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| 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 4212 | . . 3 ⊢ (𝐴 = if(𝐴 ∈ Cℋ , 𝐴, ℋ) → (𝐴 ∩ 𝐵) = (if(𝐴 ∈ Cℋ , 𝐴, ℋ) ∩ 𝐵)) | |
| 2 | 1 | eleq1d 2825 | . 2 ⊢ (𝐴 = if(𝐴 ∈ Cℋ , 𝐴, ℋ) → ((𝐴 ∩ 𝐵) ∈ Cℋ ↔ (if(𝐴 ∈ Cℋ , 𝐴, ℋ) ∩ 𝐵) ∈ Cℋ )) | 
| 3 | ineq2 4213 | . . 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 2825 | . 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 31264 | . . 3 ⊢ if(𝐴 ∈ Cℋ , 𝐴, ℋ) ∈ Cℋ | |
| 6 | ifchhv 31264 | . . 3 ⊢ if(𝐵 ∈ Cℋ , 𝐵, ℋ) ∈ Cℋ | |
| 7 | 5, 6 | chincli 31480 | . 2 ⊢ (if(𝐴 ∈ Cℋ , 𝐴, ℋ) ∩ if(𝐵 ∈ Cℋ , 𝐵, ℋ)) ∈ Cℋ | 
| 8 | 2, 4, 7 | dedth2h 4584 | 1 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ∩ 𝐵) ∈ Cℋ ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∩ cin 3949 ifcif 4524 ℋchba 30939 Cℋ cch 30949 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-1cn 11214 ax-addcl 11216 ax-hilex 31019 ax-hfvadd 31020 ax-hv0cl 31023 ax-hfvmul 31025 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-map 8869 df-nn 12268 df-hlim 30992 df-sh 31227 df-ch 31241 | 
| This theorem is referenced by: chabs1 31536 chdmj1 31549 fh1 31638 fh2 31639 cm2j 31640 mdbr2 32316 mdbr3 32317 mdbr4 32318 dmdmd 32320 dmdbr2 32323 dmdbr5 32328 mddmd2 32329 mdsl0 32330 mdsl3 32336 mdsl2i 32342 mdslmd1i 32349 cvp 32395 atomli 32402 atordi 32404 atcvat3i 32416 atcvat4i 32417 mdsymlem1 32423 mdsymlem3 32425 mdsymlem5 32427 mdsymlem6 32428 sumdmdii 32435 dmdbr5ati 32442 | 
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