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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 29027 | . . . 4 ⊢ 0ℋ ∈ Sℋ | |
2 | shocel 29053 | . . . 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 28868 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (𝑥 ·ih 0ℎ) = 0) | |
5 | df-ral 3143 | . . . . . 6 ⊢ (∀𝑦 ∈ 0ℋ (𝑥 ·ih 𝑦) = 0 ↔ ∀𝑦(𝑦 ∈ 0ℋ → (𝑥 ·ih 𝑦) = 0)) | |
6 | elch0 29025 | . . . . . . . . 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 1816 | . . . . . . 7 ⊢ (∀𝑦(𝑦 ∈ 0ℋ → (𝑥 ·ih 𝑦) = 0) ↔ ∀𝑦(𝑦 = 0ℎ → (𝑥 ·ih 𝑦) = 0)) |
9 | ax-hv0cl 28774 | . . . . . . . . 9 ⊢ 0ℎ ∈ ℋ | |
10 | 9 | elexi 3514 | . . . . . . . 8 ⊢ 0ℎ ∈ V |
11 | oveq2 7158 | . . . . . . . . 9 ⊢ (𝑦 = 0ℎ → (𝑥 ·ih 𝑦) = (𝑥 ·ih 0ℎ)) | |
12 | 11 | eqeq1d 2823 | . . . . . . . 8 ⊢ (𝑦 = 0ℎ → ((𝑥 ·ih 𝑦) = 0 ↔ (𝑥 ·ih 0ℎ) = 0)) |
13 | 10, 12 | ceqsalv 3533 | . . . . . . 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 2818 | 1 ⊢ (⊥‘0ℋ) = ℋ |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1531 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ‘cfv 6350 (class class class)co 7150 0cc0 10531 ℋchba 28690 ·ih csp 28693 0ℎc0v 28695 Sℋ csh 28699 ⊥cort 28701 0ℋc0h 28706 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611 ax-hilex 28770 ax-hfvadd 28771 ax-hvcom 28772 ax-hvass 28773 ax-hv0cl 28774 ax-hvaddid 28775 ax-hfvmul 28776 ax-hvmulid 28777 ax-hvmulass 28778 ax-hvdistr1 28779 ax-hvdistr2 28780 ax-hvmul0 28781 ax-hfi 28850 ax-his1 28853 ax-his2 28854 ax-his3 28855 ax-his4 28856 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-map 8402 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-n0 11892 df-z 11976 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-icc 12739 df-seq 13364 df-exp 13424 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-topgen 16711 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-top 21496 df-topon 21513 df-bases 21548 df-lm 21831 df-haus 21917 df-grpo 28264 df-gid 28265 df-ginv 28266 df-gdiv 28267 df-ablo 28316 df-vc 28330 df-nv 28363 df-va 28366 df-ba 28367 df-sm 28368 df-0v 28369 df-vs 28370 df-nmcv 28371 df-ims 28372 df-hnorm 28739 df-hvsub 28742 df-hlim 28743 df-sh 28978 df-ch 28992 df-oc 29023 df-ch0 29024 |
This theorem is referenced by: choc1 29098 ssjo 29218 qlaxr3i 29407 riesz3i 29833 chirredi 30165 mdsymi 30182 |
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