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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 6891 | . . . 4 ⊢ (𝐴 = if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) → (⊥‘𝐴) = (⊥‘if(𝐴 ∈ Cℋ , 𝐴, 0ℋ))) | |
3 | 1, 2 | ineq12d 4213 | . . 3 ⊢ (𝐴 = if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) → (𝐴 ∩ (⊥‘𝐴)) = (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∩ (⊥‘if(𝐴 ∈ Cℋ , 𝐴, 0ℋ)))) |
4 | 3 | eqeq1d 2733 | . 2 ⊢ (𝐴 = if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) → ((𝐴 ∩ (⊥‘𝐴)) = 0ℋ ↔ (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∩ (⊥‘if(𝐴 ∈ Cℋ , 𝐴, 0ℋ))) = 0ℋ)) |
5 | h0elch 30940 | . . . 4 ⊢ 0ℋ ∈ Cℋ | |
6 | 5 | elimel 4597 | . . 3 ⊢ if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∈ Cℋ |
7 | 6 | chocini 31139 | . 2 ⊢ (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∩ (⊥‘if(𝐴 ∈ Cℋ , 𝐴, 0ℋ))) = 0ℋ |
8 | 4, 7 | dedth 4586 | 1 ⊢ (𝐴 ∈ Cℋ → (𝐴 ∩ (⊥‘𝐴)) = 0ℋ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 ∩ cin 3947 ifcif 4528 ‘cfv 6543 Cℋ cch 30614 ⊥cort 30615 0ℋc0h 30620 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 ax-addf 11195 ax-mulf 11196 ax-hilex 30684 ax-hfvadd 30685 ax-hvcom 30686 ax-hvass 30687 ax-hv0cl 30688 ax-hvaddid 30689 ax-hfvmul 30690 ax-hvmulid 30691 ax-hvmulass 30692 ax-hvdistr1 30693 ax-hvdistr2 30694 ax-hvmul0 30695 ax-hfi 30764 ax-his1 30767 ax-his2 30768 ax-his3 30769 ax-his4 30770 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-map 8828 df-pm 8829 df-en 8946 df-dom 8947 df-sdom 8948 df-sup 9443 df-inf 9444 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-n0 12480 df-z 12566 df-uz 12830 df-q 12940 df-rp 12982 df-xneg 13099 df-xadd 13100 df-xmul 13101 df-icc 13338 df-seq 13974 df-exp 14035 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-topgen 17396 df-psmet 21224 df-xmet 21225 df-met 21226 df-bl 21227 df-mopn 21228 df-top 22715 df-topon 22732 df-bases 22768 df-lm 23052 df-haus 23138 df-grpo 30178 df-gid 30179 df-ginv 30180 df-gdiv 30181 df-ablo 30230 df-vc 30244 df-nv 30277 df-va 30280 df-ba 30281 df-sm 30282 df-0v 30283 df-vs 30284 df-nmcv 30285 df-ims 30286 df-hnorm 30653 df-hvsub 30656 df-hlim 30657 df-sh 30892 df-ch 30906 df-oc 30937 df-ch0 30938 |
This theorem is referenced by: chssoc 31181 fh1 31303 fh2 31304 |
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