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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 29603 | . . . . . . 7 ⊢ ℋ ∈ Sℋ | |
2 | shocel 29640 | . . . . . . 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 29644 | . . . . . . . 8 ⊢ ( ℋ ∈ Sℋ → (⊥‘ ℋ) ⊆ ℋ) | |
6 | 1, 5 | ax-mp 5 | . . . . . . 7 ⊢ (⊥‘ ℋ) ⊆ ℋ |
7 | 6 | sseli 3922 | . . . . . 6 ⊢ (𝑥 ∈ (⊥‘ ℋ) → 𝑥 ∈ ℋ) |
8 | hial0 29460 | . . . . . 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 29612 | . . . 4 ⊢ (𝑥 ∈ 0ℋ ↔ 𝑥 = 0ℎ) | |
12 | 10, 11 | sylibr 233 | . . 3 ⊢ (𝑥 ∈ (⊥‘ ℋ) → 𝑥 ∈ 0ℋ) |
13 | 12 | ssriv 3930 | . 2 ⊢ (⊥‘ ℋ) ⊆ 0ℋ |
14 | h0elsh 29614 | . . . 4 ⊢ 0ℋ ∈ Sℋ | |
15 | shococss 29652 | . . . 4 ⊢ (0ℋ ∈ Sℋ → 0ℋ ⊆ (⊥‘(⊥‘0ℋ))) | |
16 | 14, 15 | ax-mp 5 | . . 3 ⊢ 0ℋ ⊆ (⊥‘(⊥‘0ℋ)) |
17 | choc0 29684 | . . . 4 ⊢ (⊥‘0ℋ) = ℋ | |
18 | 17 | fveq2i 6774 | . . 3 ⊢ (⊥‘(⊥‘0ℋ)) = (⊥‘ ℋ) |
19 | 16, 18 | sseqtri 3962 | . 2 ⊢ 0ℋ ⊆ (⊥‘ ℋ) |
20 | 13, 19 | eqssi 3942 | 1 ⊢ (⊥‘ ℋ) = 0ℋ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ∀wral 3066 ⊆ wss 3892 ‘cfv 6432 (class class class)co 7271 0cc0 10872 ℋchba 29277 ·ih csp 29280 0ℎc0v 29282 Sℋ csh 29286 ⊥cort 29288 0ℋc0h 29293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-pre-sup 10950 ax-addf 10951 ax-mulf 10952 ax-hilex 29357 ax-hfvadd 29358 ax-hvcom 29359 ax-hvass 29360 ax-hv0cl 29361 ax-hvaddid 29362 ax-hfvmul 29363 ax-hvmulid 29364 ax-hvmulass 29365 ax-hvdistr1 29366 ax-hvdistr2 29367 ax-hvmul0 29368 ax-hfi 29437 ax-his1 29440 ax-his2 29441 ax-his3 29442 ax-his4 29443 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-er 8481 df-map 8600 df-pm 8601 df-en 8717 df-dom 8718 df-sdom 8719 df-sup 9179 df-inf 9180 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 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 12582 df-q 12688 df-rp 12730 df-xneg 12847 df-xadd 12848 df-xmul 12849 df-icc 13085 df-seq 13720 df-exp 13781 df-cj 14808 df-re 14809 df-im 14810 df-sqrt 14944 df-abs 14945 df-topgen 17152 df-psmet 20587 df-xmet 20588 df-met 20589 df-bl 20590 df-mopn 20591 df-top 22041 df-topon 22058 df-bases 22094 df-lm 22378 df-haus 22464 df-grpo 28851 df-gid 28852 df-ginv 28853 df-gdiv 28854 df-ablo 28903 df-vc 28917 df-nv 28950 df-va 28953 df-ba 28954 df-sm 28955 df-0v 28956 df-vs 28957 df-nmcv 28958 df-ims 28959 df-hnorm 29326 df-hvsub 29329 df-hlim 29330 df-sh 29565 df-ch 29579 df-oc 29610 df-ch0 29611 |
This theorem is referenced by: ho0val 30108 st0 30607 |
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