Proof of Theorem flfcnp
Step | Hyp | Ref
| Expression |
1 | | simprl 771 |
. . . 4
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → 𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹)) |
2 | | flfval 22675 |
. . . . 5
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝐽 fLimf 𝐿)‘𝐹) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿))) |
3 | 2 | adantr 485 |
. . . 4
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → ((𝐽 fLimf 𝐿)‘𝐹) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿))) |
4 | 1, 3 | eleqtrd 2853 |
. . 3
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → 𝐴 ∈ (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿))) |
5 | | simprr 773 |
. . 3
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴)) |
6 | | cnpflfi 22684 |
. . 3
⊢ ((𝐴 ∈ (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐺‘𝐴) ∈ ((𝐾 fLimf ((𝑋 FilMap 𝐹)‘𝐿))‘𝐺)) |
7 | 4, 5, 6 | syl2anc 588 |
. 2
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → (𝐺‘𝐴) ∈ ((𝐾 fLimf ((𝑋 FilMap 𝐹)‘𝐿))‘𝐺)) |
8 | | cnptop2 21928 |
. . . . . . . 8
⊢ (𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴) → 𝐾 ∈ Top) |
9 | 8 | ad2antll 729 |
. . . . . . 7
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → 𝐾 ∈ Top) |
10 | | toptopon2 21603 |
. . . . . . 7
⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾)) |
11 | 9, 10 | sylib 221 |
. . . . . 6
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → 𝐾 ∈ (TopOn‘∪ 𝐾)) |
12 | | toponmax 21611 |
. . . . . 6
⊢ (𝐾 ∈ (TopOn‘∪ 𝐾)
→ ∪ 𝐾 ∈ 𝐾) |
13 | 11, 12 | syl 17 |
. . . . 5
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → ∪
𝐾 ∈ 𝐾) |
14 | | simpl1 1189 |
. . . . . 6
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → 𝐽 ∈ (TopOn‘𝑋)) |
15 | | toponmax 21611 |
. . . . . 6
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽) |
16 | 14, 15 | syl 17 |
. . . . 5
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → 𝑋 ∈ 𝐽) |
17 | | simpl2 1190 |
. . . . . 6
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → 𝐿 ∈ (Fil‘𝑌)) |
18 | | filfbas 22533 |
. . . . . 6
⊢ (𝐿 ∈ (Fil‘𝑌) → 𝐿 ∈ (fBas‘𝑌)) |
19 | 17, 18 | syl 17 |
. . . . 5
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → 𝐿 ∈ (fBas‘𝑌)) |
20 | | cnpf2 21935 |
. . . . . 6
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾)
∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐺:𝑋⟶∪ 𝐾) |
21 | 14, 11, 5, 20 | syl3anc 1369 |
. . . . 5
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → 𝐺:𝑋⟶∪ 𝐾) |
22 | | simpl3 1191 |
. . . . 5
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → 𝐹:𝑌⟶𝑋) |
23 | | fmco 22646 |
. . . . 5
⊢ (((∪ 𝐾
∈ 𝐾 ∧ 𝑋 ∈ 𝐽 ∧ 𝐿 ∈ (fBas‘𝑌)) ∧ (𝐺:𝑋⟶∪ 𝐾 ∧ 𝐹:𝑌⟶𝑋)) → ((∪
𝐾 FilMap (𝐺 ∘ 𝐹))‘𝐿) = ((∪ 𝐾 FilMap 𝐺)‘((𝑋 FilMap 𝐹)‘𝐿))) |
24 | 13, 16, 19, 21, 22, 23 | syl32anc 1376 |
. . . 4
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → ((∪
𝐾 FilMap (𝐺 ∘ 𝐹))‘𝐿) = ((∪ 𝐾 FilMap 𝐺)‘((𝑋 FilMap 𝐹)‘𝐿))) |
25 | 24 | oveq2d 7159 |
. . 3
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → (𝐾 fLim ((∪ 𝐾 FilMap (𝐺 ∘ 𝐹))‘𝐿)) = (𝐾 fLim ((∪ 𝐾 FilMap 𝐺)‘((𝑋 FilMap 𝐹)‘𝐿)))) |
26 | | fco 6509 |
. . . . 5
⊢ ((𝐺:𝑋⟶∪ 𝐾 ∧ 𝐹:𝑌⟶𝑋) → (𝐺 ∘ 𝐹):𝑌⟶∪ 𝐾) |
27 | 21, 22, 26 | syl2anc 588 |
. . . 4
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → (𝐺 ∘ 𝐹):𝑌⟶∪ 𝐾) |
28 | | flfval 22675 |
. . . 4
⊢ ((𝐾 ∈ (TopOn‘∪ 𝐾)
∧ 𝐿 ∈
(Fil‘𝑌) ∧ (𝐺 ∘ 𝐹):𝑌⟶∪ 𝐾) → ((𝐾 fLimf 𝐿)‘(𝐺 ∘ 𝐹)) = (𝐾 fLim ((∪ 𝐾 FilMap (𝐺 ∘ 𝐹))‘𝐿))) |
29 | 11, 17, 27, 28 | syl3anc 1369 |
. . 3
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → ((𝐾 fLimf 𝐿)‘(𝐺 ∘ 𝐹)) = (𝐾 fLim ((∪ 𝐾 FilMap (𝐺 ∘ 𝐹))‘𝐿))) |
30 | | fmfil 22629 |
. . . . 5
⊢ ((𝑋 ∈ 𝐽 ∧ 𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋)) |
31 | 16, 19, 22, 30 | syl3anc 1369 |
. . . 4
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋)) |
32 | | flfval 22675 |
. . . 4
⊢ ((𝐾 ∈ (TopOn‘∪ 𝐾)
∧ ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋) ∧ 𝐺:𝑋⟶∪ 𝐾) → ((𝐾 fLimf ((𝑋 FilMap 𝐹)‘𝐿))‘𝐺) = (𝐾 fLim ((∪ 𝐾 FilMap 𝐺)‘((𝑋 FilMap 𝐹)‘𝐿)))) |
33 | 11, 31, 21, 32 | syl3anc 1369 |
. . 3
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → ((𝐾 fLimf ((𝑋 FilMap 𝐹)‘𝐿))‘𝐺) = (𝐾 fLim ((∪ 𝐾 FilMap 𝐺)‘((𝑋 FilMap 𝐹)‘𝐿)))) |
34 | 25, 29, 33 | 3eqtr4d 2804 |
. 2
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → ((𝐾 fLimf 𝐿)‘(𝐺 ∘ 𝐹)) = ((𝐾 fLimf ((𝑋 FilMap 𝐹)‘𝐿))‘𝐺)) |
35 | 7, 34 | eleqtrrd 2854 |
1
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → (𝐺‘𝐴) ∈ ((𝐾 fLimf 𝐿)‘(𝐺 ∘ 𝐹))) |