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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 29607 | . . . . . . 7 ⊢ ℋ ∈ Sℋ | |
2 | shocel 29644 | . . . . . . 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 29648 | . . . . . . . 8 ⊢ ( ℋ ∈ Sℋ → (⊥‘ ℋ) ⊆ ℋ) | |
6 | 1, 5 | ax-mp 5 | . . . . . . 7 ⊢ (⊥‘ ℋ) ⊆ ℋ |
7 | 6 | sseli 3917 | . . . . . 6 ⊢ (𝑥 ∈ (⊥‘ ℋ) → 𝑥 ∈ ℋ) |
8 | hial0 29464 | . . . . . 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 231 | . . . 4 ⊢ (𝑥 ∈ (⊥‘ ℋ) → 𝑥 = 0ℎ) |
11 | elch0 29616 | . . . 4 ⊢ (𝑥 ∈ 0ℋ ↔ 𝑥 = 0ℎ) | |
12 | 10, 11 | sylibr 233 | . . 3 ⊢ (𝑥 ∈ (⊥‘ ℋ) → 𝑥 ∈ 0ℋ) |
13 | 12 | ssriv 3925 | . 2 ⊢ (⊥‘ ℋ) ⊆ 0ℋ |
14 | h0elsh 29618 | . . . 4 ⊢ 0ℋ ∈ Sℋ | |
15 | shococss 29656 | . . . 4 ⊢ (0ℋ ∈ Sℋ → 0ℋ ⊆ (⊥‘(⊥‘0ℋ))) | |
16 | 14, 15 | ax-mp 5 | . . 3 ⊢ 0ℋ ⊆ (⊥‘(⊥‘0ℋ)) |
17 | choc0 29688 | . . . 4 ⊢ (⊥‘0ℋ) = ℋ | |
18 | 17 | fveq2i 6777 | . . 3 ⊢ (⊥‘(⊥‘0ℋ)) = (⊥‘ ℋ) |
19 | 16, 18 | sseqtri 3957 | . 2 ⊢ 0ℋ ⊆ (⊥‘ ℋ) |
20 | 13, 19 | eqssi 3937 | 1 ⊢ (⊥‘ ℋ) = 0ℋ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ⊆ wss 3887 ‘cfv 6433 (class class class)co 7275 0cc0 10871 ℋchba 29281 ·ih csp 29284 0ℎc0v 29286 Sℋ csh 29290 ⊥cort 29292 0ℋc0h 29297 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951 ax-hilex 29361 ax-hfvadd 29362 ax-hvcom 29363 ax-hvass 29364 ax-hv0cl 29365 ax-hvaddid 29366 ax-hfvmul 29367 ax-hvmulid 29368 ax-hvmulass 29369 ax-hvdistr1 29370 ax-hvdistr2 29371 ax-hvmul0 29372 ax-hfi 29441 ax-his1 29444 ax-his2 29445 ax-his3 29446 ax-his4 29447 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-map 8617 df-pm 8618 df-en 8734 df-dom 8735 df-sdom 8736 df-sup 9201 df-inf 9202 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-n0 12234 df-z 12320 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-icc 13086 df-seq 13722 df-exp 13783 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-topgen 17154 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-top 22043 df-topon 22060 df-bases 22096 df-lm 22380 df-haus 22466 df-grpo 28855 df-gid 28856 df-ginv 28857 df-gdiv 28858 df-ablo 28907 df-vc 28921 df-nv 28954 df-va 28957 df-ba 28958 df-sm 28959 df-0v 28960 df-vs 28961 df-nmcv 28962 df-ims 28963 df-hnorm 29330 df-hvsub 29333 df-hlim 29334 df-sh 29569 df-ch 29583 df-oc 29614 df-ch0 29615 |
This theorem is referenced by: ho0val 30112 st0 30611 |
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