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Theorem ispsubcl2N 39121
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 2730 . . 3 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
2 eqid 2730 . . 3 (βŠ₯π‘ƒβ€˜πΎ) = (βŠ₯π‘ƒβ€˜πΎ)
3 pmapsubcl.c . . 3 𝐢 = (PSubClβ€˜πΎ)
41, 2, 3ispsubclN 39111 . 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 38535 . . . . . . . . 9 (𝐾 ∈ HL β†’ 𝐾 ∈ OP)
65adantr 479 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋 βŠ† (Atomsβ€˜πΎ)) β†’ 𝐾 ∈ OP)
7 hlclat 38531 . . . . . . . . . 10 (𝐾 ∈ HL β†’ 𝐾 ∈ CLat)
87adantr 479 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑋 βŠ† (Atomsβ€˜πΎ)) β†’ 𝐾 ∈ CLat)
91, 2polssatN 39082 . . . . . . . . . 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 38463 . . . . . . . . . 10 (Atomsβ€˜πΎ) βŠ† 𝐡
129, 11sstrdi 3993 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑋 βŠ† (Atomsβ€˜πΎ)) β†’ ((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹) βŠ† 𝐡)
13 eqid 2730 . . . . . . . . . 10 (lubβ€˜πΎ) = (lubβ€˜πΎ)
1410, 13clatlubcl 18460 . . . . . . . . 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 582 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋 βŠ† (Atomsβ€˜πΎ)) β†’ ((lubβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹)) ∈ 𝐡)
16 eqid 2730 . . . . . . . . 9 (ocβ€˜πΎ) = (ocβ€˜πΎ)
1710, 16opoccl 38367 . . . . . . . 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 582 . . . . . . 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 411 . . . . . 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 490 . . . . 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 39080 . . . . . . . . 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 589 . . . . . . . 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 411 . . . . . . 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 2734 . . . . . . . 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 409 . . . . 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 511 . . . 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 6890 . . . . 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 3632 . . . 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 38933 . . . . 5 ((𝐾 ∈ HL ∧ 𝑦 ∈ 𝐡) β†’ (π‘€β€˜π‘¦) βŠ† (Atomsβ€˜πΎ))
3410, 21, 22polpmapN 39087 . . . . 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 4006 . . . . . . 7 (𝑋 = (π‘€β€˜π‘¦) β†’ (𝑋 βŠ† (Atomsβ€˜πΎ) ↔ (π‘€β€˜π‘¦) βŠ† (Atomsβ€˜πΎ)))
36 2fveq3 6895 . . . . . . . 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 2746 . . . . . . 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 629 . . . . . 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 582 . . . 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 3153 . . 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 278 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 394   = wceq 1539   ∈ wcel 2104  βˆƒwrex 3068   βŠ† wss 3947  β€˜cfv 6542  Basecbs 17148  occoc 17209  lubclub 18266  CLatccla 18455  OPcops 38345  Atomscatm 38436  HLchlt 38523  pmapcpmap 38671  βŠ₯𝑃cpolN 39076  PSubClcpscN 39108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-proset 18252  df-poset 18270  df-plt 18287  df-lub 18303  df-glb 18304  df-join 18305  df-meet 18306  df-p0 18382  df-p1 18383  df-lat 18389  df-clat 18456  df-oposet 38349  df-ol 38351  df-oml 38352  df-covers 38439  df-ats 38440  df-atl 38471  df-cvlat 38495  df-hlat 38524  df-psubsp 38677  df-pmap 38678  df-polarityN 39077  df-psubclN 39109
This theorem is referenced by: (None)
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