Theorem tsmsf1o 24060

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 9240	. . . . . . . . . . 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 6763	. . . . . . . . . 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 7043	. . . . . . . . 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 234	. . . . . . . 8 ⊢ (𝜑 → ∀𝑎 ∈ (𝒫 𝐶 ∩ Fin)(𝐻 “ 𝑎) ∈ (𝒫 𝐴 ∩ Fin))
9		sseq1 3955	. . . . . . . . . . 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 3155	. . . . . . . . 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 7028	. . . . . . . 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 6770	. . . . . . . . 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 6738	. . . . . . . . 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 3293	. . . . . . 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 6004	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑏 → (𝐻 “ 𝑎) = (𝐻 “ 𝑏))
19	18	cbvmptv 5193	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ (𝒫 𝐶 ∩ Fin) ↦ (𝐻 “ 𝑎)) = (𝑏 ∈ (𝒫 𝐶 ∩ Fin) ↦ (𝐻 “ 𝑏))
20	19	fmpt 7043	. . . . . . . . . . . . 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 234	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑏 ∈ (𝒫 𝐶 ∩ Fin)(𝐻 “ 𝑏) ∈ (𝒫 𝐴 ∩ Fin))
22		sseq2 3956	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝐻 “ 𝑏) → ((𝐻 “ 𝑎) ⊆ 𝑧 ↔ (𝐻 “ 𝑎) ⊆ (𝐻 “ 𝑏)))
23		reseq2 5922	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝐻 “ 𝑏) → (𝐹 ↾ 𝑧) = (𝐹 ↾ (𝐻 “ 𝑏)))
24	23	oveq2d 7362	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝐻 “ 𝑏) → (𝐺 Σg (𝐹 ↾ 𝑧)) = (𝐺 Σg (𝐹 ↾ (𝐻 “ 𝑏))))
25	24	eleq1d 2816	. . . . . . . . . . . . . 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 7027	. . . . . . . . . . . 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 3292	. . . . . . . . . . 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 281	. . . . . . . . . 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 480	. . . . . . . . 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 6762	. . . . . . . . . . . . . 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 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → 𝐻:𝐶–1-1→𝐴)
35		elfpw 9238	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ (𝒫 𝐶 ∩ Fin) ↔ (𝑎 ⊆ 𝐶 ∧ 𝑎 ∈ Fin))
36	35	simplbi 497	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ (𝒫 𝐶 ∩ Fin) → 𝑎 ⊆ 𝐶)
37	36	ad2antlr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → 𝑎 ⊆ 𝐶)
38		elfpw 9238	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ (𝒫 𝐶 ∩ Fin) ↔ (𝑏 ⊆ 𝐶 ∧ 𝑏 ∈ Fin))
39	38	simplbi 497	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ (𝒫 𝐶 ∩ Fin) → 𝑏 ⊆ 𝐶)
40	39	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → 𝑏 ⊆ 𝐶)
41		f1imass 7198	. . . . . . . . . . . 12 ⊢ ((𝐻:𝐶–1-1→𝐴 ∧ (𝑎 ⊆ 𝐶 ∧ 𝑏 ⊆ 𝐶)) → ((𝐻 “ 𝑎) ⊆ (𝐻 “ 𝑏) ↔ 𝑎 ⊆ 𝑏))
42	34, 37, 40, 41	syl12anc 836	. . . . . . . . . . 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 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → 𝐺 ∈ CMnd)
47		elinel2 4149	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ∈ (𝒫 𝐶 ∩ Fin) → 𝑏 ∈ Fin)
48	47	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → 𝑏 ∈ Fin)
49		f1ores 6777	. . . . . . . . . . . . . . . . 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 6770	. . . . . . . . . . . . . . . 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 9197	. . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → 𝐹:𝐴⟶𝐵)
57		imassrn 6019	. . . . . . . . . . . . . . . 16 ⊢ (𝐻 “ 𝑏) ⊆ ran 𝐻
58	1	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → 𝐻:𝐶–1-1-onto→𝐴)
59		f1ofo 6770	. . . . . . . . . . . . . . . . 17 ⊢ (𝐻:𝐶–1-1-onto→𝐴 → 𝐻:𝐶–onto→𝐴)
60		forn 6738	. . . . . . . . . . . . . . . . 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 3972	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐻 “ 𝑏) ⊆ 𝐴)
63	56, 62	fssresd 6690	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹 ↾ (𝐻 “ 𝑏)):(𝐻 “ 𝑏)⟶𝐵)
64		fvexd 6837	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → (0g‘𝐺) ∈ V)
65	63, 54, 64	fdmfifsupp 9259	. . . . . . . . . . . . . 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 19828	. . . . . . . . . . . . 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 5627	. . . . . . . . . . . . . . . . 17 ⊢ (𝐻 “ 𝑏) = ran (𝐻 ↾ 𝑏)
68	67	eqimss2i 3991	. . . . . . . . . . . . . . . 16 ⊢ ran (𝐻 ↾ 𝑏) ⊆ (𝐻 “ 𝑏)
69		cores 6196	. . . . . . . . . . . . . . . 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 6197	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∘ 𝐻) ↾ 𝑏) = (𝐹 ∘ (𝐻 ↾ 𝑏))
72	70, 71	eqtr4i 2757	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ↾ (𝐻 “ 𝑏)) ∘ (𝐻 ↾ 𝑏)) = ((𝐹 ∘ 𝐻) ↾ 𝑏)
73	72	oveq2i 7357	. . . . . . . . . . . . 13 ⊢ (𝐺 Σg ((𝐹 ↾ (𝐻 “ 𝑏)) ∘ (𝐻 ↾ 𝑏))) = (𝐺 Σg ((𝐹 ∘ 𝐻) ↾ 𝑏))
74	66, 73	eqtrdi 2782	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐺 Σg (𝐹 ↾ (𝐻 “ 𝑏))) = (𝐺 Σg ((𝐹 ∘ 𝐻) ↾ 𝑏)))
75	74	eleq1d 2816	. . . . . . . . . . 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 3153	. . . . . . . . 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 281	. . . . . . . 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 3154	. . . . . . 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 309	. . . . . 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 3155	. . . 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 630	. . 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 24048	. . 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 7889	. . . . 5 ⊢ ((𝐻:𝐶–1-1→𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐶 ∈ V)
91	33, 87, 90	syl2anc 584	. . . 4 ⊢ (𝜑 → 𝐶 ∈ V)
92		f1of 6763	. . . . . 6 ⊢ (𝐻:𝐶–1-1-onto→𝐴 → 𝐻:𝐶⟶𝐴)
93	1, 92	syl 17	. . . . 5 ⊢ (𝜑 → 𝐻:𝐶⟶𝐴)
94		fco 6675	. . . . 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 24048	. . 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 311	. 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 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056  Vcvv 3436   ∩ cin 3896   ⊆ wss 3897  𝒫 cpw 4547   ↦ cmpt 5170  ran crn 5615   ↾ cres 5616   “ cima 5617   ∘ ccom 5618  ⟶wf 6477  –1-1→wf1 6478  –onto→wfo 6479  –1-1-onto→wf1o 6480  ‘cfv 6481  (class class class)co 7346  Fincfn 8869  Basecbs 17120  TopOpenctopn 17325  0_gc0g 17343   Σ_g cgsu 17344  CMndccmn 19692  TopSpctps 22847   tsums ctsu 24041

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-isom 6490  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-supp 8091  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-er 8622  df-map 8752  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-fsupp 9246  df-oi 9396  df-card 9832  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-nn 12126  df-n0 12382  df-z 12469  df-uz 12733  df-fz 13408  df-fzo 13555  df-seq 13909  df-hash 14238  df-0g 17345  df-gsum 17346  df-mgm 18548  df-sgrp 18627  df-mnd 18643  df-cntz 19229  df-cmn 19694  df-fbas 21288  df-fg 21289  df-top 22809  df-topon 22826  df-topsp 22848  df-ntr 22935  df-nei 23013  df-fil 23761  df-fm 23853  df-flim 23854  df-flf 23855  df-tsms 24042

This theorem is referenced by:  esumf1o  34063