![]() |
Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > HSE Home > Th. List > choc0 | Structured version Visualization version GIF version |
Description: The orthocomplement of the zero subspace is the unit subspace. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
choc0 | ⊢ (⊥‘0ℋ) = ℋ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | h0elsh 31003 | . . . 4 ⊢ 0ℋ ∈ Sℋ | |
2 | shocel 31029 | . . . 4 ⊢ (0ℋ ∈ Sℋ → (𝑥 ∈ (⊥‘0ℋ) ↔ (𝑥 ∈ ℋ ∧ ∀𝑦 ∈ 0ℋ (𝑥 ·ih 𝑦) = 0))) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ (𝑥 ∈ (⊥‘0ℋ) ↔ (𝑥 ∈ ℋ ∧ ∀𝑦 ∈ 0ℋ (𝑥 ·ih 𝑦) = 0)) |
4 | hi02 30844 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (𝑥 ·ih 0ℎ) = 0) | |
5 | df-ral 3054 | . . . . . 6 ⊢ (∀𝑦 ∈ 0ℋ (𝑥 ·ih 𝑦) = 0 ↔ ∀𝑦(𝑦 ∈ 0ℋ → (𝑥 ·ih 𝑦) = 0)) | |
6 | elch0 31001 | . . . . . . . . 9 ⊢ (𝑦 ∈ 0ℋ ↔ 𝑦 = 0ℎ) | |
7 | 6 | imbi1i 349 | . . . . . . . 8 ⊢ ((𝑦 ∈ 0ℋ → (𝑥 ·ih 𝑦) = 0) ↔ (𝑦 = 0ℎ → (𝑥 ·ih 𝑦) = 0)) |
8 | 7 | albii 1813 | . . . . . . 7 ⊢ (∀𝑦(𝑦 ∈ 0ℋ → (𝑥 ·ih 𝑦) = 0) ↔ ∀𝑦(𝑦 = 0ℎ → (𝑥 ·ih 𝑦) = 0)) |
9 | ax-hv0cl 30750 | . . . . . . . . 9 ⊢ 0ℎ ∈ ℋ | |
10 | 9 | elexi 3486 | . . . . . . . 8 ⊢ 0ℎ ∈ V |
11 | oveq2 7410 | . . . . . . . . 9 ⊢ (𝑦 = 0ℎ → (𝑥 ·ih 𝑦) = (𝑥 ·ih 0ℎ)) | |
12 | 11 | eqeq1d 2726 | . . . . . . . 8 ⊢ (𝑦 = 0ℎ → ((𝑥 ·ih 𝑦) = 0 ↔ (𝑥 ·ih 0ℎ) = 0)) |
13 | 10, 12 | ceqsalv 3504 | . . . . . . 7 ⊢ (∀𝑦(𝑦 = 0ℎ → (𝑥 ·ih 𝑦) = 0) ↔ (𝑥 ·ih 0ℎ) = 0) |
14 | 8, 13 | bitri 275 | . . . . . 6 ⊢ (∀𝑦(𝑦 ∈ 0ℋ → (𝑥 ·ih 𝑦) = 0) ↔ (𝑥 ·ih 0ℎ) = 0) |
15 | 5, 14 | bitri 275 | . . . . 5 ⊢ (∀𝑦 ∈ 0ℋ (𝑥 ·ih 𝑦) = 0 ↔ (𝑥 ·ih 0ℎ) = 0) |
16 | 4, 15 | sylibr 233 | . . . 4 ⊢ (𝑥 ∈ ℋ → ∀𝑦 ∈ 0ℋ (𝑥 ·ih 𝑦) = 0) |
17 | abai 824 | . . . 4 ⊢ ((𝑥 ∈ ℋ ∧ ∀𝑦 ∈ 0ℋ (𝑥 ·ih 𝑦) = 0) ↔ (𝑥 ∈ ℋ ∧ (𝑥 ∈ ℋ → ∀𝑦 ∈ 0ℋ (𝑥 ·ih 𝑦) = 0))) | |
18 | 16, 17 | mpbiran2 707 | . . 3 ⊢ ((𝑥 ∈ ℋ ∧ ∀𝑦 ∈ 0ℋ (𝑥 ·ih 𝑦) = 0) ↔ 𝑥 ∈ ℋ) |
19 | 3, 18 | bitri 275 | . 2 ⊢ (𝑥 ∈ (⊥‘0ℋ) ↔ 𝑥 ∈ ℋ) |
20 | 19 | eqriv 2721 | 1 ⊢ (⊥‘0ℋ) = ℋ |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∀wal 1531 = wceq 1533 ∈ wcel 2098 ∀wral 3053 ‘cfv 6534 (class class class)co 7402 0cc0 11107 ℋchba 30666 ·ih csp 30669 0ℎc0v 30671 Sℋ csh 30675 ⊥cort 30677 0ℋc0h 30682 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 ax-addf 11186 ax-mulf 11187 ax-hilex 30746 ax-hfvadd 30747 ax-hvcom 30748 ax-hvass 30749 ax-hv0cl 30750 ax-hvaddid 30751 ax-hfvmul 30752 ax-hvmulid 30753 ax-hvmulass 30754 ax-hvdistr1 30755 ax-hvdistr2 30756 ax-hvmul0 30757 ax-hfi 30826 ax-his1 30829 ax-his2 30830 ax-his3 30831 ax-his4 30832 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-map 8819 df-pm 8820 df-en 8937 df-dom 8938 df-sdom 8939 df-sup 9434 df-inf 9435 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-n0 12472 df-z 12558 df-uz 12822 df-q 12932 df-rp 12976 df-xneg 13093 df-xadd 13094 df-xmul 13095 df-icc 13332 df-seq 13968 df-exp 14029 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-topgen 17394 df-psmet 21226 df-xmet 21227 df-met 21228 df-bl 21229 df-mopn 21230 df-top 22740 df-topon 22757 df-bases 22793 df-lm 23077 df-haus 23163 df-grpo 30240 df-gid 30241 df-ginv 30242 df-gdiv 30243 df-ablo 30292 df-vc 30306 df-nv 30339 df-va 30342 df-ba 30343 df-sm 30344 df-0v 30345 df-vs 30346 df-nmcv 30347 df-ims 30348 df-hnorm 30715 df-hvsub 30718 df-hlim 30719 df-sh 30954 df-ch 30968 df-oc 30999 df-ch0 31000 |
This theorem is referenced by: choc1 31074 ssjo 31194 qlaxr3i 31383 riesz3i 31809 chirredi 32141 mdsymi 32158 |
Copyright terms: Public domain | W3C validator |