Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > HSE Home > Th. List > chincl | Structured version Visualization version GIF version |
Description: Closure of Hilbert lattice intersection. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
chincl | ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ∩ 𝐵) ∈ Cℋ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ineq1 4180 | . . 3 ⊢ (𝐴 = if(𝐴 ∈ Cℋ , 𝐴, ℋ) → (𝐴 ∩ 𝐵) = (if(𝐴 ∈ Cℋ , 𝐴, ℋ) ∩ 𝐵)) | |
2 | 1 | eleq1d 2897 | . 2 ⊢ (𝐴 = if(𝐴 ∈ Cℋ , 𝐴, ℋ) → ((𝐴 ∩ 𝐵) ∈ Cℋ ↔ (if(𝐴 ∈ Cℋ , 𝐴, ℋ) ∩ 𝐵) ∈ Cℋ )) |
3 | ineq2 4182 | . . 3 ⊢ (𝐵 = if(𝐵 ∈ Cℋ , 𝐵, ℋ) → (if(𝐴 ∈ Cℋ , 𝐴, ℋ) ∩ 𝐵) = (if(𝐴 ∈ Cℋ , 𝐴, ℋ) ∩ if(𝐵 ∈ Cℋ , 𝐵, ℋ))) | |
4 | 3 | eleq1d 2897 | . 2 ⊢ (𝐵 = if(𝐵 ∈ Cℋ , 𝐵, ℋ) → ((if(𝐴 ∈ Cℋ , 𝐴, ℋ) ∩ 𝐵) ∈ Cℋ ↔ (if(𝐴 ∈ Cℋ , 𝐴, ℋ) ∩ if(𝐵 ∈ Cℋ , 𝐵, ℋ)) ∈ Cℋ )) |
5 | ifchhv 29015 | . . 3 ⊢ if(𝐴 ∈ Cℋ , 𝐴, ℋ) ∈ Cℋ | |
6 | ifchhv 29015 | . . 3 ⊢ if(𝐵 ∈ Cℋ , 𝐵, ℋ) ∈ Cℋ | |
7 | 5, 6 | chincli 29231 | . 2 ⊢ (if(𝐴 ∈ Cℋ , 𝐴, ℋ) ∩ if(𝐵 ∈ Cℋ , 𝐵, ℋ)) ∈ Cℋ |
8 | 2, 4, 7 | dedth2h 4523 | 1 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ∩ 𝐵) ∈ Cℋ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∩ cin 3934 ifcif 4466 ℋchba 28690 Cℋ cch 28700 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-1cn 10589 ax-addcl 10591 ax-hilex 28770 ax-hfvadd 28771 ax-hv0cl 28774 ax-hfvmul 28776 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-map 8402 df-nn 11633 df-hlim 28743 df-sh 28978 df-ch 28992 |
This theorem is referenced by: chabs1 29287 chdmj1 29300 fh1 29389 fh2 29390 cm2j 29391 mdbr2 30067 mdbr3 30068 mdbr4 30069 dmdmd 30071 dmdbr2 30074 dmdbr5 30079 mddmd2 30080 mdsl0 30081 mdsl3 30087 mdsl2i 30093 mdslmd1i 30100 cvp 30146 atomli 30153 atordi 30155 atcvat3i 30167 atcvat4i 30168 mdsymlem1 30174 mdsymlem3 30176 mdsymlem5 30178 mdsymlem6 30179 sumdmdii 30186 dmdbr5ati 30193 |
Copyright terms: Public domain | W3C validator |