Theorem tsmsf1o 24372

Description: Re-index an infinite group sum using a bijection. (Contributed by Mario Carneiro, 18-Sep-2015.)

Hypotheses

Ref	Expression
tsmsf1o.b	⊢ 𝐵 = (Base‘𝐺)
tsmsf1o.1	⊢ (𝜑 → 𝐺 ∈ CMnd)
tsmsf1o.2	⊢ (𝜑 → 𝐺 ∈ TopSp)
tsmsf1o.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
tsmsf1o.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
tsmsf1o.s	⊢ (𝜑 → 𝐻:𝐶–1-1-onto→𝐴)

Assertion

Ref	Expression
tsmsf1o	⊢ (𝜑 → (𝐺 tsums 𝐹) = (𝐺 tsums (𝐹 ∘ 𝐻)))

Proof of Theorem tsmsf1o

Dummy variables 𝑎 𝑏 𝑢 𝑦 𝑧 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tsmsf1o.s	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐻:𝐶–1-1-onto→𝐴)
2		f1opwfi 9326	. . . . . . . . . . 11 ⊢ (𝐻:𝐶–1-1-onto→𝐴 → (𝑎 ∈ (𝒫 𝐶 ∩ Fin) ↦ (𝐻 “ 𝑎)):(𝒫 𝐶 ∩ Fin)–1-1-onto→(𝒫 𝐴 ∩ Fin))
3	1, 2	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (𝑎 ∈ (𝒫 𝐶 ∩ Fin) ↦ (𝐻 “ 𝑎)):(𝒫 𝐶 ∩ Fin)–1-1-onto→(𝒫 𝐴 ∩ Fin))
4		f1of 6821	. . . . . . . . . 10 ⊢ ((𝑎 ∈ (𝒫 𝐶 ∩ Fin) ↦ (𝐻 “ 𝑎)):(𝒫 𝐶 ∩ Fin)–1-1-onto→(𝒫 𝐴 ∩ Fin) → (𝑎 ∈ (𝒫 𝐶 ∩ Fin) ↦ (𝐻 “ 𝑎)):(𝒫 𝐶 ∩ Fin)⟶(𝒫 𝐴 ∩ Fin))
5	3, 4	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝑎 ∈ (𝒫 𝐶 ∩ Fin) ↦ (𝐻 “ 𝑎)):(𝒫 𝐶 ∩ Fin)⟶(𝒫 𝐴 ∩ Fin))
6		eqid 2762	. . . . . . . . . 10 ⊢ (𝑎 ∈ (𝒫 𝐶 ∩ Fin) ↦ (𝐻 “ 𝑎)) = (𝑎 ∈ (𝒫 𝐶 ∩ Fin) ↦ (𝐻 “ 𝑎))
7	6	fmpt 7106	. . . . . . . . 9 ⊢ (∀𝑎 ∈ (𝒫 𝐶 ∩ Fin)(𝐻 “ 𝑎) ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑎 ∈ (𝒫 𝐶 ∩ Fin) ↦ (𝐻 “ 𝑎)):(𝒫 𝐶 ∩ Fin)⟶(𝒫 𝐴 ∩ Fin))
8	5, 7	sylibr 237	. . . . . . . 8 ⊢ (𝜑 → ∀𝑎 ∈ (𝒫 𝐶 ∩ Fin)(𝐻 “ 𝑎) ∈ (𝒫 𝐴 ∩ Fin))
9		sseq1 3959	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐻 “ 𝑎) → (𝑦 ⊆ 𝑧 ↔ (𝐻 “ 𝑎) ⊆ 𝑧))
10	9	imbi1d 344	. . . . . . . . . 10 ⊢ (𝑦 = (𝐻 “ 𝑎) → ((𝑦 ⊆ 𝑧 → (𝐺 Σg (𝐹 ↾ 𝑧)) ∈ 𝑢) ↔ ((𝐻 “ 𝑎) ⊆ 𝑧 → (𝐺 Σg (𝐹 ↾ 𝑧)) ∈ 𝑢)))
11	10	ralbidv 3187	. . . . . . . . 9 ⊢ (𝑦 = (𝐻 “ 𝑎) → (∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σg (𝐹 ↾ 𝑧)) ∈ 𝑢) ↔ ∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)((𝐻 “ 𝑎) ⊆ 𝑧 → (𝐺 Σg (𝐹 ↾ 𝑧)) ∈ 𝑢)))
12	6, 11	rexrnmptw 7091	. . . . . . . 8 ⊢ (∀𝑎 ∈ (𝒫 𝐶 ∩ Fin)(𝐻 “ 𝑎) ∈ (𝒫 𝐴 ∩ Fin) → (∃𝑦 ∈ ran (𝑎 ∈ (𝒫 𝐶 ∩ Fin) ↦ (𝐻 “ 𝑎))∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σg (𝐹 ↾ 𝑧)) ∈ 𝑢) ↔ ∃𝑎 ∈ (𝒫 𝐶 ∩ Fin)∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)((𝐻 “ 𝑎) ⊆ 𝑧 → (𝐺 Σg (𝐹 ↾ 𝑧)) ∈ 𝑢)))
13	8, 12	syl 18	. . . . . . 7 ⊢ (𝜑 → (∃𝑦 ∈ ran (𝑎 ∈ (𝒫 𝐶 ∩ Fin) ↦ (𝐻 “ 𝑎))∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σg (𝐹 ↾ 𝑧)) ∈ 𝑢) ↔ ∃𝑎 ∈ (𝒫 𝐶 ∩ Fin)∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)((𝐻 “ 𝑎) ⊆ 𝑧 → (𝐺 Σg (𝐹 ↾ 𝑧)) ∈ 𝑢)))
14		f1ofo 6829	. . . . . . . . 9 ⊢ ((𝑎 ∈ (𝒫 𝐶 ∩ Fin) ↦ (𝐻 “ 𝑎)):(𝒫 𝐶 ∩ Fin)–1-1-onto→(𝒫 𝐴 ∩ Fin) → (𝑎 ∈ (𝒫 𝐶 ∩ Fin) ↦ (𝐻 “ 𝑎)):(𝒫 𝐶 ∩ Fin)–onto→(𝒫 𝐴 ∩ Fin))
15		forn 6796	. . . . . . . . 9 ⊢ ((𝑎 ∈ (𝒫 𝐶 ∩ Fin) ↦ (𝐻 “ 𝑎)):(𝒫 𝐶 ∩ Fin)–onto→(𝒫 𝐴 ∩ Fin) → ran (𝑎 ∈ (𝒫 𝐶 ∩ Fin) ↦ (𝐻 “ 𝑎)) = (𝒫 𝐴 ∩ Fin))
16	3, 14, 15	3syl 19	. . . . . . . 8 ⊢ (𝜑 → ran (𝑎 ∈ (𝒫 𝐶 ∩ Fin) ↦ (𝐻 “ 𝑎)) = (𝒫 𝐴 ∩ Fin))
17	16	rexeqdv 3322	. . . . . . 7 ⊢ (𝜑 → (∃𝑦 ∈ ran (𝑎 ∈ (𝒫 𝐶 ∩ Fin) ↦ (𝐻 “ 𝑎))∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σg (𝐹 ↾ 𝑧)) ∈ 𝑢) ↔ ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σg (𝐹 ↾ 𝑧)) ∈ 𝑢)))
18		imaeq2 6056	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑏 → (𝐻 “ 𝑎) = (𝐻 “ 𝑏))
19	18	cbvmptv 5213	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ (𝒫 𝐶 ∩ Fin) ↦ (𝐻 “ 𝑎)) = (𝑏 ∈ (𝒫 𝐶 ∩ Fin) ↦ (𝐻 “ 𝑏))
20	19	fmpt 7106	. . . . . . . . . . . . 13 ⊢ (∀𝑏 ∈ (𝒫 𝐶 ∩ Fin)(𝐻 “ 𝑏) ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑎 ∈ (𝒫 𝐶 ∩ Fin) ↦ (𝐻 “ 𝑎)):(𝒫 𝐶 ∩ Fin)⟶(𝒫 𝐴 ∩ Fin))
21	5, 20	sylibr 237	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑏 ∈ (𝒫 𝐶 ∩ Fin)(𝐻 “ 𝑏) ∈ (𝒫 𝐴 ∩ Fin))
22		sseq2 3960	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝐻 “ 𝑏) → ((𝐻 “ 𝑎) ⊆ 𝑧 ↔ (𝐻 “ 𝑎) ⊆ (𝐻 “ 𝑏)))
23		reseq2 5971	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝐻 “ 𝑏) → (𝐹 ↾ 𝑧) = (𝐹 ↾ (𝐻 “ 𝑏)))
24	23	oveq2d 7432	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝐻 “ 𝑏) → (𝐺 Σg (𝐹 ↾ 𝑧)) = (𝐺 Σg (𝐹 ↾ (𝐻 “ 𝑏))))
25	24	eleq1d 2847	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝐻 “ 𝑏) → ((𝐺 Σg (𝐹 ↾ 𝑧)) ∈ 𝑢 ↔ (𝐺 Σg (𝐹 ↾ (𝐻 “ 𝑏))) ∈ 𝑢))
26	22, 25	imbi12d 347	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐻 “ 𝑏) → (((𝐻 “ 𝑎) ⊆ 𝑧 → (𝐺 Σg (𝐹 ↾ 𝑧)) ∈ 𝑢) ↔ ((𝐻 “ 𝑎) ⊆ (𝐻 “ 𝑏) → (𝐺 Σg (𝐹 ↾ (𝐻 “ 𝑏))) ∈ 𝑢)))
27	19, 26	ralrnmptw 7090	. . . . . . . . . . . 12 ⊢ (∀𝑏 ∈ (𝒫 𝐶 ∩ Fin)(𝐻 “ 𝑏) ∈ (𝒫 𝐴 ∩ Fin) → (∀𝑧 ∈ ran (𝑎 ∈ (𝒫 𝐶 ∩ Fin) ↦ (𝐻 “ 𝑎))((𝐻 “ 𝑎) ⊆ 𝑧 → (𝐺 Σg (𝐹 ↾ 𝑧)) ∈ 𝑢) ↔ ∀𝑏 ∈ (𝒫 𝐶 ∩ Fin)((𝐻 “ 𝑎) ⊆ (𝐻 “ 𝑏) → (𝐺 Σg (𝐹 ↾ (𝐻 “ 𝑏))) ∈ 𝑢)))
28	21, 27	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (∀𝑧 ∈ ran (𝑎 ∈ (𝒫 𝐶 ∩ Fin) ↦ (𝐻 “ 𝑎))((𝐻 “ 𝑎) ⊆ 𝑧 → (𝐺 Σg (𝐹 ↾ 𝑧)) ∈ 𝑢) ↔ ∀𝑏 ∈ (𝒫 𝐶 ∩ Fin)((𝐻 “ 𝑎) ⊆ (𝐻 “ 𝑏) → (𝐺 Σg (𝐹 ↾ (𝐻 “ 𝑏))) ∈ 𝑢)))
29	16	raleqdv 3321	. . . . . . . . . . 11 ⊢ (𝜑 → (∀𝑧 ∈ ran (𝑎 ∈ (𝒫 𝐶 ∩ Fin) ↦ (𝐻 “ 𝑎))((𝐻 “ 𝑎) ⊆ 𝑧 → (𝐺 Σg (𝐹 ↾ 𝑧)) ∈ 𝑢) ↔ ∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)((𝐻 “ 𝑎) ⊆ 𝑧 → (𝐺 Σg (𝐹 ↾ 𝑧)) ∈ 𝑢)))
30	28, 29	bitr3d 284	. . . . . . . . . 10 ⊢ (𝜑 → (∀𝑏 ∈ (𝒫 𝐶 ∩ Fin)((𝐻 “ 𝑎) ⊆ (𝐻 “ 𝑏) → (𝐺 Σg (𝐹 ↾ (𝐻 “ 𝑏))) ∈ 𝑢) ↔ ∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)((𝐻 “ 𝑎) ⊆ 𝑧 → (𝐺 Σg (𝐹 ↾ 𝑧)) ∈ 𝑢)))
31	30	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) → (∀𝑏 ∈ (𝒫 𝐶 ∩ Fin)((𝐻 “ 𝑎) ⊆ (𝐻 “ 𝑏) → (𝐺 Σg (𝐹 ↾ (𝐻 “ 𝑏))) ∈ 𝑢) ↔ ∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)((𝐻 “ 𝑎) ⊆ 𝑧 → (𝐺 Σg (𝐹 ↾ 𝑧)) ∈ 𝑢)))
32		f1of1 6820	. . . . . . . . . . . . . 14 ⊢ (𝐻:𝐶–1-1-onto→𝐴 → 𝐻:𝐶–1-1→𝐴)
33	1, 32	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐻:𝐶–1-1→𝐴)
34	33	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → 𝐻:𝐶–1-1→𝐴)
35		elfpw 9324	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ (𝒫 𝐶 ∩ Fin) ↔ (𝑎 ⊆ 𝐶 ∧ 𝑎 ∈ Fin))
36	35	simplbi 502	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ (𝒫 𝐶 ∩ Fin) → 𝑎 ⊆ 𝐶)
37	36	ad2antlr 740	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → 𝑎 ⊆ 𝐶)
38		elfpw 9324	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ (𝒫 𝐶 ∩ Fin) ↔ (𝑏 ⊆ 𝐶 ∧ 𝑏 ∈ Fin))
39	38	simplbi 502	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ (𝒫 𝐶 ∩ Fin) → 𝑏 ⊆ 𝐶)
40	39	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → 𝑏 ⊆ 𝐶)
41		f1imass 7264	. . . . . . . . . . . 12 ⊢ ((𝐻:𝐶–1-1→𝐴 ∧ (𝑎 ⊆ 𝐶 ∧ 𝑏 ⊆ 𝐶)) → ((𝐻 “ 𝑎) ⊆ (𝐻 “ 𝑏) ↔ 𝑎 ⊆ 𝑏))
42	34, 37, 40, 41	syl12anc 850	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → ((𝐻 “ 𝑎) ⊆ (𝐻 “ 𝑏) ↔ 𝑎 ⊆ 𝑏))
43		tsmsf1o.b	. . . . . . . . . . . . . 14 ⊢ 𝐵 = (Base‘𝐺)
44		eqid 2762	. . . . . . . . . . . . . 14 ⊢ (0g‘𝐺) = (0g‘𝐺)
45		tsmsf1o.1	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺 ∈ CMnd)
46	45	ad2antrr 739	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → 𝐺 ∈ CMnd)
47		elinel2 4151	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ∈ (𝒫 𝐶 ∩ Fin) → 𝑏 ∈ Fin)
48	47	adantl 487	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → 𝑏 ∈ Fin)
49		f1ores 6836	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐻:𝐶–1-1→𝐴 ∧ 𝑏 ⊆ 𝐶) → (𝐻 ↾ 𝑏):𝑏–1-1-onto→(𝐻 “ 𝑏))
50	34, 40, 49	syl2anc 596	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐻 ↾ 𝑏):𝑏–1-1-onto→(𝐻 “ 𝑏))
51		f1ofo 6829	. . . . . . . . . . . . . . . 16 ⊢ ((𝐻 ↾ 𝑏):𝑏–1-1-onto→(𝐻 “ 𝑏) → (𝐻 ↾ 𝑏):𝑏–onto→(𝐻 “ 𝑏))
52	50, 51	syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐻 ↾ 𝑏):𝑏–onto→(𝐻 “ 𝑏))
53		fofi 9286	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ∈ Fin ∧ (𝐻 ↾ 𝑏):𝑏–onto→(𝐻 “ 𝑏)) → (𝐻 “ 𝑏) ∈ Fin)
54	48, 52, 53	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐻 “ 𝑏) ∈ Fin)
55		tsmsf1o.f	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
56	55	ad2antrr 739	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → 𝐹:𝐴⟶𝐵)
57		imassrn 6071	. . . . . . . . . . . . . . . 16 ⊢ (𝐻 “ 𝑏) ⊆ ran 𝐻
58	1	ad2antrr 739	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → 𝐻:𝐶–1-1-onto→𝐴)
59		f1ofo 6829	. . . . . . . . . . . . . . . . 17 ⊢ (𝐻:𝐶–1-1-onto→𝐴 → 𝐻:𝐶–onto→𝐴)
60		forn 6796	. . . . . . . . . . . . . . . . 17 ⊢ (𝐻:𝐶–onto→𝐴 → ran 𝐻 = 𝐴)
61	58, 59, 60	3syl 19	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → ran 𝐻 = 𝐴)
62	57, 61	sseqtrid 3976	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐻 “ 𝑏) ⊆ 𝐴)
63	56, 62	fssresd 6746	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹 ↾ (𝐻 “ 𝑏)):(𝐻 “ 𝑏)⟶𝐵)
64		fvexd 6897	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → (0g‘𝐺) ∈ V)
65	63, 54, 64	fdmfifsupp 9348	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹 ↾ (𝐻 “ 𝑏)) finSupp (0g‘𝐺))
66	43, 44, 46, 54, 63, 65, 50	gsumf1o 20044	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐺 Σg (𝐹 ↾ (𝐻 “ 𝑏))) = (𝐺 Σg ((𝐹 ↾ (𝐻 “ 𝑏)) ∘ (𝐻 ↾ 𝑏))))
67		df-ima 5672	. . . . . . . . . . . . . . . . 17 ⊢ (𝐻 “ 𝑏) = ran (𝐻 ↾ 𝑏)
68	67	eqimss2i 3995	. . . . . . . . . . . . . . . 16 ⊢ ran (𝐻 ↾ 𝑏) ⊆ (𝐻 “ 𝑏)
69		cores 6249	. . . . . . . . . . . . . . . 16 ⊢ (ran (𝐻 ↾ 𝑏) ⊆ (𝐻 “ 𝑏) → ((𝐹 ↾ (𝐻 “ 𝑏)) ∘ (𝐻 ↾ 𝑏)) = (𝐹 ∘ (𝐻 ↾ 𝑏)))
70	68, 69	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ↾ (𝐻 “ 𝑏)) ∘ (𝐻 ↾ 𝑏)) = (𝐹 ∘ (𝐻 ↾ 𝑏))
71		resco 6250	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∘ 𝐻) ↾ 𝑏) = (𝐹 ∘ (𝐻 ↾ 𝑏))
72	70, 71	eqtr4i 2788	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ↾ (𝐻 “ 𝑏)) ∘ (𝐻 ↾ 𝑏)) = ((𝐹 ∘ 𝐻) ↾ 𝑏)
73	72	oveq2i 7427	. . . . . . . . . . . . 13 ⊢ (𝐺 Σg ((𝐹 ↾ (𝐻 “ 𝑏)) ∘ (𝐻 ↾ 𝑏))) = (𝐺 Σg ((𝐹 ∘ 𝐻) ↾ 𝑏))
74	66, 73	eqtrdi 2813	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐺 Σg (𝐹 ↾ (𝐻 “ 𝑏))) = (𝐺 Σg ((𝐹 ∘ 𝐻) ↾ 𝑏)))
75	74	eleq1d 2847	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → ((𝐺 Σg (𝐹 ↾ (𝐻 “ 𝑏))) ∈ 𝑢 ↔ (𝐺 Σg ((𝐹 ∘ 𝐻) ↾ 𝑏)) ∈ 𝑢))
76	42, 75	imbi12d 347	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → (((𝐻 “ 𝑎) ⊆ (𝐻 “ 𝑏) → (𝐺 Σg (𝐹 ↾ (𝐻 “ 𝑏))) ∈ 𝑢) ↔ (𝑎 ⊆ 𝑏 → (𝐺 Σg ((𝐹 ∘ 𝐻) ↾ 𝑏)) ∈ 𝑢)))
77	76	ralbidva 3185	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) → (∀𝑏 ∈ (𝒫 𝐶 ∩ Fin)((𝐻 “ 𝑎) ⊆ (𝐻 “ 𝑏) → (𝐺 Σg (𝐹 ↾ (𝐻 “ 𝑏))) ∈ 𝑢) ↔ ∀𝑏 ∈ (𝒫 𝐶 ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σg ((𝐹 ∘ 𝐻) ↾ 𝑏)) ∈ 𝑢)))
78	31, 77	bitr3d 284	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) → (∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)((𝐻 “ 𝑎) ⊆ 𝑧 → (𝐺 Σg (𝐹 ↾ 𝑧)) ∈ 𝑢) ↔ ∀𝑏 ∈ (𝒫 𝐶 ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σg ((𝐹 ∘ 𝐻) ↾ 𝑏)) ∈ 𝑢)))
79	78	rexbidva 3186	. . . . . . 7 ⊢ (𝜑 → (∃𝑎 ∈ (𝒫 𝐶 ∩ Fin)∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)((𝐻 “ 𝑎) ⊆ 𝑧 → (𝐺 Σg (𝐹 ↾ 𝑧)) ∈ 𝑢) ↔ ∃𝑎 ∈ (𝒫 𝐶 ∩ Fin)∀𝑏 ∈ (𝒫 𝐶 ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σg ((𝐹 ∘ 𝐻) ↾ 𝑏)) ∈ 𝑢)))
80	13, 17, 79	3bitr3d 312	. . . . . 6 ⊢ (𝜑 → (∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σg (𝐹 ↾ 𝑧)) ∈ 𝑢) ↔ ∃𝑎 ∈ (𝒫 𝐶 ∩ Fin)∀𝑏 ∈ (𝒫 𝐶 ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σg ((𝐹 ∘ 𝐻) ↾ 𝑏)) ∈ 𝑢)))
81	80	imbi2d 343	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑢 → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σg (𝐹 ↾ 𝑧)) ∈ 𝑢)) ↔ (𝑥 ∈ 𝑢 → ∃𝑎 ∈ (𝒫 𝐶 ∩ Fin)∀𝑏 ∈ (𝒫 𝐶 ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σg ((𝐹 ∘ 𝐻) ↾ 𝑏)) ∈ 𝑢))))
82	81	ralbidv 3187	. . . 4 ⊢ (𝜑 → (∀𝑢 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑢 → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σg (𝐹 ↾ 𝑧)) ∈ 𝑢)) ↔ ∀𝑢 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑢 → ∃𝑎 ∈ (𝒫 𝐶 ∩ Fin)∀𝑏 ∈ (𝒫 𝐶 ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σg ((𝐹 ∘ 𝐻) ↾ 𝑏)) ∈ 𝑢))))
83	82	anbi2d 642	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ ∀𝑢 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑢 → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σg (𝐹 ↾ 𝑧)) ∈ 𝑢))) ↔ (𝑥 ∈ 𝐵 ∧ ∀𝑢 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑢 → ∃𝑎 ∈ (𝒫 𝐶 ∩ Fin)∀𝑏 ∈ (𝒫 𝐶 ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σg ((𝐹 ∘ 𝐻) ↾ 𝑏)) ∈ 𝑢)))))
84		eqid 2762	. . . 4 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
85		eqid 2762	. . . 4 ⊢ (𝒫 𝐴 ∩ Fin) = (𝒫 𝐴 ∩ Fin)
86		tsmsf1o.2	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TopSp)
87		tsmsf1o.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
88	43, 84, 85, 45, 86, 87, 55	eltsms 24360	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐺 tsums 𝐹) ↔ (𝑥 ∈ 𝐵 ∧ ∀𝑢 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑢 → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σg (𝐹 ↾ 𝑧)) ∈ 𝑢)))))
89		eqid 2762	. . . 4 ⊢ (𝒫 𝐶 ∩ Fin) = (𝒫 𝐶 ∩ Fin)
90		f1dmex 7957	. . . . 5 ⊢ ((𝐻:𝐶–1-1→𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐶 ∈ V)
91	33, 87, 90	syl2anc 596	. . . 4 ⊢ (𝜑 → 𝐶 ∈ V)
92		f1of 6821	. . . . . 6 ⊢ (𝐻:𝐶–1-1-onto→𝐴 → 𝐻:𝐶⟶𝐴)
93	1, 92	syl 18	. . . . 5 ⊢ (𝜑 → 𝐻:𝐶⟶𝐴)
94		fco 6731	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐻:𝐶⟶𝐴) → (𝐹 ∘ 𝐻):𝐶⟶𝐵)
95	55, 93, 94	syl2anc 596	. . . 4 ⊢ (𝜑 → (𝐹 ∘ 𝐻):𝐶⟶𝐵)
96	43, 84, 89, 45, 86, 91, 95	eltsms 24360	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐺 tsums (𝐹 ∘ 𝐻)) ↔ (𝑥 ∈ 𝐵 ∧ ∀𝑢 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑢 → ∃𝑎 ∈ (𝒫 𝐶 ∩ Fin)∀𝑏 ∈ (𝒫 𝐶 ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σg ((𝐹 ∘ 𝐻) ↾ 𝑏)) ∈ 𝑢)))))
97	83, 88, 96	3bitr4d 314	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐺 tsums 𝐹) ↔ 𝑥 ∈ (𝐺 tsums (𝐹 ∘ 𝐻))))
98	97	eqrdv 2760	1 ⊢ (𝜑 → (𝐺 tsums 𝐹) = (𝐺 tsums (𝐹 ∘ 𝐻)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078  ∃wrex 3088  Vcvv 3453   ∩ cin 3901   ⊆ wss 3902  𝒫 cpw 4560   ↦ cmpt 5190  ran crn 5660   ↾ cres 5661   “ cima 5662   ∘ ccom 5663  ⟶wf 6533  –1-1→wf1 6534  –onto→wfo 6535  –1-1-onto→wf1o 6536  ‘cfv 6537  (class class class)co 7416  Fincfn 8955  Basecbs 17305  TopOpenctopn 17510  0_gc0g 17528   Σ_g cgsu 17529  CMndccmn 19908  TopSpctps 23158   tsums ctsu 24353

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-supp 8162  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-er 8699  df-map 8831  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-fsupp 9335  df-oi 9485  df-card 9947  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-nn 12261  df-n0 12532  df-z 12619  df-uz 12891  df-fz 13564  df-fzo 13712  df-seq 14068  df-hash 14397  df-0g 17530  df-gsum 17531  df-mgm 18734  df-sgrp 18823  df-mnd 18839  df-cntz 19445  df-cmn 19910  df-fbas 21583  df-fg 21584  df-top 23120  df-topon 23137  df-topsp 23159  df-ntr 23246  df-nei 23324  df-fil 24073  df-fm 24165  df-flim 24166  df-flf 24167  df-tsms 24354

This theorem is used by:  esumf1o  34547