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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 29025 | . . . 4 ⊢ 0ℋ ∈ Sℋ | |
2 | shocel 29051 | . . . 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 28866 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (𝑥 ·ih 0ℎ) = 0) | |
5 | df-ral 3141 | . . . . . 6 ⊢ (∀𝑦 ∈ 0ℋ (𝑥 ·ih 𝑦) = 0 ↔ ∀𝑦(𝑦 ∈ 0ℋ → (𝑥 ·ih 𝑦) = 0)) | |
6 | elch0 29023 | . . . . . . . . 9 ⊢ (𝑦 ∈ 0ℋ ↔ 𝑦 = 0ℎ) | |
7 | 6 | imbi1i 352 | . . . . . . . 8 ⊢ ((𝑦 ∈ 0ℋ → (𝑥 ·ih 𝑦) = 0) ↔ (𝑦 = 0ℎ → (𝑥 ·ih 𝑦) = 0)) |
8 | 7 | albii 1814 | . . . . . . 7 ⊢ (∀𝑦(𝑦 ∈ 0ℋ → (𝑥 ·ih 𝑦) = 0) ↔ ∀𝑦(𝑦 = 0ℎ → (𝑥 ·ih 𝑦) = 0)) |
9 | ax-hv0cl 28772 | . . . . . . . . 9 ⊢ 0ℎ ∈ ℋ | |
10 | 9 | elexi 3512 | . . . . . . . 8 ⊢ 0ℎ ∈ V |
11 | oveq2 7156 | . . . . . . . . 9 ⊢ (𝑦 = 0ℎ → (𝑥 ·ih 𝑦) = (𝑥 ·ih 0ℎ)) | |
12 | 11 | eqeq1d 2821 | . . . . . . . 8 ⊢ (𝑦 = 0ℎ → ((𝑥 ·ih 𝑦) = 0 ↔ (𝑥 ·ih 0ℎ) = 0)) |
13 | 10, 12 | ceqsalv 3531 | . . . . . . 7 ⊢ (∀𝑦(𝑦 = 0ℎ → (𝑥 ·ih 𝑦) = 0) ↔ (𝑥 ·ih 0ℎ) = 0) |
14 | 8, 13 | bitri 277 | . . . . . 6 ⊢ (∀𝑦(𝑦 ∈ 0ℋ → (𝑥 ·ih 𝑦) = 0) ↔ (𝑥 ·ih 0ℎ) = 0) |
15 | 5, 14 | bitri 277 | . . . . 5 ⊢ (∀𝑦 ∈ 0ℋ (𝑥 ·ih 𝑦) = 0 ↔ (𝑥 ·ih 0ℎ) = 0) |
16 | 4, 15 | sylibr 236 | . . . 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 708 | . . 3 ⊢ ((𝑥 ∈ ℋ ∧ ∀𝑦 ∈ 0ℋ (𝑥 ·ih 𝑦) = 0) ↔ 𝑥 ∈ ℋ) |
19 | 3, 18 | bitri 277 | . 2 ⊢ (𝑥 ∈ (⊥‘0ℋ) ↔ 𝑥 ∈ ℋ) |
20 | 19 | eqriv 2816 | 1 ⊢ (⊥‘0ℋ) = ℋ |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1529 = wceq 1531 ∈ wcel 2108 ∀wral 3136 ‘cfv 6348 (class class class)co 7148 0cc0 10529 ℋchba 28688 ·ih csp 28691 0ℎc0v 28693 Sℋ csh 28697 ⊥cort 28699 0ℋc0h 28704 |
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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-pre-sup 10607 ax-addf 10608 ax-mulf 10609 ax-hilex 28768 ax-hfvadd 28769 ax-hvcom 28770 ax-hvass 28771 ax-hv0cl 28772 ax-hvaddid 28773 ax-hfvmul 28774 ax-hvmulid 28775 ax-hvmulass 28776 ax-hvdistr1 28777 ax-hvdistr2 28778 ax-hvmul0 28779 ax-hfi 28848 ax-his1 28851 ax-his2 28852 ax-his3 28853 ax-his4 28854 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-er 8281 df-map 8400 df-pm 8401 df-en 8502 df-dom 8503 df-sdom 8504 df-sup 8898 df-inf 8899 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-n0 11890 df-z 11974 df-uz 12236 df-q 12341 df-rp 12382 df-xneg 12499 df-xadd 12500 df-xmul 12501 df-icc 12737 df-seq 13362 df-exp 13422 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-topgen 16709 df-psmet 20529 df-xmet 20530 df-met 20531 df-bl 20532 df-mopn 20533 df-top 21494 df-topon 21511 df-bases 21546 df-lm 21829 df-haus 21915 df-grpo 28262 df-gid 28263 df-ginv 28264 df-gdiv 28265 df-ablo 28314 df-vc 28328 df-nv 28361 df-va 28364 df-ba 28365 df-sm 28366 df-0v 28367 df-vs 28368 df-nmcv 28369 df-ims 28370 df-hnorm 28737 df-hvsub 28740 df-hlim 28741 df-sh 28976 df-ch 28990 df-oc 29021 df-ch0 29022 |
This theorem is referenced by: choc1 29096 ssjo 29216 qlaxr3i 29405 riesz3i 29831 chirredi 30163 mdsymi 30180 |
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