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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 3909 | . . . 4 ⊢ ℋ ⊆ ℋ | |
2 | ax-hv0cl 29038 | . . . 4 ⊢ 0ℎ ∈ ℋ | |
3 | 1, 2 | pm3.2i 474 | . . 3 ⊢ ( ℋ ⊆ ℋ ∧ 0ℎ ∈ ℋ) |
4 | hvaddcl 29047 | . . . . 5 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 +ℎ 𝑦) ∈ ℋ) | |
5 | 4 | rgen2 3114 | . . . 4 ⊢ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 +ℎ 𝑦) ∈ ℋ |
6 | hvmulcl 29048 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ 𝑦) ∈ ℋ) | |
7 | 6 | rgen2 3114 | . . . 4 ⊢ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ (𝑥 ·ℎ 𝑦) ∈ ℋ |
8 | 5, 7 | pm3.2i 474 | . . 3 ⊢ (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 +ℎ 𝑦) ∈ ℋ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ (𝑥 ·ℎ 𝑦) ∈ ℋ) |
9 | issh2 29244 | . . 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 711 | . 2 ⊢ ℋ ∈ Sℋ |
11 | vex 3402 | . . . . 5 ⊢ 𝑥 ∈ V | |
12 | 11 | hlimveci 29225 | . . . 4 ⊢ (𝑓 ⇝𝑣 𝑥 → 𝑥 ∈ ℋ) |
13 | 12 | adantl 485 | . . 3 ⊢ ((𝑓:ℕ⟶ ℋ ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ ℋ) |
14 | 13 | gen2 1804 | . 2 ⊢ ∀𝑓∀𝑥((𝑓:ℕ⟶ ℋ ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ ℋ) |
15 | isch2 29258 | . 2 ⊢ ( ℋ ∈ Cℋ ↔ ( ℋ ∈ Sℋ ∧ ∀𝑓∀𝑥((𝑓:ℕ⟶ ℋ ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ ℋ))) | |
16 | 10, 14, 15 | mpbir2an 711 | 1 ⊢ ℋ ∈ Cℋ |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∀wal 1541 ∈ wcel 2112 ∀wral 3051 ⊆ wss 3853 class class class wbr 5039 ⟶wf 6354 (class class class)co 7191 ℂcc 10692 ℕcn 11795 ℋchba 28954 +ℎ cva 28955 ·ℎ csm 28956 0ℎc0v 28959 ⇝𝑣 chli 28962 Sℋ csh 28963 Cℋ cch 28964 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-1cn 10752 ax-addcl 10754 ax-hilex 29034 ax-hfvadd 29035 ax-hv0cl 29038 ax-hfvmul 29040 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-map 8488 df-nn 11796 df-hlim 29007 df-sh 29242 df-ch 29256 |
This theorem is referenced by: ifchhv 29279 helsh 29280 ococin 29443 chj1i 29524 hne0 29582 pjch1 29705 pjo 29706 pjsslem 29714 ho0val 29785 dfiop2 29788 hoid1i 29824 hoid1ri 29825 pjtoi 30214 pjoci 30215 pjclem3 30232 hst0 30268 st0 30284 strlem3a 30287 hstrlem3a 30295 stcltr2i 30310 |
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