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Mirrors > Home > HSE Home > Th. List > choc1 | Structured version Visualization version GIF version |
Description: The orthocomplement of the unit subspace is the zero subspace. Does not require Axiom of Choice. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
choc1 | ⊢ (⊥‘ ℋ) = 0ℋ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | helsh 28949 | . . . . . . 7 ⊢ ℋ ∈ Sℋ | |
2 | shocel 28986 | . . . . . . 7 ⊢ ( ℋ ∈ Sℋ → (𝑥 ∈ (⊥‘ ℋ) ↔ (𝑥 ∈ ℋ ∧ ∀𝑦 ∈ ℋ (𝑥 ·ih 𝑦) = 0))) | |
3 | 1, 2 | ax-mp 5 | . . . . . 6 ⊢ (𝑥 ∈ (⊥‘ ℋ) ↔ (𝑥 ∈ ℋ ∧ ∀𝑦 ∈ ℋ (𝑥 ·ih 𝑦) = 0)) |
4 | 3 | simprbi 497 | . . . . 5 ⊢ (𝑥 ∈ (⊥‘ ℋ) → ∀𝑦 ∈ ℋ (𝑥 ·ih 𝑦) = 0) |
5 | shocss 28990 | . . . . . . . 8 ⊢ ( ℋ ∈ Sℋ → (⊥‘ ℋ) ⊆ ℋ) | |
6 | 1, 5 | ax-mp 5 | . . . . . . 7 ⊢ (⊥‘ ℋ) ⊆ ℋ |
7 | 6 | sseli 3960 | . . . . . 6 ⊢ (𝑥 ∈ (⊥‘ ℋ) → 𝑥 ∈ ℋ) |
8 | hial0 28806 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → (∀𝑦 ∈ ℋ (𝑥 ·ih 𝑦) = 0 ↔ 𝑥 = 0ℎ)) | |
9 | 7, 8 | syl 17 | . . . . 5 ⊢ (𝑥 ∈ (⊥‘ ℋ) → (∀𝑦 ∈ ℋ (𝑥 ·ih 𝑦) = 0 ↔ 𝑥 = 0ℎ)) |
10 | 4, 9 | mpbid 233 | . . . 4 ⊢ (𝑥 ∈ (⊥‘ ℋ) → 𝑥 = 0ℎ) |
11 | elch0 28958 | . . . 4 ⊢ (𝑥 ∈ 0ℋ ↔ 𝑥 = 0ℎ) | |
12 | 10, 11 | sylibr 235 | . . 3 ⊢ (𝑥 ∈ (⊥‘ ℋ) → 𝑥 ∈ 0ℋ) |
13 | 12 | ssriv 3968 | . 2 ⊢ (⊥‘ ℋ) ⊆ 0ℋ |
14 | h0elsh 28960 | . . . 4 ⊢ 0ℋ ∈ Sℋ | |
15 | shococss 28998 | . . . 4 ⊢ (0ℋ ∈ Sℋ → 0ℋ ⊆ (⊥‘(⊥‘0ℋ))) | |
16 | 14, 15 | ax-mp 5 | . . 3 ⊢ 0ℋ ⊆ (⊥‘(⊥‘0ℋ)) |
17 | choc0 29030 | . . . 4 ⊢ (⊥‘0ℋ) = ℋ | |
18 | 17 | fveq2i 6666 | . . 3 ⊢ (⊥‘(⊥‘0ℋ)) = (⊥‘ ℋ) |
19 | 16, 18 | sseqtri 4000 | . 2 ⊢ 0ℋ ⊆ (⊥‘ ℋ) |
20 | 13, 19 | eqssi 3980 | 1 ⊢ (⊥‘ ℋ) = 0ℋ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ⊆ wss 3933 ‘cfv 6348 (class class class)co 7145 0cc0 10525 ℋchba 28623 ·ih csp 28626 0ℎc0v 28628 Sℋ csh 28632 ⊥cort 28634 0ℋc0h 28639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 ax-hilex 28703 ax-hfvadd 28704 ax-hvcom 28705 ax-hvass 28706 ax-hv0cl 28707 ax-hvaddid 28708 ax-hfvmul 28709 ax-hvmulid 28710 ax-hvmulass 28711 ax-hvdistr1 28712 ax-hvdistr2 28713 ax-hvmul0 28714 ax-hfi 28783 ax-his1 28786 ax-his2 28787 ax-his3 28788 ax-his4 28789 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 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 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-map 8397 df-pm 8398 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-icc 12733 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-topgen 16705 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-top 21430 df-topon 21447 df-bases 21482 df-lm 21765 df-haus 21851 df-grpo 28197 df-gid 28198 df-ginv 28199 df-gdiv 28200 df-ablo 28249 df-vc 28263 df-nv 28296 df-va 28299 df-ba 28300 df-sm 28301 df-0v 28302 df-vs 28303 df-nmcv 28304 df-ims 28305 df-hnorm 28672 df-hvsub 28675 df-hlim 28676 df-sh 28911 df-ch 28925 df-oc 28956 df-ch0 28957 |
This theorem is referenced by: ho0val 29454 st0 29953 |
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