Theorem tsmsf1o 23533

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 9307	. . . . . . . . . . 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 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑎 ∈ (𝒫 𝐶 ∩ Fin) ↦ (𝐻 “ 𝑎)):(𝒫 𝐶 ∩ Fin)–1-1-onto→(𝒫 𝐴 ∩ Fin))
4		f1of 6789	. . . . . . . . . 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 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑎 ∈ (𝒫 𝐶 ∩ Fin) ↦ (𝐻 “ 𝑎)):(𝒫 𝐶 ∩ Fin)⟶(𝒫 𝐴 ∩ Fin))
6		eqid 2731	. . . . . . . . . 10 ⊢ (𝑎 ∈ (𝒫 𝐶 ∩ Fin) ↦ (𝐻 “ 𝑎)) = (𝑎 ∈ (𝒫 𝐶 ∩ Fin) ↦ (𝐻 “ 𝑎))
7	6	fmpt 7063	. . . . . . . . 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 233	. . . . . . . 8 ⊢ (𝜑 → ∀𝑎 ∈ (𝒫 𝐶 ∩ Fin)(𝐻 “ 𝑎) ∈ (𝒫 𝐴 ∩ Fin))
9		sseq1 3972	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐻 “ 𝑎) → (𝑦 ⊆ 𝑧 ↔ (𝐻 “ 𝑎) ⊆ 𝑧))
10	9	imbi1d 341	. . . . . . . . . 10 ⊢ (𝑦 = (𝐻 “ 𝑎) → ((𝑦 ⊆ 𝑧 → (𝐺 Σg (𝐹 ↾ 𝑧)) ∈ 𝑢) ↔ ((𝐻 “ 𝑎) ⊆ 𝑧 → (𝐺 Σg (𝐹 ↾ 𝑧)) ∈ 𝑢)))
11	10	ralbidv 3170	. . . . . . . . 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 7050	. . . . . . . 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 17	. . . . . . 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 6796	. . . . . . . . 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 6764	. . . . . . . . 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 18	. . . . . . . 8 ⊢ (𝜑 → ran (𝑎 ∈ (𝒫 𝐶 ∩ Fin) ↦ (𝐻 “ 𝑎)) = (𝒫 𝐴 ∩ Fin))
17	16	rexeqdv 3312	. . . . . . 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 6014	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑏 → (𝐻 “ 𝑎) = (𝐻 “ 𝑏))
19	18	cbvmptv 5223	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ (𝒫 𝐶 ∩ Fin) ↦ (𝐻 “ 𝑎)) = (𝑏 ∈ (𝒫 𝐶 ∩ Fin) ↦ (𝐻 “ 𝑏))
20	19	fmpt 7063	. . . . . . . . . . . . 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 233	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑏 ∈ (𝒫 𝐶 ∩ Fin)(𝐻 “ 𝑏) ∈ (𝒫 𝐴 ∩ Fin))
22		sseq2 3973	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝐻 “ 𝑏) → ((𝐻 “ 𝑎) ⊆ 𝑧 ↔ (𝐻 “ 𝑎) ⊆ (𝐻 “ 𝑏)))
23		reseq2 5937	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝐻 “ 𝑏) → (𝐹 ↾ 𝑧) = (𝐹 ↾ (𝐻 “ 𝑏)))
24	23	oveq2d 7378	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝐻 “ 𝑏) → (𝐺 Σg (𝐹 ↾ 𝑧)) = (𝐺 Σg (𝐹 ↾ (𝐻 “ 𝑏))))
25	24	eleq1d 2817	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝐻 “ 𝑏) → ((𝐺 Σg (𝐹 ↾ 𝑧)) ∈ 𝑢 ↔ (𝐺 Σg (𝐹 ↾ (𝐻 “ 𝑏))) ∈ 𝑢))
26	22, 25	imbi12d 344	. . . . . . . . . . . . 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 7049	. . . . . . . . . . . 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 17	. . . . . . . . . . 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 3311	. . . . . . . . . . 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 280	. . . . . . . . . 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 481	. . . . . . . . 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 6788	. . . . . . . . . . . . . 14 ⊢ (𝐻:𝐶–1-1-onto→𝐴 → 𝐻:𝐶–1-1→𝐴)
33	1, 32	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐻:𝐶–1-1→𝐴)
34	33	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → 𝐻:𝐶–1-1→𝐴)
35		elfpw 9305	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ (𝒫 𝐶 ∩ Fin) ↔ (𝑎 ⊆ 𝐶 ∧ 𝑎 ∈ Fin))
36	35	simplbi 498	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ (𝒫 𝐶 ∩ Fin) → 𝑎 ⊆ 𝐶)
37	36	ad2antlr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → 𝑎 ⊆ 𝐶)
38		elfpw 9305	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ (𝒫 𝐶 ∩ Fin) ↔ (𝑏 ⊆ 𝐶 ∧ 𝑏 ∈ Fin))
39	38	simplbi 498	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ (𝒫 𝐶 ∩ Fin) → 𝑏 ⊆ 𝐶)
40	39	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → 𝑏 ⊆ 𝐶)
41		f1imass 7216	. . . . . . . . . . . 12 ⊢ ((𝐻:𝐶–1-1→𝐴 ∧ (𝑎 ⊆ 𝐶 ∧ 𝑏 ⊆ 𝐶)) → ((𝐻 “ 𝑎) ⊆ (𝐻 “ 𝑏) ↔ 𝑎 ⊆ 𝑏))
42	34, 37, 40, 41	syl12anc 835	. . . . . . . . . . 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 2731	. . . . . . . . . . . . . 14 ⊢ (0g‘𝐺) = (0g‘𝐺)
45		tsmsf1o.1	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺 ∈ CMnd)
46	45	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → 𝐺 ∈ CMnd)
47		elinel2 4161	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ∈ (𝒫 𝐶 ∩ Fin) → 𝑏 ∈ Fin)
48	47	adantl 482	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → 𝑏 ∈ Fin)
49		f1ores 6803	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐻:𝐶–1-1→𝐴 ∧ 𝑏 ⊆ 𝐶) → (𝐻 ↾ 𝑏):𝑏–1-1-onto→(𝐻 “ 𝑏))
50	34, 40, 49	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐻 ↾ 𝑏):𝑏–1-1-onto→(𝐻 “ 𝑏))
51		f1ofo 6796	. . . . . . . . . . . . . . . 16 ⊢ ((𝐻 ↾ 𝑏):𝑏–1-1-onto→(𝐻 “ 𝑏) → (𝐻 ↾ 𝑏):𝑏–onto→(𝐻 “ 𝑏))
52	50, 51	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐻 ↾ 𝑏):𝑏–onto→(𝐻 “ 𝑏))
53		fofi 9289	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ∈ Fin ∧ (𝐻 ↾ 𝑏):𝑏–onto→(𝐻 “ 𝑏)) → (𝐻 “ 𝑏) ∈ Fin)
54	48, 52, 53	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐻 “ 𝑏) ∈ Fin)
55		tsmsf1o.f	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
56	55	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → 𝐹:𝐴⟶𝐵)
57		imassrn 6029	. . . . . . . . . . . . . . . 16 ⊢ (𝐻 “ 𝑏) ⊆ ran 𝐻
58	1	ad2antrr 724	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → 𝐻:𝐶–1-1-onto→𝐴)
59		f1ofo 6796	. . . . . . . . . . . . . . . . 17 ⊢ (𝐻:𝐶–1-1-onto→𝐴 → 𝐻:𝐶–onto→𝐴)
60		forn 6764	. . . . . . . . . . . . . . . . 17 ⊢ (𝐻:𝐶–onto→𝐴 → ran 𝐻 = 𝐴)
61	58, 59, 60	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → ran 𝐻 = 𝐴)
62	57, 61	sseqtrid 3999	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐻 “ 𝑏) ⊆ 𝐴)
63	56, 62	fssresd 6714	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹 ↾ (𝐻 “ 𝑏)):(𝐻 “ 𝑏)⟶𝐵)
64		fvexd 6862	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → (0g‘𝐺) ∈ V)
65	63, 54, 64	fdmfifsupp 9324	. . . . . . . . . . . . . 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 19707	. . . . . . . . . . . . 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 5651	. . . . . . . . . . . . . . . . 17 ⊢ (𝐻 “ 𝑏) = ran (𝐻 ↾ 𝑏)
68	67	eqimss2i 4008	. . . . . . . . . . . . . . . 16 ⊢ ran (𝐻 ↾ 𝑏) ⊆ (𝐻 “ 𝑏)
69		cores 6206	. . . . . . . . . . . . . . . 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 6207	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∘ 𝐻) ↾ 𝑏) = (𝐹 ∘ (𝐻 ↾ 𝑏))
72	70, 71	eqtr4i 2762	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ↾ (𝐻 “ 𝑏)) ∘ (𝐻 ↾ 𝑏)) = ((𝐹 ∘ 𝐻) ↾ 𝑏)
73	72	oveq2i 7373	. . . . . . . . . . . . 13 ⊢ (𝐺 Σg ((𝐹 ↾ (𝐻 “ 𝑏)) ∘ (𝐻 ↾ 𝑏))) = (𝐺 Σg ((𝐹 ∘ 𝐻) ↾ 𝑏))
74	66, 73	eqtrdi 2787	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐺 Σg (𝐹 ↾ (𝐻 “ 𝑏))) = (𝐺 Σg ((𝐹 ∘ 𝐻) ↾ 𝑏)))
75	74	eleq1d 2817	. . . . . . . . . . 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 344	. . . . . . . . . 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 3168	. . . . . . . . 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 280	. . . . . . . 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 3169	. . . . . . 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 308	. . . . . 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 340	. . . . 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 3170	. . . 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 629	. . 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 2731	. . . 4 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
85		eqid 2731	. . . 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 23521	. . 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 2731	. . . 4 ⊢ (𝒫 𝐶 ∩ Fin) = (𝒫 𝐶 ∩ Fin)
90		f1dmex 7894	. . . . 5 ⊢ ((𝐻:𝐶–1-1→𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐶 ∈ V)
91	33, 87, 90	syl2anc 584	. . . 4 ⊢ (𝜑 → 𝐶 ∈ V)
92		f1of 6789	. . . . . 6 ⊢ (𝐻:𝐶–1-1-onto→𝐴 → 𝐻:𝐶⟶𝐴)
93	1, 92	syl 17	. . . . 5 ⊢ (𝜑 → 𝐻:𝐶⟶𝐴)
94		fco 6697	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐻:𝐶⟶𝐴) → (𝐹 ∘ 𝐻):𝐶⟶𝐵)
95	55, 93, 94	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐹 ∘ 𝐻):𝐶⟶𝐵)
96	43, 84, 89, 45, 86, 91, 95	eltsms 23521	. . 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 310	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐺 tsums 𝐹) ↔ 𝑥 ∈ (𝐺 tsums (𝐹 ∘ 𝐻))))
98	97	eqrdv 2729	1 ⊢ (𝜑 → (𝐺 tsums 𝐹) = (𝐺 tsums (𝐹 ∘ 𝐻)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3060  ∃wrex 3069  Vcvv 3446   ∩ cin 3912   ⊆ wss 3913  𝒫 cpw 4565   ↦ cmpt 5193  ran crn 5639   ↾ cres 5640   “ cima 5641   ∘ ccom 5642  ⟶wf 6497  –1-1→wf1 6498  –onto→wfo 6499  –1-1-onto→wf1o 6500  ‘cfv 6501  (class class class)co 7362  Fincfn 8890  Basecbs 17094  TopOpenctopn 17317  0_gc0g 17335   Σ_g cgsu 17336  CMndccmn 19576  TopSpctps 22318   tsums ctsu 23514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11116  ax-resscn 11117  ax-1cn 11118  ax-icn 11119  ax-addcl 11120  ax-addrcl 11121  ax-mulcl 11122  ax-mulrcl 11123  ax-mulcom 11124  ax-addass 11125  ax-mulass 11126  ax-distr 11127  ax-i2m1 11128  ax-1ne0 11129  ax-1rid 11130  ax-rnegex 11131  ax-rrecex 11132  ax-cnre 11133  ax-pre-lttri 11134  ax-pre-lttrn 11135  ax-pre-ltadd 11136  ax-pre-mulgt0 11137

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-supp 8098  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-er 8655  df-map 8774  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-fsupp 9313  df-oi 9455  df-card 9884  df-pnf 11200  df-mnf 11201  df-xr 11202  df-ltxr 11203  df-le 11204  df-sub 11396  df-neg 11397  df-nn 12163  df-n0 12423  df-z 12509  df-uz 12773  df-fz 13435  df-fzo 13578  df-seq 13917  df-hash 14241  df-0g 17337  df-gsum 17338  df-mgm 18511  df-sgrp 18560  df-mnd 18571  df-cntz 19111  df-cmn 19578  df-fbas 20830  df-fg 20831  df-top 22280  df-topon 22297  df-topsp 22319  df-ntr 22408  df-nei 22486  df-fil 23234  df-fm 23326  df-flim 23327  df-flf 23328  df-tsms 23515

This theorem is referenced by:  esumf1o  32738