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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 3988 | . . . 4 ⊢ ℋ ⊆ ℋ | |
2 | ax-hv0cl 28779 | . . . 4 ⊢ 0ℎ ∈ ℋ | |
3 | 1, 2 | pm3.2i 473 | . . 3 ⊢ ( ℋ ⊆ ℋ ∧ 0ℎ ∈ ℋ) |
4 | hvaddcl 28788 | . . . . 5 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 +ℎ 𝑦) ∈ ℋ) | |
5 | 4 | rgen2 3203 | . . . 4 ⊢ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 +ℎ 𝑦) ∈ ℋ |
6 | hvmulcl 28789 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ 𝑦) ∈ ℋ) | |
7 | 6 | rgen2 3203 | . . . 4 ⊢ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ (𝑥 ·ℎ 𝑦) ∈ ℋ |
8 | 5, 7 | pm3.2i 473 | . . 3 ⊢ (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 +ℎ 𝑦) ∈ ℋ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ (𝑥 ·ℎ 𝑦) ∈ ℋ) |
9 | issh2 28985 | . . 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 709 | . 2 ⊢ ℋ ∈ Sℋ |
11 | vex 3497 | . . . . 5 ⊢ 𝑥 ∈ V | |
12 | 11 | hlimveci 28966 | . . . 4 ⊢ (𝑓 ⇝𝑣 𝑥 → 𝑥 ∈ ℋ) |
13 | 12 | adantl 484 | . . 3 ⊢ ((𝑓:ℕ⟶ ℋ ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ ℋ) |
14 | 13 | gen2 1793 | . 2 ⊢ ∀𝑓∀𝑥((𝑓:ℕ⟶ ℋ ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ ℋ) |
15 | isch2 28999 | . 2 ⊢ ( ℋ ∈ Cℋ ↔ ( ℋ ∈ Sℋ ∧ ∀𝑓∀𝑥((𝑓:ℕ⟶ ℋ ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ ℋ))) | |
16 | 10, 14, 15 | mpbir2an 709 | 1 ⊢ ℋ ∈ Cℋ |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∀wal 1531 ∈ wcel 2110 ∀wral 3138 ⊆ wss 3935 class class class wbr 5065 ⟶wf 6350 (class class class)co 7155 ℂcc 10534 ℕcn 11637 ℋchba 28695 +ℎ cva 28696 ·ℎ csm 28697 0ℎc0v 28700 ⇝𝑣 chli 28703 Sℋ csh 28704 Cℋ cch 28705 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-1cn 10594 ax-addcl 10596 ax-hilex 28775 ax-hfvadd 28776 ax-hv0cl 28779 ax-hfvmul 28781 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-map 8407 df-nn 11638 df-hlim 28748 df-sh 28983 df-ch 28997 |
This theorem is referenced by: ifchhv 29020 helsh 29021 ococin 29184 chj1i 29265 hne0 29323 pjch1 29446 pjo 29447 pjsslem 29455 ho0val 29526 dfiop2 29529 hoid1i 29565 hoid1ri 29566 pjtoi 29955 pjoci 29956 pjclem3 29973 hst0 30009 st0 30025 strlem3a 30028 hstrlem3a 30036 stcltr2i 30051 |
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