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| Mirrors > Home > HSE Home > Th. List > chintcli | Structured version Visualization version GIF version | ||
| Description: The intersection of a nonempty set of closed subspaces is a closed subspace. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| chintcl.1 | ⊢ (𝐴 ⊆ Cℋ ∧ 𝐴 ≠ ∅) |
| Ref | Expression |
|---|---|
| chintcli | ⊢ ∩ 𝐴 ∈ Cℋ |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | chintcl.1 | . . . . . 6 ⊢ (𝐴 ⊆ Cℋ ∧ 𝐴 ≠ ∅) | |
| 2 | 1 | simpli 483 | . . . . 5 ⊢ 𝐴 ⊆ Cℋ |
| 3 | chsssh 31160 | . . . . 5 ⊢ Cℋ ⊆ Sℋ | |
| 4 | 2, 3 | sstri 3958 | . . . 4 ⊢ 𝐴 ⊆ Sℋ |
| 5 | 1 | simpri 485 | . . . 4 ⊢ 𝐴 ≠ ∅ |
| 6 | 4, 5 | pm3.2i 470 | . . 3 ⊢ (𝐴 ⊆ Sℋ ∧ 𝐴 ≠ ∅) |
| 7 | 6 | shintcli 31264 | . 2 ⊢ ∩ 𝐴 ∈ Sℋ |
| 8 | 2 | sseli 3944 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ Cℋ ) |
| 9 | vex 3454 | . . . . . . . . . . 11 ⊢ 𝑥 ∈ V | |
| 10 | 9 | chlimi 31169 | . . . . . . . . . 10 ⊢ ((𝑦 ∈ Cℋ ∧ 𝑓:ℕ⟶𝑦 ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ 𝑦) |
| 11 | 10 | 3exp 1119 | . . . . . . . . 9 ⊢ (𝑦 ∈ Cℋ → (𝑓:ℕ⟶𝑦 → (𝑓 ⇝𝑣 𝑥 → 𝑥 ∈ 𝑦))) |
| 12 | 11 | com3r 87 | . . . . . . . 8 ⊢ (𝑓 ⇝𝑣 𝑥 → (𝑦 ∈ Cℋ → (𝑓:ℕ⟶𝑦 → 𝑥 ∈ 𝑦))) |
| 13 | 8, 12 | syl5 34 | . . . . . . 7 ⊢ (𝑓 ⇝𝑣 𝑥 → (𝑦 ∈ 𝐴 → (𝑓:ℕ⟶𝑦 → 𝑥 ∈ 𝑦))) |
| 14 | 13 | imp 406 | . . . . . 6 ⊢ ((𝑓 ⇝𝑣 𝑥 ∧ 𝑦 ∈ 𝐴) → (𝑓:ℕ⟶𝑦 → 𝑥 ∈ 𝑦)) |
| 15 | 14 | ralimdva 3146 | . . . . 5 ⊢ (𝑓 ⇝𝑣 𝑥 → (∀𝑦 ∈ 𝐴 𝑓:ℕ⟶𝑦 → ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)) |
| 16 | 5 | fint 6741 | . . . . 5 ⊢ (𝑓:ℕ⟶∩ 𝐴 ↔ ∀𝑦 ∈ 𝐴 𝑓:ℕ⟶𝑦) |
| 17 | 9 | elint2 4919 | . . . . 5 ⊢ (𝑥 ∈ ∩ 𝐴 ↔ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) |
| 18 | 15, 16, 17 | 3imtr4g 296 | . . . 4 ⊢ (𝑓 ⇝𝑣 𝑥 → (𝑓:ℕ⟶∩ 𝐴 → 𝑥 ∈ ∩ 𝐴)) |
| 19 | 18 | impcom 407 | . . 3 ⊢ ((𝑓:ℕ⟶∩ 𝐴 ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ ∩ 𝐴) |
| 20 | 19 | gen2 1796 | . 2 ⊢ ∀𝑓∀𝑥((𝑓:ℕ⟶∩ 𝐴 ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ ∩ 𝐴) |
| 21 | isch2 31158 | . 2 ⊢ (∩ 𝐴 ∈ Cℋ ↔ (∩ 𝐴 ∈ Sℋ ∧ ∀𝑓∀𝑥((𝑓:ℕ⟶∩ 𝐴 ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ ∩ 𝐴))) | |
| 22 | 7, 20, 21 | mpbir2an 711 | 1 ⊢ ∩ 𝐴 ∈ Cℋ |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∀wal 1538 ∈ wcel 2109 ≠ wne 2926 ∀wral 3045 ⊆ wss 3916 ∅c0 4298 ∩ cint 4912 class class class wbr 5109 ⟶wf 6509 ℕcn 12187 ⇝𝑣 chli 30862 Sℋ csh 30863 Cℋ cch 30864 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5236 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713 ax-cnex 11130 ax-1cn 11132 ax-addcl 11134 ax-hilex 30934 ax-hfvadd 30935 ax-hv0cl 30938 ax-hfvmul 30940 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3756 df-csb 3865 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-pss 3936 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-int 4913 df-iun 4959 df-br 5110 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6276 df-ord 6337 df-on 6338 df-lim 6339 df-suc 6340 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521 df-ov 7392 df-oprab 7393 df-mpo 7394 df-om 7845 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8380 df-map 8803 df-nn 12188 df-sh 31142 df-ch 31156 |
| This theorem is referenced by: chintcl 31267 chincli 31395 |
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