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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 3991 | . . . 4 ⊢ ℋ ⊆ ℋ | |
2 | ax-hv0cl 28782 | . . . 4 ⊢ 0ℎ ∈ ℋ | |
3 | 1, 2 | pm3.2i 473 | . . 3 ⊢ ( ℋ ⊆ ℋ ∧ 0ℎ ∈ ℋ) |
4 | hvaddcl 28791 | . . . . 5 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 +ℎ 𝑦) ∈ ℋ) | |
5 | 4 | rgen2 3205 | . . . 4 ⊢ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 +ℎ 𝑦) ∈ ℋ |
6 | hvmulcl 28792 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ 𝑦) ∈ ℋ) | |
7 | 6 | rgen2 3205 | . . . 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 28988 | . . 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 3499 | . . . . 5 ⊢ 𝑥 ∈ V | |
12 | 11 | hlimveci 28969 | . . . 4 ⊢ (𝑓 ⇝𝑣 𝑥 → 𝑥 ∈ ℋ) |
13 | 12 | adantl 484 | . . 3 ⊢ ((𝑓:ℕ⟶ ℋ ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ ℋ) |
14 | 13 | gen2 1797 | . 2 ⊢ ∀𝑓∀𝑥((𝑓:ℕ⟶ ℋ ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ ℋ) |
15 | isch2 29002 | . 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 1535 ∈ wcel 2114 ∀wral 3140 ⊆ wss 3938 class class class wbr 5068 ⟶wf 6353 (class class class)co 7158 ℂcc 10537 ℕcn 11640 ℋchba 28698 +ℎ cva 28699 ·ℎ csm 28700 0ℎc0v 28703 ⇝𝑣 chli 28706 Sℋ csh 28707 Cℋ cch 28708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-1cn 10597 ax-addcl 10599 ax-hilex 28778 ax-hfvadd 28779 ax-hv0cl 28782 ax-hfvmul 28784 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-map 8410 df-nn 11641 df-hlim 28751 df-sh 28986 df-ch 29000 |
This theorem is referenced by: ifchhv 29023 helsh 29024 ococin 29187 chj1i 29268 hne0 29326 pjch1 29449 pjo 29450 pjsslem 29458 ho0val 29529 dfiop2 29532 hoid1i 29568 hoid1ri 29569 pjtoi 29958 pjoci 29959 pjclem3 29976 hst0 30012 st0 30028 strlem3a 30031 hstrlem3a 30039 stcltr2i 30054 |
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