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Theorem psubclinN 39332
Description: The intersection of two closed subspaces is closed. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
Hypothesis
Ref Expression
psubclin.c 𝐢 = (PSubClβ€˜πΎ)
Assertion
Ref Expression
psubclinN ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ π‘Œ ∈ 𝐢) β†’ (𝑋 ∩ π‘Œ) ∈ 𝐢)

Proof of Theorem psubclinN
StepHypRef Expression
1 simp1 1133 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ π‘Œ ∈ 𝐢) β†’ 𝐾 ∈ HL)
2 hlclat 38741 . . . . . 6 (𝐾 ∈ HL β†’ 𝐾 ∈ CLat)
323ad2ant1 1130 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ π‘Œ ∈ 𝐢) β†’ 𝐾 ∈ CLat)
4 eqid 2726 . . . . . . . 8 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
5 psubclin.c . . . . . . . 8 𝐢 = (PSubClβ€˜πΎ)
64, 5psubclssatN 39325 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢) β†’ 𝑋 βŠ† (Atomsβ€˜πΎ))
763adant3 1129 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ π‘Œ ∈ 𝐢) β†’ 𝑋 βŠ† (Atomsβ€˜πΎ))
8 eqid 2726 . . . . . . 7 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
98, 4atssbase 38673 . . . . . 6 (Atomsβ€˜πΎ) βŠ† (Baseβ€˜πΎ)
107, 9sstrdi 3989 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ π‘Œ ∈ 𝐢) β†’ 𝑋 βŠ† (Baseβ€˜πΎ))
11 eqid 2726 . . . . . 6 (lubβ€˜πΎ) = (lubβ€˜πΎ)
128, 11clatlubcl 18468 . . . . 5 ((𝐾 ∈ CLat ∧ 𝑋 βŠ† (Baseβ€˜πΎ)) β†’ ((lubβ€˜πΎ)β€˜π‘‹) ∈ (Baseβ€˜πΎ))
133, 10, 12syl2anc 583 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ π‘Œ ∈ 𝐢) β†’ ((lubβ€˜πΎ)β€˜π‘‹) ∈ (Baseβ€˜πΎ))
144, 5psubclssatN 39325 . . . . . . 7 ((𝐾 ∈ HL ∧ π‘Œ ∈ 𝐢) β†’ π‘Œ βŠ† (Atomsβ€˜πΎ))
15143adant2 1128 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ π‘Œ ∈ 𝐢) β†’ π‘Œ βŠ† (Atomsβ€˜πΎ))
1615, 9sstrdi 3989 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ π‘Œ ∈ 𝐢) β†’ π‘Œ βŠ† (Baseβ€˜πΎ))
178, 11clatlubcl 18468 . . . . 5 ((𝐾 ∈ CLat ∧ π‘Œ βŠ† (Baseβ€˜πΎ)) β†’ ((lubβ€˜πΎ)β€˜π‘Œ) ∈ (Baseβ€˜πΎ))
183, 16, 17syl2anc 583 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ π‘Œ ∈ 𝐢) β†’ ((lubβ€˜πΎ)β€˜π‘Œ) ∈ (Baseβ€˜πΎ))
19 eqid 2726 . . . . 5 (meetβ€˜πΎ) = (meetβ€˜πΎ)
20 eqid 2726 . . . . 5 (pmapβ€˜πΎ) = (pmapβ€˜πΎ)
218, 19, 4, 20pmapmeet 39157 . . . 4 ((𝐾 ∈ HL ∧ ((lubβ€˜πΎ)β€˜π‘‹) ∈ (Baseβ€˜πΎ) ∧ ((lubβ€˜πΎ)β€˜π‘Œ) ∈ (Baseβ€˜πΎ)) β†’ ((pmapβ€˜πΎ)β€˜(((lubβ€˜πΎ)β€˜π‘‹)(meetβ€˜πΎ)((lubβ€˜πΎ)β€˜π‘Œ))) = (((pmapβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‹)) ∩ ((pmapβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘Œ))))
221, 13, 18, 21syl3anc 1368 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ π‘Œ ∈ 𝐢) β†’ ((pmapβ€˜πΎ)β€˜(((lubβ€˜πΎ)β€˜π‘‹)(meetβ€˜πΎ)((lubβ€˜πΎ)β€˜π‘Œ))) = (((pmapβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‹)) ∩ ((pmapβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘Œ))))
2311, 20, 5pmapidclN 39326 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢) β†’ ((pmapβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‹)) = 𝑋)
24233adant3 1129 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ π‘Œ ∈ 𝐢) β†’ ((pmapβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‹)) = 𝑋)
2511, 20, 5pmapidclN 39326 . . . . 5 ((𝐾 ∈ HL ∧ π‘Œ ∈ 𝐢) β†’ ((pmapβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘Œ)) = π‘Œ)
26253adant2 1128 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ π‘Œ ∈ 𝐢) β†’ ((pmapβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘Œ)) = π‘Œ)
2724, 26ineq12d 4208 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ π‘Œ ∈ 𝐢) β†’ (((pmapβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‹)) ∩ ((pmapβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘Œ))) = (𝑋 ∩ π‘Œ))
2822, 27eqtrd 2766 . 2 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ π‘Œ ∈ 𝐢) β†’ ((pmapβ€˜πΎ)β€˜(((lubβ€˜πΎ)β€˜π‘‹)(meetβ€˜πΎ)((lubβ€˜πΎ)β€˜π‘Œ))) = (𝑋 ∩ π‘Œ))
29 hllat 38746 . . . . 5 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
30293ad2ant1 1130 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ π‘Œ ∈ 𝐢) β†’ 𝐾 ∈ Lat)
318, 19latmcl 18405 . . . 4 ((𝐾 ∈ Lat ∧ ((lubβ€˜πΎ)β€˜π‘‹) ∈ (Baseβ€˜πΎ) ∧ ((lubβ€˜πΎ)β€˜π‘Œ) ∈ (Baseβ€˜πΎ)) β†’ (((lubβ€˜πΎ)β€˜π‘‹)(meetβ€˜πΎ)((lubβ€˜πΎ)β€˜π‘Œ)) ∈ (Baseβ€˜πΎ))
3230, 13, 18, 31syl3anc 1368 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ π‘Œ ∈ 𝐢) β†’ (((lubβ€˜πΎ)β€˜π‘‹)(meetβ€˜πΎ)((lubβ€˜πΎ)β€˜π‘Œ)) ∈ (Baseβ€˜πΎ))
338, 20, 5pmapsubclN 39330 . . 3 ((𝐾 ∈ HL ∧ (((lubβ€˜πΎ)β€˜π‘‹)(meetβ€˜πΎ)((lubβ€˜πΎ)β€˜π‘Œ)) ∈ (Baseβ€˜πΎ)) β†’ ((pmapβ€˜πΎ)β€˜(((lubβ€˜πΎ)β€˜π‘‹)(meetβ€˜πΎ)((lubβ€˜πΎ)β€˜π‘Œ))) ∈ 𝐢)
341, 32, 33syl2anc 583 . 2 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ π‘Œ ∈ 𝐢) β†’ ((pmapβ€˜πΎ)β€˜(((lubβ€˜πΎ)β€˜π‘‹)(meetβ€˜πΎ)((lubβ€˜πΎ)β€˜π‘Œ))) ∈ 𝐢)
3528, 34eqeltrrd 2828 1 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ π‘Œ ∈ 𝐢) β†’ (𝑋 ∩ π‘Œ) ∈ 𝐢)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ∩ cin 3942   βŠ† wss 3943  β€˜cfv 6537  (class class class)co 7405  Basecbs 17153  lubclub 18274  meetcmee 18277  Latclat 18396  CLatccla 18463  Atomscatm 38646  HLchlt 38733  pmapcpmap 38881  PSubClcpscN 39318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-proset 18260  df-poset 18278  df-plt 18295  df-lub 18311  df-glb 18312  df-join 18313  df-meet 18314  df-p0 18390  df-p1 18391  df-lat 18397  df-clat 18464  df-oposet 38559  df-ol 38561  df-oml 38562  df-covers 38649  df-ats 38650  df-atl 38681  df-cvlat 38705  df-hlat 38734  df-pmap 38888  df-polarityN 39287  df-psubclN 39319
This theorem is referenced by:  osumcllem9N  39348
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