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Theorem tsmsres 22435

Description: Extend an infinite group sum by padding outside with zeroes. (Contributed by Mario Carneiro, 18-Sep-2015.) (Revised by AV, 25-Jul-2019.)

**Hypotheses**
Ref	Expression
tsmsres.b	⊢ 𝐵 = (Base‘𝐺)
tsmsres.z	⊢ 0 = (0_g‘𝐺)
tsmsres.1	⊢ (𝜑 → 𝐺 ∈ CMnd)
tsmsres.2	⊢ (𝜑 → 𝐺 ∈ TopSp)
tsmsres.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
tsmsres.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
tsmsres.s	⊢ (𝜑 → (𝐹 supp 0 ) ⊆ 𝑊)

**Assertion**
Ref	Expression
tsmsres	⊢ (𝜑 → (𝐺 tsums (𝐹 ↾ 𝑊)) = (𝐺 tsums 𝐹))

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

Step	Hyp	Ref	Expression
1		inss1 4127	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ 𝑊) ⊆ 𝐴
2		sspwb 5236	. . . . . . . . . . . 12 ⊢ ((𝐴 ∩ 𝑊) ⊆ 𝐴 ↔ 𝒫 (𝐴 ∩ 𝑊) ⊆ 𝒫 𝐴)
3	1, 2	mpbi 231	. . . . . . . . . . 11 ⊢ 𝒫 (𝐴 ∩ 𝑊) ⊆ 𝒫 𝐴
4		ssrin 4132	. . . . . . . . . . 11 ⊢ (𝒫 (𝐴 ∩ 𝑊) ⊆ 𝒫 𝐴 → (𝒫 (𝐴 ∩ 𝑊) ∩ Fin) ⊆ (𝒫 𝐴 ∩ Fin))
5	3, 4	ax-mp 5	. . . . . . . . . 10 ⊢ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin) ⊆ (𝒫 𝐴 ∩ Fin)
6		simpr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)) → 𝑎 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin))
7	5, 6	sseldi 3889	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)) → 𝑎 ∈ (𝒫 𝐴 ∩ Fin))
8		elfpw 8675	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑧 ⊆ 𝐴 ∧ 𝑧 ∈ Fin))
9	8	simplbi 498	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ (𝒫 𝐴 ∩ Fin) → 𝑧 ⊆ 𝐴)
10	9	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)) ∧ 𝑧 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑧 ⊆ 𝐴)
11	10	ssrind 4134	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)) ∧ 𝑧 ∈ (𝒫 𝐴 ∩ Fin)) → (𝑧 ∩ 𝑊) ⊆ (𝐴 ∩ 𝑊))
12		elinel2 4096	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ (𝒫 𝐴 ∩ Fin) → 𝑧 ∈ Fin)
13	12	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)) ∧ 𝑧 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑧 ∈ Fin)
14		inss1 4127	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∩ 𝑊) ⊆ 𝑧
15		ssfi 8587	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ Fin ∧ (𝑧 ∩ 𝑊) ⊆ 𝑧) → (𝑧 ∩ 𝑊) ∈ Fin)
16	13, 14, 15	sylancl 586	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)) ∧ 𝑧 ∈ (𝒫 𝐴 ∩ Fin)) → (𝑧 ∩ 𝑊) ∈ Fin)
17		elfpw 8675	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∩ 𝑊) ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin) ↔ ((𝑧 ∩ 𝑊) ⊆ (𝐴 ∩ 𝑊) ∧ (𝑧 ∩ 𝑊) ∈ Fin))
18	11, 16, 17	sylanbrc 583	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)) ∧ 𝑧 ∈ (𝒫 𝐴 ∩ Fin)) → (𝑧 ∩ 𝑊) ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin))
19		sseq2 3916	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (𝑧 ∩ 𝑊) → (𝑎 ⊆ 𝑏 ↔ 𝑎 ⊆ (𝑧 ∩ 𝑊)))
20		ssin 4129	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ⊆ 𝑧 ∧ 𝑎 ⊆ 𝑊) ↔ 𝑎 ⊆ (𝑧 ∩ 𝑊))
21	19, 20	syl6bbr 290	. . . . . . . . . . . . . 14 ⊢ (𝑏 = (𝑧 ∩ 𝑊) → (𝑎 ⊆ 𝑏 ↔ (𝑎 ⊆ 𝑧 ∧ 𝑎 ⊆ 𝑊)))
22		reseq2 5732	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = (𝑧 ∩ 𝑊) → ((𝐹 ↾ 𝑊) ↾ 𝑏) = ((𝐹 ↾ 𝑊) ↾ (𝑧 ∩ 𝑊)))
23		inss2 4128	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∩ 𝑊) ⊆ 𝑊
24		resabs1 5767	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∩ 𝑊) ⊆ 𝑊 → ((𝐹 ↾ 𝑊) ↾ (𝑧 ∩ 𝑊)) = (𝐹 ↾ (𝑧 ∩ 𝑊)))
25	23, 24	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 ↾ 𝑊) ↾ (𝑧 ∩ 𝑊)) = (𝐹 ↾ (𝑧 ∩ 𝑊))
26	22, 25	syl6eq 2846	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝑧 ∩ 𝑊) → ((𝐹 ↾ 𝑊) ↾ 𝑏) = (𝐹 ↾ (𝑧 ∩ 𝑊)))
27	26	oveq2d 7035	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (𝑧 ∩ 𝑊) → (𝐺 Σ_g ((𝐹 ↾ 𝑊) ↾ 𝑏)) = (𝐺 Σ_g (𝐹 ↾ (𝑧 ∩ 𝑊))))
28	27	eleq1d 2866	. . . . . . . . . . . . . 14 ⊢ (𝑏 = (𝑧 ∩ 𝑊) → ((𝐺 Σ_g ((𝐹 ↾ 𝑊) ↾ 𝑏)) ∈ 𝑢 ↔ (𝐺 Σ_g (𝐹 ↾ (𝑧 ∩ 𝑊))) ∈ 𝑢))
29	21, 28	imbi12d 346	. . . . . . . . . . . . 13 ⊢ (𝑏 = (𝑧 ∩ 𝑊) → ((𝑎 ⊆ 𝑏 → (𝐺 Σ_g ((𝐹 ↾ 𝑊) ↾ 𝑏)) ∈ 𝑢) ↔ ((𝑎 ⊆ 𝑧 ∧ 𝑎 ⊆ 𝑊) → (𝐺 Σ_g (𝐹 ↾ (𝑧 ∩ 𝑊))) ∈ 𝑢)))
30	29	rspcv 3553	. . . . . . . . . . . 12 ⊢ ((𝑧 ∩ 𝑊) ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin) → (∀𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σ_g ((𝐹 ↾ 𝑊) ↾ 𝑏)) ∈ 𝑢) → ((𝑎 ⊆ 𝑧 ∧ 𝑎 ⊆ 𝑊) → (𝐺 Σ_g (𝐹 ↾ (𝑧 ∩ 𝑊))) ∈ 𝑢)))
31	18, 30	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)) ∧ 𝑧 ∈ (𝒫 𝐴 ∩ Fin)) → (∀𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σ_g ((𝐹 ↾ 𝑊) ↾ 𝑏)) ∈ 𝑢) → ((𝑎 ⊆ 𝑧 ∧ 𝑎 ⊆ 𝑊) → (𝐺 Σ_g (𝐹 ↾ (𝑧 ∩ 𝑊))) ∈ 𝑢)))
32		elfpw 8675	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin) ↔ (𝑎 ⊆ (𝐴 ∩ 𝑊) ∧ 𝑎 ∈ Fin))
33	32	simplbi 498	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin) → 𝑎 ⊆ (𝐴 ∩ 𝑊))
34	33	ad2antlr 723	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)) ∧ 𝑧 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑎 ⊆ (𝐴 ∩ 𝑊))
35		inss2 4128	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∩ 𝑊) ⊆ 𝑊
36	34, 35	syl6ss 3903	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)) ∧ 𝑧 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑎 ⊆ 𝑊)
37	36	biantrud 532	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)) ∧ 𝑧 ∈ (𝒫 𝐴 ∩ Fin)) → (𝑎 ⊆ 𝑧 ↔ (𝑎 ⊆ 𝑧 ∧ 𝑎 ⊆ 𝑊)))
38		resres 5750	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ↾ 𝑧) ↾ 𝑊) = (𝐹 ↾ (𝑧 ∩ 𝑊))
39	38	oveq2i 7030	. . . . . . . . . . . . . 14 ⊢ (𝐺 Σ_g ((𝐹 ↾ 𝑧) ↾ 𝑊)) = (𝐺 Σ_g (𝐹 ↾ (𝑧 ∩ 𝑊)))
40		tsmsres.b	. . . . . . . . . . . . . . 15 ⊢ 𝐵 = (Base‘𝐺)
41		tsmsres.z	. . . . . . . . . . . . . . 15 ⊢ 0 = (0_g‘𝐺)
42		tsmsres.1	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐺 ∈ CMnd)
43	42	ad2antrr 722	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)) ∧ 𝑧 ∈ (𝒫 𝐴 ∩ Fin)) → 𝐺 ∈ CMnd)
44		tsmsres.f	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
45	44	ad2antrr 722	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)) ∧ 𝑧 ∈ (𝒫 𝐴 ∩ Fin)) → 𝐹:𝐴⟶𝐵)
46	45, 10	fssresd 6416	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)) ∧ 𝑧 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 ↾ 𝑧):𝑧⟶𝐵)
47		tsmsres.a	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
48		fex 6858	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V)
49	44, 47, 48	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹 ∈ V)
50	49	ad2antrr 722	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)) ∧ 𝑧 ∈ (𝒫 𝐴 ∩ Fin)) → 𝐹 ∈ V)
51	41	fvexi 6555	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ V
52		ressuppss 7703	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 ∈ V ∧ 0 ∈ V) → ((𝐹 ↾ 𝑧) supp 0 ) ⊆ (𝐹 supp 0 ))
53	50, 51, 52	sylancl 586	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)) ∧ 𝑧 ∈ (𝒫 𝐴 ∩ Fin)) → ((𝐹 ↾ 𝑧) supp 0 ) ⊆ (𝐹 supp 0 ))
54		tsmsres.s	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ 𝑊)
55	54	ad2antrr 722	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)) ∧ 𝑧 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 supp 0 ) ⊆ 𝑊)
56	53, 55	sstrd 3901	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)) ∧ 𝑧 ∈ (𝒫 𝐴 ∩ Fin)) → ((𝐹 ↾ 𝑧) supp 0 ) ⊆ 𝑊)
57	51	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)) ∧ 𝑧 ∈ (𝒫 𝐴 ∩ Fin)) → 0 ∈ V)
58	46, 13, 57	fdmfifsupp 8692	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)) ∧ 𝑧 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 ↾ 𝑧) finSupp 0 )
59	40, 41, 43, 13, 46, 56, 58	gsumres 18754	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)) ∧ 𝑧 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐺 Σ_g ((𝐹 ↾ 𝑧) ↾ 𝑊)) = (𝐺 Σ_g (𝐹 ↾ 𝑧)))
60	39, 59	syl5reqr 2845	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)) ∧ 𝑧 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐺 Σ_g (𝐹 ↾ 𝑧)) = (𝐺 Σ_g (𝐹 ↾ (𝑧 ∩ 𝑊))))
61	60	eleq1d 2866	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)) ∧ 𝑧 ∈ (𝒫 𝐴 ∩ Fin)) → ((𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢 ↔ (𝐺 Σ_g (𝐹 ↾ (𝑧 ∩ 𝑊))) ∈ 𝑢))
62	37, 61	imbi12d 346	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)) ∧ 𝑧 ∈ (𝒫 𝐴 ∩ Fin)) → ((𝑎 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢) ↔ ((𝑎 ⊆ 𝑧 ∧ 𝑎 ⊆ 𝑊) → (𝐺 Σ_g (𝐹 ↾ (𝑧 ∩ 𝑊))) ∈ 𝑢)))
63	31, 62	sylibrd 260	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)) ∧ 𝑧 ∈ (𝒫 𝐴 ∩ Fin)) → (∀𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σ_g ((𝐹 ↾ 𝑊) ↾ 𝑏)) ∈ 𝑢) → (𝑎 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢)))
64	63	ralrimdva 3155	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)) → (∀𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σ_g ((𝐹 ↾ 𝑊) ↾ 𝑏)) ∈ 𝑢) → ∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑎 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢)))
65		sseq1 3915	. . . . . . . . . 10 ⊢ (𝑦 = 𝑎 → (𝑦 ⊆ 𝑧 ↔ 𝑎 ⊆ 𝑧))
66	65	rspceaimv 3565	. . . . . . . . 9 ⊢ ((𝑎 ∈ (𝒫 𝐴 ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑎 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢)) → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢))
67	7, 64, 66	syl6an 680	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)) → (∀𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σ_g ((𝐹 ↾ 𝑊) ↾ 𝑏)) ∈ 𝑢) → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢)))
68	67	rexlimdva 3246	. . . . . . 7 ⊢ (𝜑 → (∃𝑎 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)∀𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σ_g ((𝐹 ↾ 𝑊) ↾ 𝑏)) ∈ 𝑢) → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢)))
69		elfpw 8675	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑦 ⊆ 𝐴 ∧ 𝑦 ∈ Fin))
70	69	simplbi 498	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝒫 𝐴 ∩ Fin) → 𝑦 ⊆ 𝐴)
71	70	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦 ⊆ 𝐴)
72	71	ssrind 4134	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝑦 ∩ 𝑊) ⊆ (𝐴 ∩ 𝑊))
73		elinel2 4096	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝒫 𝐴 ∩ Fin) → 𝑦 ∈ Fin)
74	73	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦 ∈ Fin)
75		inss1 4127	. . . . . . . . . . 11 ⊢ (𝑦 ∩ 𝑊) ⊆ 𝑦
76		ssfi 8587	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ Fin ∧ (𝑦 ∩ 𝑊) ⊆ 𝑦) → (𝑦 ∩ 𝑊) ∈ Fin)
77	74, 75, 76	sylancl 586	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝑦 ∩ 𝑊) ∈ Fin)
78		elfpw 8675	. . . . . . . . . 10 ⊢ ((𝑦 ∩ 𝑊) ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin) ↔ ((𝑦 ∩ 𝑊) ⊆ (𝐴 ∩ 𝑊) ∧ (𝑦 ∩ 𝑊) ∈ Fin))
79	72, 77, 78	sylanbrc 583	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝑦 ∩ 𝑊) ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin))
80	70	ad2antlr 723	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)) → 𝑦 ⊆ 𝐴)
81		elfpw 8675	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin) ↔ (𝑏 ⊆ (𝐴 ∩ 𝑊) ∧ 𝑏 ∈ Fin))
82	81	simplbi 498	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin) → 𝑏 ⊆ (𝐴 ∩ 𝑊))
83	82	adantl 482	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)) → 𝑏 ⊆ (𝐴 ∩ 𝑊))
84	83, 1	syl6ss 3903	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)) → 𝑏 ⊆ 𝐴)
85	80, 84	unssd 4085	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)) → (𝑦 ∪ 𝑏) ⊆ 𝐴)
86		elinel2 4096	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin) → 𝑏 ∈ Fin)
87		unfi 8634	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ Fin ∧ 𝑏 ∈ Fin) → (𝑦 ∪ 𝑏) ∈ Fin)
88	74, 86, 87	syl2an 595	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)) → (𝑦 ∪ 𝑏) ∈ Fin)
89		elfpw 8675	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∪ 𝑏) ∈ (𝒫 𝐴 ∩ Fin) ↔ ((𝑦 ∪ 𝑏) ⊆ 𝐴 ∧ (𝑦 ∪ 𝑏) ∈ Fin))
90	85, 88, 89	sylanbrc 583	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)) → (𝑦 ∪ 𝑏) ∈ (𝒫 𝐴 ∩ Fin))
91		ssun1 4071	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ⊆ (𝑦 ∪ 𝑏)
92		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑦 ∪ 𝑏) → 𝑧 = (𝑦 ∪ 𝑏))
93	91, 92	sseqtrrid 3943	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑦 ∪ 𝑏) → 𝑦 ⊆ 𝑧)
94		pm5.5 363	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ⊆ 𝑧 → ((𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢) ↔ (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢))
95	93, 94	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑦 ∪ 𝑏) → ((𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢) ↔ (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢))
96		reseq2 5732	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑦 ∪ 𝑏) → (𝐹 ↾ 𝑧) = (𝐹 ↾ (𝑦 ∪ 𝑏)))
97	96	oveq2d 7035	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑦 ∪ 𝑏) → (𝐺 Σ_g (𝐹 ↾ 𝑧)) = (𝐺 Σ_g (𝐹 ↾ (𝑦 ∪ 𝑏))))
98	97	eleq1d 2866	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑦 ∪ 𝑏) → ((𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢 ↔ (𝐺 Σ_g (𝐹 ↾ (𝑦 ∪ 𝑏))) ∈ 𝑢))
99	95, 98	bitrd 280	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝑦 ∪ 𝑏) → ((𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢) ↔ (𝐺 Σ_g (𝐹 ↾ (𝑦 ∪ 𝑏))) ∈ 𝑢))
100	99	rspcv 3553	. . . . . . . . . . . 12 ⊢ ((𝑦 ∪ 𝑏) ∈ (𝒫 𝐴 ∩ Fin) → (∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢) → (𝐺 Σ_g (𝐹 ↾ (𝑦 ∪ 𝑏))) ∈ 𝑢))
101	90, 100	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)) → (∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢) → (𝐺 Σ_g (𝐹 ↾ (𝑦 ∪ 𝑏))) ∈ 𝑢))
102	42	ad2antrr 722	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin) ∧ (𝑦 ∩ 𝑊) ⊆ 𝑏)) → 𝐺 ∈ CMnd)
103	88	adantrr 713	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin) ∧ (𝑦 ∩ 𝑊) ⊆ 𝑏)) → (𝑦 ∪ 𝑏) ∈ Fin)
104	44	ad2antrr 722	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin) ∧ (𝑦 ∩ 𝑊) ⊆ 𝑏)) → 𝐹:𝐴⟶𝐵)
105	85	adantrr 713	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin) ∧ (𝑦 ∩ 𝑊) ⊆ 𝑏)) → (𝑦 ∪ 𝑏) ⊆ 𝐴)
106	104, 105	fssresd 6416	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin) ∧ (𝑦 ∩ 𝑊) ⊆ 𝑏)) → (𝐹 ↾ (𝑦 ∪ 𝑏)):(𝑦 ∪ 𝑏)⟶𝐵)
107	49, 51	jctir 521	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐹 ∈ V ∧ 0 ∈ V))
108	107	ad2antrr 722	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin) ∧ (𝑦 ∩ 𝑊) ⊆ 𝑏)) → (𝐹 ∈ V ∧ 0 ∈ V))
109		ressuppss 7703	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹 ∈ V ∧ 0 ∈ V) → ((𝐹 ↾ (𝑦 ∪ 𝑏)) supp 0 ) ⊆ (𝐹 supp 0 ))
110	108, 109	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin) ∧ (𝑦 ∩ 𝑊) ⊆ 𝑏)) → ((𝐹 ↾ (𝑦 ∪ 𝑏)) supp 0 ) ⊆ (𝐹 supp 0 ))
111	54	ad2antrr 722	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin) ∧ (𝑦 ∩ 𝑊) ⊆ 𝑏)) → (𝐹 supp 0 ) ⊆ 𝑊)
112	110, 111	sstrd 3901	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin) ∧ (𝑦 ∩ 𝑊) ⊆ 𝑏)) → ((𝐹 ↾ (𝑦 ∪ 𝑏)) supp 0 ) ⊆ 𝑊)
113	51	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin) ∧ (𝑦 ∩ 𝑊) ⊆ 𝑏)) → 0 ∈ V)
114	106, 103, 113	fdmfifsupp 8692	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin) ∧ (𝑦 ∩ 𝑊) ⊆ 𝑏)) → (𝐹 ↾ (𝑦 ∪ 𝑏)) finSupp 0 )
115	40, 41, 102, 103, 106, 112, 114	gsumres 18754	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin) ∧ (𝑦 ∩ 𝑊) ⊆ 𝑏)) → (𝐺 Σ_g ((𝐹 ↾ (𝑦 ∪ 𝑏)) ↾ 𝑊)) = (𝐺 Σ_g (𝐹 ↾ (𝑦 ∪ 𝑏))))
116		resres 5750	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹 ↾ (𝑦 ∪ 𝑏)) ↾ 𝑊) = (𝐹 ↾ ((𝑦 ∪ 𝑏) ∩ 𝑊))
117		indir 4174	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∪ 𝑏) ∩ 𝑊) = ((𝑦 ∩ 𝑊) ∪ (𝑏 ∩ 𝑊))
118	83, 35	syl6ss 3903	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)) → 𝑏 ⊆ 𝑊)
119	118	adantrr 713	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin) ∧ (𝑦 ∩ 𝑊) ⊆ 𝑏)) → 𝑏 ⊆ 𝑊)
120		df-ss 3876	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑏 ⊆ 𝑊 ↔ (𝑏 ∩ 𝑊) = 𝑏)
121	119, 120	sylib 219	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin) ∧ (𝑦 ∩ 𝑊) ⊆ 𝑏)) → (𝑏 ∩ 𝑊) = 𝑏)
122	121	uneq2d 4062	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin) ∧ (𝑦 ∩ 𝑊) ⊆ 𝑏)) → ((𝑦 ∩ 𝑊) ∪ (𝑏 ∩ 𝑊)) = ((𝑦 ∩ 𝑊) ∪ 𝑏))
123		simprr 769	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin) ∧ (𝑦 ∩ 𝑊) ⊆ 𝑏)) → (𝑦 ∩ 𝑊) ⊆ 𝑏)
124		ssequn1 4079	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑦 ∩ 𝑊) ⊆ 𝑏 ↔ ((𝑦 ∩ 𝑊) ∪ 𝑏) = 𝑏)
125	123, 124	sylib 219	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin) ∧ (𝑦 ∩ 𝑊) ⊆ 𝑏)) → ((𝑦 ∩ 𝑊) ∪ 𝑏) = 𝑏)
126	122, 125	eqtrd 2830	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin) ∧ (𝑦 ∩ 𝑊) ⊆ 𝑏)) → ((𝑦 ∩ 𝑊) ∪ (𝑏 ∩ 𝑊)) = 𝑏)
127	117, 126	syl5eq 2842	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin) ∧ (𝑦 ∩ 𝑊) ⊆ 𝑏)) → ((𝑦 ∪ 𝑏) ∩ 𝑊) = 𝑏)
128	127	reseq2d 5737	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin) ∧ (𝑦 ∩ 𝑊) ⊆ 𝑏)) → (𝐹 ↾ ((𝑦 ∪ 𝑏) ∩ 𝑊)) = (𝐹 ↾ 𝑏))
129	116, 128	syl5eq 2842	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin) ∧ (𝑦 ∩ 𝑊) ⊆ 𝑏)) → ((𝐹 ↾ (𝑦 ∪ 𝑏)) ↾ 𝑊) = (𝐹 ↾ 𝑏))
130	119	resabs1d 5768	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin) ∧ (𝑦 ∩ 𝑊) ⊆ 𝑏)) → ((𝐹 ↾ 𝑊) ↾ 𝑏) = (𝐹 ↾ 𝑏))
131	129, 130	eqtr4d 2833	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin) ∧ (𝑦 ∩ 𝑊) ⊆ 𝑏)) → ((𝐹 ↾ (𝑦 ∪ 𝑏)) ↾ 𝑊) = ((𝐹 ↾ 𝑊) ↾ 𝑏))
132	131	oveq2d 7035	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin) ∧ (𝑦 ∩ 𝑊) ⊆ 𝑏)) → (𝐺 Σ_g ((𝐹 ↾ (𝑦 ∪ 𝑏)) ↾ 𝑊)) = (𝐺 Σ_g ((𝐹 ↾ 𝑊) ↾ 𝑏)))
133	115, 132	eqtr3d 2832	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin) ∧ (𝑦 ∩ 𝑊) ⊆ 𝑏)) → (𝐺 Σ_g (𝐹 ↾ (𝑦 ∪ 𝑏))) = (𝐺 Σ_g ((𝐹 ↾ 𝑊) ↾ 𝑏)))
134	133	eleq1d 2866	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin) ∧ (𝑦 ∩ 𝑊) ⊆ 𝑏)) → ((𝐺 Σ_g (𝐹 ↾ (𝑦 ∪ 𝑏))) ∈ 𝑢 ↔ (𝐺 Σ_g ((𝐹 ↾ 𝑊) ↾ 𝑏)) ∈ 𝑢))
135	134	biimpd 230	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin) ∧ (𝑦 ∩ 𝑊) ⊆ 𝑏)) → ((𝐺 Σ_g (𝐹 ↾ (𝑦 ∪ 𝑏))) ∈ 𝑢 → (𝐺 Σ_g ((𝐹 ↾ 𝑊) ↾ 𝑏)) ∈ 𝑢))
136	135	expr 457	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)) → ((𝑦 ∩ 𝑊) ⊆ 𝑏 → ((𝐺 Σ_g (𝐹 ↾ (𝑦 ∪ 𝑏))) ∈ 𝑢 → (𝐺 Σ_g ((𝐹 ↾ 𝑊) ↾ 𝑏)) ∈ 𝑢)))
137	136	com23 86	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)) → ((𝐺 Σ_g (𝐹 ↾ (𝑦 ∪ 𝑏))) ∈ 𝑢 → ((𝑦 ∩ 𝑊) ⊆ 𝑏 → (𝐺 Σ_g ((𝐹 ↾ 𝑊) ↾ 𝑏)) ∈ 𝑢)))
138	101, 137	syld 47	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)) → (∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢) → ((𝑦 ∩ 𝑊) ⊆ 𝑏 → (𝐺 Σ_g ((𝐹 ↾ 𝑊) ↾ 𝑏)) ∈ 𝑢)))
139	138	ralrimdva 3155	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢) → ∀𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)((𝑦 ∩ 𝑊) ⊆ 𝑏 → (𝐺 Σ_g ((𝐹 ↾ 𝑊) ↾ 𝑏)) ∈ 𝑢)))
140		sseq1 3915	. . . . . . . . . 10 ⊢ (𝑎 = (𝑦 ∩ 𝑊) → (𝑎 ⊆ 𝑏 ↔ (𝑦 ∩ 𝑊) ⊆ 𝑏))
141	140	rspceaimv 3565	. . . . . . . . 9 ⊢ (((𝑦 ∩ 𝑊) ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin) ∧ ∀𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)((𝑦 ∩ 𝑊) ⊆ 𝑏 → (𝐺 Σ_g ((𝐹 ↾ 𝑊) ↾ 𝑏)) ∈ 𝑢)) → ∃𝑎 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)∀𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σ_g ((𝐹 ↾ 𝑊) ↾ 𝑏)) ∈ 𝑢))
142	79, 139, 141	syl6an 680	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢) → ∃𝑎 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)∀𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σ_g ((𝐹 ↾ 𝑊) ↾ 𝑏)) ∈ 𝑢)))
143	142	rexlimdva 3246	. . . . . . 7 ⊢ (𝜑 → (∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢) → ∃𝑎 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)∀𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σ_g ((𝐹 ↾ 𝑊) ↾ 𝑏)) ∈ 𝑢)))
144	68, 143	impbid 213	. . . . . 6 ⊢ (𝜑 → (∃𝑎 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)∀𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σ_g ((𝐹 ↾ 𝑊) ↾ 𝑏)) ∈ 𝑢) ↔ ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢)))
145	144	imbi2d 342	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑢 → ∃𝑎 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)∀𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σ_g ((𝐹 ↾ 𝑊) ↾ 𝑏)) ∈ 𝑢)) ↔ (𝑥 ∈ 𝑢 → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢))))
146	145	ralbidv 3163	. . . 4 ⊢ (𝜑 → (∀𝑢 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑢 → ∃𝑎 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)∀𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σ_g ((𝐹 ↾ 𝑊) ↾ 𝑏)) ∈ 𝑢)) ↔ ∀𝑢 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑢 → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢))))
147	146	anbi2d 628	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ ∀𝑢 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑢 → ∃𝑎 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)∀𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σ_g ((𝐹 ↾ 𝑊) ↾ 𝑏)) ∈ 𝑢))) ↔ (𝑥 ∈ 𝐵 ∧ ∀𝑢 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑢 → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢)))))
148		eqid 2794	. . . 4 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
149		eqid 2794	. . . 4 ⊢ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin) = (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)
150		tsmsres.2	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TopSp)
151		inex1g 5117	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝑊) ∈ V)
152	47, 151	syl 17	. . . 4 ⊢ (𝜑 → (𝐴 ∩ 𝑊) ∈ V)
153		fssres 6415	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐴 ∩ 𝑊) ⊆ 𝐴) → (𝐹 ↾ (𝐴 ∩ 𝑊)):(𝐴 ∩ 𝑊)⟶𝐵)
154	44, 1, 153	sylancl 586	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ (𝐴 ∩ 𝑊)):(𝐴 ∩ 𝑊)⟶𝐵)
155		resres 5750	. . . . . . 7 ⊢ ((𝐹 ↾ 𝐴) ↾ 𝑊) = (𝐹 ↾ (𝐴 ∩ 𝑊))
156		ffn 6385	. . . . . . . . 9 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
157		fnresdm 6339	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
158	44, 156, 157	3syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ↾ 𝐴) = 𝐹)
159	158	reseq1d 5736	. . . . . . 7 ⊢ (𝜑 → ((𝐹 ↾ 𝐴) ↾ 𝑊) = (𝐹 ↾ 𝑊))
160	155, 159	syl5eqr 2844	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ (𝐴 ∩ 𝑊)) = (𝐹 ↾ 𝑊))
161	160	feq1d 6370	. . . . 5 ⊢ (𝜑 → ((𝐹 ↾ (𝐴 ∩ 𝑊)):(𝐴 ∩ 𝑊)⟶𝐵 ↔ (𝐹 ↾ 𝑊):(𝐴 ∩ 𝑊)⟶𝐵))
162	154, 161	mpbid 233	. . . 4 ⊢ (𝜑 → (𝐹 ↾ 𝑊):(𝐴 ∩ 𝑊)⟶𝐵)
163	40, 148, 149, 42, 150, 152, 162	eltsms 22424	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐺 tsums (𝐹 ↾ 𝑊)) ↔ (𝑥 ∈ 𝐵 ∧ ∀𝑢 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑢 → ∃𝑎 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)∀𝑏 ∈ (𝒫 (𝐴 ∩ 𝑊) ∩ Fin)(𝑎 ⊆ 𝑏 → (𝐺 Σ_g ((𝐹 ↾ 𝑊) ↾ 𝑏)) ∈ 𝑢)))))
164		eqid 2794	. . . 4 ⊢ (𝒫 𝐴 ∩ Fin) = (𝒫 𝐴 ∩ Fin)
165	40, 148, 164, 42, 150, 47, 44	eltsms 22424	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐺 tsums 𝐹) ↔ (𝑥 ∈ 𝐵 ∧ ∀𝑢 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑢 → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)∀𝑧 ∈ (𝒫 𝐴 ∩ Fin)(𝑦 ⊆ 𝑧 → (𝐺 Σ_g (𝐹 ↾ 𝑧)) ∈ 𝑢)))))
166	147, 163, 165	3bitr4d 312	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐺 tsums (𝐹 ↾ 𝑊)) ↔ 𝑥 ∈ (𝐺 tsums 𝐹)))
167	166	eqrdv 2792	1 ⊢ (𝜑 → (𝐺 tsums (𝐹 ↾ 𝑊)) = (𝐺 tsums 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1522 ∈ wcel 2080 ∀wral 3104 ∃wrex 3105 Vcvv 3436 ∪ cun 3859 ∩ cin 3860 ⊆ wss 3861 𝒫 cpw 4455 ↾ cres 5448 Fn wfn 6223 ⟶wf 6224 ‘cfv 6228 (class class class)co 7019 supp csupp 7684 Fincfn 8360 Basecbs 16312 TopOpenctopn 16524 0_gc0g 16542 Σ_g cgsu 16543 CMndccmn 18633 TopSpctps 21224 tsums ctsu 22417

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 ax-rep 5084 ax-sep 5097 ax-nul 5104 ax-pow 5160 ax-pr 5224 ax-un 7322 ax-cnex 10442 ax-resscn 10443 ax-1cn 10444 ax-icn 10445 ax-addcl 10446 ax-addrcl 10447 ax-mulcl 10448 ax-mulrcl 10449 ax-mulcom 10450 ax-addass 10451 ax-mulass 10452 ax-distr 10453 ax-i2m1 10454 ax-1ne0 10455 ax-1rid 10456 ax-rnegex 10457 ax-rrecex 10458 ax-cnre 10459 ax-pre-lttri 10460 ax-pre-lttrn 10461 ax-pre-ltadd 10462 ax-pre-mulgt0 10463

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ne 2984 df-nel 3090 df-ral 3109 df-rex 3110 df-reu 3111 df-rmo 3112 df-rab 3113 df-v 3438 df-sbc 3708 df-csb 3814 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-pss 3878 df-nul 4214 df-if 4384 df-pw 4457 df-sn 4475 df-pr 4477 df-tp 4479 df-op 4481 df-uni 4748 df-int 4785 df-iun 4829 df-br 4965 df-opab 5027 df-mpt 5044 df-tr 5067 df-id 5351 df-eprel 5356 df-po 5365 df-so 5366 df-fr 5405 df-se 5406 df-we 5407 df-xp 5452 df-rel 5453 df-cnv 5454 df-co 5455 df-dm 5456 df-rn 5457 df-res 5458 df-ima 5459 df-pred 6026 df-ord 6072 df-on 6073 df-lim 6074 df-suc 6075 df-iota 6192 df-fun 6230 df-fn 6231 df-f 6232 df-f1 6233 df-fo 6234 df-f1o 6235 df-fv 6236 df-isom 6237 df-riota 6980 df-ov 7022 df-oprab 7023 df-mpo 7024 df-om 7440 df-1st 7548 df-2nd 7549 df-supp 7685 df-wrecs 7801 df-recs 7863 df-rdg 7901 df-1o 7956 df-oadd 7960 df-er 8142 df-map 8261 df-en 8361 df-dom 8362 df-sdom 8363 df-fin 8364 df-fsupp 8683 df-oi 8823 df-card 9217 df-pnf 10526 df-mnf 10527 df-xr 10528 df-ltxr 10529 df-le 10530 df-sub 10721 df-neg 10722 df-nn 11489 df-n0 11748 df-z 11832 df-uz 12094 df-fz 12743 df-fzo 12884 df-seq 13220 df-hash 13541 df-0g 16544 df-gsum 16545 df-mgm 17681 df-sgrp 17723 df-mnd 17734 df-cntz 18188 df-cmn 18635 df-fbas 20224 df-fg 20225 df-top 21186 df-topon 21203 df-topsp 21225 df-ntr 21312 df-nei 21390 df-fil 22138 df-fm 22230 df-flim 22231 df-flf 22232 df-tsms 22418

This theorem is referenced by: tsmssplit 22443 esumss 30940

W3C validator