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Mirrors > Home > HSE Home > Th. List > helch | Structured version Visualization version GIF version |
Description: The Hilbert lattice one (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 4002 | . . . 4 ⊢ ℋ ⊆ ℋ | |
2 | ax-hv0cl 30936 | . . . 4 ⊢ 0ℎ ∈ ℋ | |
3 | 1, 2 | pm3.2i 469 | . . 3 ⊢ ( ℋ ⊆ ℋ ∧ 0ℎ ∈ ℋ) |
4 | hvaddcl 30945 | . . . . 5 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 +ℎ 𝑦) ∈ ℋ) | |
5 | 4 | rgen2 3188 | . . . 4 ⊢ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 +ℎ 𝑦) ∈ ℋ |
6 | hvmulcl 30946 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ 𝑦) ∈ ℋ) | |
7 | 6 | rgen2 3188 | . . . 4 ⊢ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ (𝑥 ·ℎ 𝑦) ∈ ℋ |
8 | 5, 7 | pm3.2i 469 | . . 3 ⊢ (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 +ℎ 𝑦) ∈ ℋ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ (𝑥 ·ℎ 𝑦) ∈ ℋ) |
9 | issh2 31142 | . . 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 3466 | . . . . 5 ⊢ 𝑥 ∈ V | |
12 | 11 | hlimveci 31123 | . . . 4 ⊢ (𝑓 ⇝𝑣 𝑥 → 𝑥 ∈ ℋ) |
13 | 12 | adantl 480 | . . 3 ⊢ ((𝑓:ℕ⟶ ℋ ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ ℋ) |
14 | 13 | gen2 1791 | . 2 ⊢ ∀𝑓∀𝑥((𝑓:ℕ⟶ ℋ ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ ℋ) |
15 | isch2 31156 | . 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 394 ∀wal 1532 ∈ wcel 2099 ∀wral 3051 ⊆ wss 3947 class class class wbr 5153 ⟶wf 6550 (class class class)co 7424 ℂcc 11156 ℕcn 12264 ℋchba 30852 +ℎ cva 30853 ·ℎ csm 30854 0ℎc0v 30857 ⇝𝑣 chli 30860 Sℋ csh 30861 Cℋ cch 30862 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-1cn 11216 ax-addcl 11218 ax-hilex 30932 ax-hfvadd 30933 ax-hv0cl 30936 ax-hfvmul 30938 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-map 8857 df-nn 12265 df-hlim 30905 df-sh 31140 df-ch 31154 |
This theorem is referenced by: ifchhv 31177 helsh 31178 ococin 31341 chj1i 31422 hne0 31480 pjch1 31603 pjo 31604 pjsslem 31612 ho0val 31683 dfiop2 31686 hoid1i 31722 hoid1ri 31723 pjtoi 32112 pjoci 32113 pjclem3 32130 hst0 32166 st0 32182 strlem3a 32185 hstrlem3a 32193 stcltr2i 32208 |
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