Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > HSE Home > Th. List > chocin | Structured version Visualization version GIF version |
Description: Intersection of a closed subspace and its orthocomplement. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 13-Jun-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
chocin | ⊢ (𝐴 ∈ Cℋ → (𝐴 ∩ (⊥‘𝐴)) = 0ℋ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . 4 ⊢ (𝐴 = if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) → 𝐴 = if(𝐴 ∈ Cℋ , 𝐴, 0ℋ)) | |
2 | fveq2 6686 | . . . 4 ⊢ (𝐴 = if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) → (⊥‘𝐴) = (⊥‘if(𝐴 ∈ Cℋ , 𝐴, 0ℋ))) | |
3 | 1, 2 | ineq12d 4114 | . . 3 ⊢ (𝐴 = if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) → (𝐴 ∩ (⊥‘𝐴)) = (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∩ (⊥‘if(𝐴 ∈ Cℋ , 𝐴, 0ℋ)))) |
4 | 3 | eqeq1d 2741 | . 2 ⊢ (𝐴 = if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) → ((𝐴 ∩ (⊥‘𝐴)) = 0ℋ ↔ (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∩ (⊥‘if(𝐴 ∈ Cℋ , 𝐴, 0ℋ))) = 0ℋ)) |
5 | h0elch 29202 | . . . 4 ⊢ 0ℋ ∈ Cℋ | |
6 | 5 | elimel 4493 | . . 3 ⊢ if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∈ Cℋ |
7 | 6 | chocini 29401 | . 2 ⊢ (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∩ (⊥‘if(𝐴 ∈ Cℋ , 𝐴, 0ℋ))) = 0ℋ |
8 | 4, 7 | dedth 4482 | 1 ⊢ (𝐴 ∈ Cℋ → (𝐴 ∩ (⊥‘𝐴)) = 0ℋ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ∩ cin 3852 ifcif 4424 ‘cfv 6349 Cℋ cch 28876 ⊥cort 28877 0ℋc0h 28882 |
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 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-cnex 10683 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 ax-pre-mulgt0 10704 ax-pre-sup 10705 ax-addf 10706 ax-mulf 10707 ax-hilex 28946 ax-hfvadd 28947 ax-hvcom 28948 ax-hvass 28949 ax-hv0cl 28950 ax-hvaddid 28951 ax-hfvmul 28952 ax-hvmulid 28953 ax-hvmulass 28954 ax-hvdistr1 28955 ax-hvdistr2 28956 ax-hvmul0 28957 ax-hfi 29026 ax-his1 29029 ax-his2 29030 ax-his3 29031 ax-his4 29032 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6185 df-on 6186 df-lim 6187 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-om 7612 df-1st 7726 df-2nd 7727 df-wrecs 7988 df-recs 8049 df-rdg 8087 df-er 8332 df-map 8451 df-pm 8452 df-en 8568 df-dom 8569 df-sdom 8570 df-sup 8991 df-inf 8992 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 df-div 11388 df-nn 11729 df-2 11791 df-3 11792 df-4 11793 df-n0 11989 df-z 12075 df-uz 12337 df-q 12443 df-rp 12485 df-xneg 12602 df-xadd 12603 df-xmul 12604 df-icc 12840 df-seq 13473 df-exp 13534 df-cj 14560 df-re 14561 df-im 14562 df-sqrt 14696 df-abs 14697 df-topgen 16832 df-psmet 20221 df-xmet 20222 df-met 20223 df-bl 20224 df-mopn 20225 df-top 21657 df-topon 21674 df-bases 21709 df-lm 21992 df-haus 22078 df-grpo 28440 df-gid 28441 df-ginv 28442 df-gdiv 28443 df-ablo 28492 df-vc 28506 df-nv 28539 df-va 28542 df-ba 28543 df-sm 28544 df-0v 28545 df-vs 28546 df-nmcv 28547 df-ims 28548 df-hnorm 28915 df-hvsub 28918 df-hlim 28919 df-sh 29154 df-ch 29168 df-oc 29199 df-ch0 29200 |
This theorem is referenced by: chssoc 29443 fh1 29565 fh2 29566 |
Copyright terms: Public domain | W3C validator |