐻) “ (V ∖ { 0 })))
108	106, 17, 107	sylancl 694	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐹 ∘ 𝐻) supp 0 ) = (^◡(𝐹 ∘ 𝐻) “ (V ∖ { 0 })))
109	108	sseq1d 3632	. . . . . . . . 9 ⊢ (𝜑 → (((𝐹 ∘ 𝐻) supp 0 ) ⊆ ran (^◡𝐻 ∘ 𝑓) ↔ (^◡(𝐹 ∘ 𝐻) “ (V ∖ { 0 })) ⊆ ran (^◡𝐻 ∘ 𝑓)))
110	109	adantr 481	. . . . . . . 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 247	. . . . . . 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 2622	. . . . . . 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 18308	. . . . . 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 2666	. . . . 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 643	. . . 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 1861	. . 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 629	. 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 8281	. . . 4 ⊢ (𝐹 finSupp 0 → (Fun 𝐹 ∧ (𝐹 supp 0 ) ∈ Fin))
120	119	simprd 479	. . 3 ⊢ (𝐹 finSupp 0 → (𝐹 supp 0 ) ∈ Fin)
121		fz1f1o 14441	. . 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 396	1 ⊢ (𝜑 → (𝐺 Σ_g 𝐹) = (𝐺 Σ_g (𝐹 ∘ 𝐻)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 Vcvv 3200 ∖ cdif 3571 ⊆ wss 3574 ∅c0 3915 {csn 4177 class class class wbr 4653 ↦ cmpt 4729 I cid 5023 ^◡ccnv 5113 dom cdm 5114 ran crn 5115 ↾ cres 5116 “ cima 5117 ∘ ccom 5118 Fun wfun 5882 ⟶wf 5884 –1-1→wf1 5885 –onto→wfo 5886 –1-1-onto→wf1o 5887 ‘cfv 5888 (class class class)co 6650 supp csupp 7295 Fincfn 7955 finSupp cfsupp 8275 1c1 9937 ℕcn 11020 ...cfz 12326 seqcseq 12801 #chash 13117 Basecbs 15857 +_gcplusg 15941 0_gc0g 16100 Σ_g cgsu 16101 Mndcmnd 17294 Cntzccntz 17748

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-0g 16102 df-gsum 16103 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-cntz 17750

This theorem is referenced by: gsumf1o 18317 smadiadetlem3 20474

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