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Theorem ispsubcl2N 38818
Description: Alternate predicate for "is a closed projective subspace". Remark in [Holland95] p. 223. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
pmapsubcl.b 𝐡 = (Baseβ€˜πΎ)
pmapsubcl.m 𝑀 = (pmapβ€˜πΎ)
pmapsubcl.c 𝐢 = (PSubClβ€˜πΎ)
Assertion
Ref Expression
ispsubcl2N (𝐾 ∈ HL β†’ (𝑋 ∈ 𝐢 ↔ βˆƒπ‘¦ ∈ 𝐡 𝑋 = (π‘€β€˜π‘¦)))
Distinct variable groups:   𝑦,𝐡   𝑦,𝐾   𝑦,𝑀   𝑦,𝑋
Allowed substitution hint:   𝐢(𝑦)

Proof of Theorem ispsubcl2N
StepHypRef Expression
1 eqid 2733 . . 3 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
2 eqid 2733 . . 3 (βŠ₯π‘ƒβ€˜πΎ) = (βŠ₯π‘ƒβ€˜πΎ)
3 pmapsubcl.c . . 3 𝐢 = (PSubClβ€˜πΎ)
41, 2, 3ispsubclN 38808 . 2 (𝐾 ∈ HL β†’ (𝑋 ∈ 𝐢 ↔ (𝑋 βŠ† (Atomsβ€˜πΎ) ∧ ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹)) = 𝑋)))
5 hlop 38232 . . . . . . . . 9 (𝐾 ∈ HL β†’ 𝐾 ∈ OP)
65adantr 482 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋 βŠ† (Atomsβ€˜πΎ)) β†’ 𝐾 ∈ OP)
7 hlclat 38228 . . . . . . . . . 10 (𝐾 ∈ HL β†’ 𝐾 ∈ CLat)
87adantr 482 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑋 βŠ† (Atomsβ€˜πΎ)) β†’ 𝐾 ∈ CLat)
91, 2polssatN 38779 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑋 βŠ† (Atomsβ€˜πΎ)) β†’ ((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹) βŠ† (Atomsβ€˜πΎ))
10 pmapsubcl.b . . . . . . . . . . 11 𝐡 = (Baseβ€˜πΎ)
1110, 1atssbase 38160 . . . . . . . . . 10 (Atomsβ€˜πΎ) βŠ† 𝐡
129, 11sstrdi 3995 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑋 βŠ† (Atomsβ€˜πΎ)) β†’ ((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹) βŠ† 𝐡)
13 eqid 2733 . . . . . . . . . 10 (lubβ€˜πΎ) = (lubβ€˜πΎ)
1410, 13clatlubcl 18456 . . . . . . . . 9 ((𝐾 ∈ CLat ∧ ((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹) βŠ† 𝐡) β†’ ((lubβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹)) ∈ 𝐡)
158, 12, 14syl2anc 585 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋 βŠ† (Atomsβ€˜πΎ)) β†’ ((lubβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹)) ∈ 𝐡)
16 eqid 2733 . . . . . . . . 9 (ocβ€˜πΎ) = (ocβ€˜πΎ)
1710, 16opoccl 38064 . . . . . . . 8 ((𝐾 ∈ OP ∧ ((lubβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹)) ∈ 𝐡) β†’ ((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹))) ∈ 𝐡)
186, 15, 17syl2anc 585 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋 βŠ† (Atomsβ€˜πΎ)) β†’ ((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹))) ∈ 𝐡)
1918ex 414 . . . . . 6 (𝐾 ∈ HL β†’ (𝑋 βŠ† (Atomsβ€˜πΎ) β†’ ((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹))) ∈ 𝐡))
2019adantrd 493 . . . . 5 (𝐾 ∈ HL β†’ ((𝑋 βŠ† (Atomsβ€˜πΎ) ∧ ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹)) = 𝑋) β†’ ((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹))) ∈ 𝐡))
21 pmapsubcl.m . . . . . . . . . 10 𝑀 = (pmapβ€˜πΎ)
2213, 16, 1, 21, 2polval2N 38777 . . . . . . . . 9 ((𝐾 ∈ HL ∧ ((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹) βŠ† (Atomsβ€˜πΎ)) β†’ ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹)) = (π‘€β€˜((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹)))))
239, 22syldan 592 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋 βŠ† (Atomsβ€˜πΎ)) β†’ ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹)) = (π‘€β€˜((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹)))))
2423ex 414 . . . . . . 7 (𝐾 ∈ HL β†’ (𝑋 βŠ† (Atomsβ€˜πΎ) β†’ ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹)) = (π‘€β€˜((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹))))))
25 eqeq1 2737 . . . . . . . 8 (((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹)) = 𝑋 β†’ (((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹)) = (π‘€β€˜((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹)))) ↔ 𝑋 = (π‘€β€˜((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹))))))
2625biimpcd 248 . . . . . . 7 (((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹)) = (π‘€β€˜((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹)))) β†’ (((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹)) = 𝑋 β†’ 𝑋 = (π‘€β€˜((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹))))))
2724, 26syl6 35 . . . . . 6 (𝐾 ∈ HL β†’ (𝑋 βŠ† (Atomsβ€˜πΎ) β†’ (((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹)) = 𝑋 β†’ 𝑋 = (π‘€β€˜((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹)))))))
2827impd 412 . . . . 5 (𝐾 ∈ HL β†’ ((𝑋 βŠ† (Atomsβ€˜πΎ) ∧ ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹)) = 𝑋) β†’ 𝑋 = (π‘€β€˜((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹))))))
2920, 28jcad 514 . . . 4 (𝐾 ∈ HL β†’ ((𝑋 βŠ† (Atomsβ€˜πΎ) ∧ ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹)) = 𝑋) β†’ (((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹))) ∈ 𝐡 ∧ 𝑋 = (π‘€β€˜((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹)))))))
30 fveq2 6892 . . . . 5 (𝑦 = ((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹))) β†’ (π‘€β€˜π‘¦) = (π‘€β€˜((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹)))))
3130rspceeqv 3634 . . . 4 ((((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹))) ∈ 𝐡 ∧ 𝑋 = (π‘€β€˜((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹))))) β†’ βˆƒπ‘¦ ∈ 𝐡 𝑋 = (π‘€β€˜π‘¦))
3229, 31syl6 35 . . 3 (𝐾 ∈ HL β†’ ((𝑋 βŠ† (Atomsβ€˜πΎ) ∧ ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹)) = 𝑋) β†’ βˆƒπ‘¦ ∈ 𝐡 𝑋 = (π‘€β€˜π‘¦)))
3310, 1, 21pmapssat 38630 . . . . 5 ((𝐾 ∈ HL ∧ 𝑦 ∈ 𝐡) β†’ (π‘€β€˜π‘¦) βŠ† (Atomsβ€˜πΎ))
3410, 21, 22polpmapN 38784 . . . . 5 ((𝐾 ∈ HL ∧ 𝑦 ∈ 𝐡) β†’ ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜(π‘€β€˜π‘¦))) = (π‘€β€˜π‘¦))
35 sseq1 4008 . . . . . . 7 (𝑋 = (π‘€β€˜π‘¦) β†’ (𝑋 βŠ† (Atomsβ€˜πΎ) ↔ (π‘€β€˜π‘¦) βŠ† (Atomsβ€˜πΎ)))
36 2fveq3 6897 . . . . . . . 8 (𝑋 = (π‘€β€˜π‘¦) β†’ ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹)) = ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜(π‘€β€˜π‘¦))))
37 id 22 . . . . . . . 8 (𝑋 = (π‘€β€˜π‘¦) β†’ 𝑋 = (π‘€β€˜π‘¦))
3836, 37eqeq12d 2749 . . . . . . 7 (𝑋 = (π‘€β€˜π‘¦) β†’ (((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹)) = 𝑋 ↔ ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜(π‘€β€˜π‘¦))) = (π‘€β€˜π‘¦)))
3935, 38anbi12d 632 . . . . . 6 (𝑋 = (π‘€β€˜π‘¦) β†’ ((𝑋 βŠ† (Atomsβ€˜πΎ) ∧ ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹)) = 𝑋) ↔ ((π‘€β€˜π‘¦) βŠ† (Atomsβ€˜πΎ) ∧ ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜(π‘€β€˜π‘¦))) = (π‘€β€˜π‘¦))))
4039biimprcd 249 . . . . 5 (((π‘€β€˜π‘¦) βŠ† (Atomsβ€˜πΎ) ∧ ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜(π‘€β€˜π‘¦))) = (π‘€β€˜π‘¦)) β†’ (𝑋 = (π‘€β€˜π‘¦) β†’ (𝑋 βŠ† (Atomsβ€˜πΎ) ∧ ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹)) = 𝑋)))
4133, 34, 40syl2anc 585 . . . 4 ((𝐾 ∈ HL ∧ 𝑦 ∈ 𝐡) β†’ (𝑋 = (π‘€β€˜π‘¦) β†’ (𝑋 βŠ† (Atomsβ€˜πΎ) ∧ ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹)) = 𝑋)))
4241rexlimdva 3156 . . 3 (𝐾 ∈ HL β†’ (βˆƒπ‘¦ ∈ 𝐡 𝑋 = (π‘€β€˜π‘¦) β†’ (𝑋 βŠ† (Atomsβ€˜πΎ) ∧ ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹)) = 𝑋)))
4332, 42impbid 211 . 2 (𝐾 ∈ HL β†’ ((𝑋 βŠ† (Atomsβ€˜πΎ) ∧ ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹)) = 𝑋) ↔ βˆƒπ‘¦ ∈ 𝐡 𝑋 = (π‘€β€˜π‘¦)))
444, 43bitrd 279 1 (𝐾 ∈ HL β†’ (𝑋 ∈ 𝐢 ↔ βˆƒπ‘¦ ∈ 𝐡 𝑋 = (π‘€β€˜π‘¦)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3071   βŠ† wss 3949  β€˜cfv 6544  Basecbs 17144  occoc 17205  lubclub 18262  CLatccla 18451  OPcops 38042  Atomscatm 38133  HLchlt 38220  pmapcpmap 38368  βŠ₯𝑃cpolN 38773  PSubClcpscN 38805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-proset 18248  df-poset 18266  df-plt 18283  df-lub 18299  df-glb 18300  df-join 18301  df-meet 18302  df-p0 18378  df-p1 18379  df-lat 18385  df-clat 18452  df-oposet 38046  df-ol 38048  df-oml 38049  df-covers 38136  df-ats 38137  df-atl 38168  df-cvlat 38192  df-hlat 38221  df-psubsp 38374  df-pmap 38375  df-polarityN 38774  df-psubclN 38806
This theorem is referenced by: (None)
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