Theorem tsmsf1o 24089

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 9256	. . . . . . . . . . 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 6774	. . . . . . . . . 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 2736	. . . . . . . . . 10 ⊢ (𝑎 ∈ (𝒫 𝐶 ∩ Fin) ↦ (𝐻 “ 𝑎)) = (𝑎 ∈ (𝒫 𝐶 ∩ Fin) ↦ (𝐻 “ 𝑎))
7	6	fmpt 7055	. . . . . . . . 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 3959	. . . . . . . . . . 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 3159	. . . . . . . . 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 7040	. . . . . . . 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 6781	. . . . . . . . 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 6749	. . . . . . . . 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 3297	. . . . . . 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 6015	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑏 → (𝐻 “ 𝑎) = (𝐻 “ 𝑏))
19	18	cbvmptv 5202	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ (𝒫 𝐶 ∩ Fin) ↦ (𝐻 “ 𝑎)) = (𝑏 ∈ (𝒫 𝐶 ∩ Fin) ↦ (𝐻 “ 𝑏))
20	19	fmpt 7055	. . . . . . . . . . . . 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 3960	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝐻 “ 𝑏) → ((𝐻 “ 𝑎) ⊆ 𝑧 ↔ (𝐻 “ 𝑎) ⊆ (𝐻 “ 𝑏)))
23		reseq2 5933	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝐻 “ 𝑏) → (𝐹 ↾ 𝑧) = (𝐹 ↾ (𝐻 “ 𝑏)))
24	23	oveq2d 7374	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝐻 “ 𝑏) → (𝐺 Σg (𝐹 ↾ 𝑧)) = (𝐺 Σg (𝐹 ↾ (𝐻 “ 𝑏))))
25	24	eleq1d 2821	. . . . . . . . . . . . . 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 7039	. . . . . . . . . . . 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 3296	. . . . . . . . . . 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 6773	. . . . . . . . . . . . . 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 9254	. . . . . . . . . . . . . 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 9254	. . . . . . . . . . . . . 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 7210	. . . . . . . . . . . 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 2736	. . . . . . . . . . . . . 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 4154	. . . . . . . . . . . . . . . 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 6788	. . . . . . . . . . . . . . . . 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 6781	. . . . . . . . . . . . . . . 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 9213	. . . . . . . . . . . . . . 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 6030	. . . . . . . . . . . . . . . 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 6781	. . . . . . . . . . . . . . . . 17 ⊢ (𝐻:𝐶–1-1-onto→𝐴 → 𝐻:𝐶–onto→𝐴)
60		forn 6749	. . . . . . . . . . . . . . . . 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 3976	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐻 “ 𝑏) ⊆ 𝐴)
63	56, 62	fssresd 6701	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐹 ↾ (𝐻 “ 𝑏)):(𝐻 “ 𝑏)⟶𝐵)
64		fvexd 6849	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 𝐶 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 𝐶 ∩ Fin)) → (0g‘𝐺) ∈ V)
65	63, 54, 64	fdmfifsupp 9278	. . . . . . . . . . . . . 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 19845	. . . . . . . . . . . . 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 5637	. . . . . . . . . . . . . . . . 17 ⊢ (𝐻 “ 𝑏) = ran (𝐻 ↾ 𝑏)
68	67	eqimss2i 3995	. . . . . . . . . . . . . . . 16 ⊢ ran (𝐻 ↾ 𝑏) ⊆ (𝐻 “ 𝑏)
69		cores 6207	. . . . . . . . . . . . . . . 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 6208	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∘ 𝐻) ↾ 𝑏) = (𝐹 ∘ (𝐻 ↾ 𝑏))
72	70, 71	eqtr4i 2762	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ↾ (𝐻 “ 𝑏)) ∘ (𝐻 ↾ 𝑏)) = ((𝐹 ∘ 𝐻) ↾ 𝑏)
73	72	oveq2i 7369	. . . . . . . . . . . . 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 2821	. . . . . . . . . . 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 3157	. . . . . . . . 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 3158	. . . . . . 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 3159	. . . 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 2736	. . . 4 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
85		eqid 2736	. . . 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 24077	. . 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 2736	. . . 4 ⊢ (𝒫 𝐶 ∩ Fin) = (𝒫 𝐶 ∩ Fin)
90		f1dmex 7901	. . . . 5 ⊢ ((𝐻:𝐶–1-1→𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐶 ∈ V)
91	33, 87, 90	syl2anc 584	. . . 4 ⊢ (𝜑 → 𝐶 ∈ V)
92		f1of 6774	. . . . . 6 ⊢ (𝐻:𝐶–1-1-onto→𝐴 → 𝐻:𝐶⟶𝐴)
93	1, 92	syl 17	. . . . 5 ⊢ (𝜑 → 𝐻:𝐶⟶𝐴)
94		fco 6686	. . . . 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 24077	. . 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 2734	1 ⊢ (𝜑 → (𝐺 tsums 𝐹) = (𝐺 tsums (𝐹 ∘ 𝐻)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3051  ∃wrex 3060  Vcvv 3440   ∩ cin 3900   ⊆ wss 3901  𝒫 cpw 4554   ↦ cmpt 5179  ran crn 5625   ↾ cres 5626   “ cima 5627   ∘ ccom 5628  ⟶wf 6488  –1-1→wf1 6489  –onto→wfo 6490  –1-1-onto→wf1o 6491  ‘cfv 6492  (class class class)co 7358  Fincfn 8883  Basecbs 17136  TopOpenctopn 17341  0_gc0g 17359   Σ_g cgsu 17360  CMndccmn 19709  TopSpctps 22876   tsums ctsu 24070

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-supp 8103  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-er 8635  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9265  df-oi 9415  df-card 9851  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-n0 12402  df-z 12489  df-uz 12752  df-fz 13424  df-fzo 13571  df-seq 13925  df-hash 14254  df-0g 17361  df-gsum 17362  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-cntz 19246  df-cmn 19711  df-fbas 21306  df-fg 21307  df-top 22838  df-topon 22855  df-topsp 22877  df-ntr 22964  df-nei 23042  df-fil 23790  df-fm 23882  df-flim 23883  df-flf 23884  df-tsms 24071

This theorem is referenced by:  esumf1o  34207