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Mirrors > Home > HSE Home > Th. List > helch | Structured version Visualization version GIF version |
Description: The unit Hilbert lattice element (which is all of Hilbert space) belongs to the Hilbert lattice. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 6-Sep-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
helch | ⊢ ℋ ∈ Cℋ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3819 | . . . 4 ⊢ ℋ ⊆ ℋ | |
2 | ax-hv0cl 28385 | . . . 4 ⊢ 0ℎ ∈ ℋ | |
3 | 1, 2 | pm3.2i 463 | . . 3 ⊢ ( ℋ ⊆ ℋ ∧ 0ℎ ∈ ℋ) |
4 | hvaddcl 28394 | . . . . 5 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 +ℎ 𝑦) ∈ ℋ) | |
5 | 4 | rgen2a 3158 | . . . 4 ⊢ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 +ℎ 𝑦) ∈ ℋ |
6 | hvmulcl 28395 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ 𝑦) ∈ ℋ) | |
7 | 6 | rgen2 3156 | . . . 4 ⊢ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ (𝑥 ·ℎ 𝑦) ∈ ℋ |
8 | 5, 7 | pm3.2i 463 | . . 3 ⊢ (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 +ℎ 𝑦) ∈ ℋ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ (𝑥 ·ℎ 𝑦) ∈ ℋ) |
9 | issh2 28591 | . . 3 ⊢ ( ℋ ∈ Sℋ ↔ (( ℋ ⊆ ℋ ∧ 0ℎ ∈ ℋ) ∧ (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 +ℎ 𝑦) ∈ ℋ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ (𝑥 ·ℎ 𝑦) ∈ ℋ))) | |
10 | 3, 8, 9 | mpbir2an 703 | . 2 ⊢ ℋ ∈ Sℋ |
11 | vex 3388 | . . . . 5 ⊢ 𝑥 ∈ V | |
12 | 11 | hlimveci 28572 | . . . 4 ⊢ (𝑓 ⇝𝑣 𝑥 → 𝑥 ∈ ℋ) |
13 | 12 | adantl 474 | . . 3 ⊢ ((𝑓:ℕ⟶ ℋ ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ ℋ) |
14 | 13 | gen2 1892 | . 2 ⊢ ∀𝑓∀𝑥((𝑓:ℕ⟶ ℋ ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ ℋ) |
15 | isch2 28605 | . 2 ⊢ ( ℋ ∈ Cℋ ↔ ( ℋ ∈ Sℋ ∧ ∀𝑓∀𝑥((𝑓:ℕ⟶ ℋ ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ ℋ))) | |
16 | 10, 14, 15 | mpbir2an 703 | 1 ⊢ ℋ ∈ Cℋ |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 ∀wal 1651 ∈ wcel 2157 ∀wral 3089 ⊆ wss 3769 class class class wbr 4843 ⟶wf 6097 (class class class)co 6878 ℂcc 10222 ℕcn 11312 ℋchba 28301 +ℎ cva 28302 ·ℎ csm 28303 0ℎc0v 28306 ⇝𝑣 chli 28309 Sℋ csh 28310 Cℋ cch 28311 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-1cn 10282 ax-addcl 10284 ax-hilex 28381 ax-hfvadd 28382 ax-hv0cl 28385 ax-hfvmul 28387 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-map 8097 df-nn 11313 df-hlim 28354 df-sh 28589 df-ch 28603 |
This theorem is referenced by: ifchhv 28626 helsh 28627 ococin 28792 chj1i 28873 hne0 28931 pjch1 29054 pjo 29055 pjsslem 29063 ho0val 29134 dfiop2 29137 hoid1i 29173 hoid1ri 29174 pjtoi 29563 pjoci 29564 pjclem3 29581 hst0 29617 st0 29633 strlem3a 29636 hstrlem3a 29644 stcltr2i 29659 |
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