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Theorem ispsubcl2N 37961
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 𝐵 = (Base‘𝐾)
pmapsubcl.m 𝑀 = (pmap‘𝐾)
pmapsubcl.c 𝐶 = (PSubCl‘𝐾)
Assertion
Ref Expression
ispsubcl2N (𝐾 ∈ HL → (𝑋𝐶 ↔ ∃𝑦𝐵 𝑋 = (𝑀𝑦)))
Distinct variable groups:   𝑦,𝐵   𝑦,𝐾   𝑦,𝑀   𝑦,𝑋
Allowed substitution hint:   𝐶(𝑦)

Proof of Theorem ispsubcl2N
StepHypRef Expression
1 eqid 2738 . . 3 (Atoms‘𝐾) = (Atoms‘𝐾)
2 eqid 2738 . . 3 (⊥𝑃𝐾) = (⊥𝑃𝐾)
3 pmapsubcl.c . . 3 𝐶 = (PSubCl‘𝐾)
41, 2, 3ispsubclN 37951 . 2 (𝐾 ∈ HL → (𝑋𝐶 ↔ (𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋)))
5 hlop 37376 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ OP)
65adantr 481 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾)) → 𝐾 ∈ OP)
7 hlclat 37372 . . . . . . . . . 10 (𝐾 ∈ HL → 𝐾 ∈ CLat)
87adantr 481 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾)) → 𝐾 ∈ CLat)
91, 2polssatN 37922 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾)) → ((⊥𝑃𝐾)‘𝑋) ⊆ (Atoms‘𝐾))
10 pmapsubcl.b . . . . . . . . . . 11 𝐵 = (Base‘𝐾)
1110, 1atssbase 37304 . . . . . . . . . 10 (Atoms‘𝐾) ⊆ 𝐵
129, 11sstrdi 3933 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾)) → ((⊥𝑃𝐾)‘𝑋) ⊆ 𝐵)
13 eqid 2738 . . . . . . . . . 10 (lub‘𝐾) = (lub‘𝐾)
1410, 13clatlubcl 18221 . . . . . . . . 9 ((𝐾 ∈ CLat ∧ ((⊥𝑃𝐾)‘𝑋) ⊆ 𝐵) → ((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋)) ∈ 𝐵)
158, 12, 14syl2anc 584 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾)) → ((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋)) ∈ 𝐵)
16 eqid 2738 . . . . . . . . 9 (oc‘𝐾) = (oc‘𝐾)
1710, 16opoccl 37208 . . . . . . . 8 ((𝐾 ∈ OP ∧ ((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋)) ∈ 𝐵) → ((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))) ∈ 𝐵)
186, 15, 17syl2anc 584 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾)) → ((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))) ∈ 𝐵)
1918ex 413 . . . . . 6 (𝐾 ∈ HL → (𝑋 ⊆ (Atoms‘𝐾) → ((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))) ∈ 𝐵))
2019adantrd 492 . . . . 5 (𝐾 ∈ HL → ((𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋) → ((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))) ∈ 𝐵))
21 pmapsubcl.m . . . . . . . . . 10 𝑀 = (pmap‘𝐾)
2213, 16, 1, 21, 2polval2N 37920 . . . . . . . . 9 ((𝐾 ∈ HL ∧ ((⊥𝑃𝐾)‘𝑋) ⊆ (Atoms‘𝐾)) → ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋)))))
239, 22syldan 591 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾)) → ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋)))))
2423ex 413 . . . . . . 7 (𝐾 ∈ HL → (𝑋 ⊆ (Atoms‘𝐾) → ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))))))
25 eqeq1 2742 . . . . . . . 8 (((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋 → (((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋)))) ↔ 𝑋 = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))))))
2625biimpcd 248 . . . . . . 7 (((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋)))) → (((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋𝑋 = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))))))
2724, 26syl6 35 . . . . . 6 (𝐾 ∈ HL → (𝑋 ⊆ (Atoms‘𝐾) → (((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋𝑋 = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋)))))))
2827impd 411 . . . . 5 (𝐾 ∈ HL → ((𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋) → 𝑋 = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))))))
2920, 28jcad 513 . . . 4 (𝐾 ∈ HL → ((𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋) → (((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))) ∈ 𝐵𝑋 = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋)))))))
30 fveq2 6774 . . . . 5 (𝑦 = ((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))) → (𝑀𝑦) = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋)))))
3130rspceeqv 3575 . . . 4 ((((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))) ∈ 𝐵𝑋 = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))))) → ∃𝑦𝐵 𝑋 = (𝑀𝑦))
3229, 31syl6 35 . . 3 (𝐾 ∈ HL → ((𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋) → ∃𝑦𝐵 𝑋 = (𝑀𝑦)))
3310, 1, 21pmapssat 37773 . . . . 5 ((𝐾 ∈ HL ∧ 𝑦𝐵) → (𝑀𝑦) ⊆ (Atoms‘𝐾))
3410, 21, 22polpmapN 37927 . . . . 5 ((𝐾 ∈ HL ∧ 𝑦𝐵) → ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘(𝑀𝑦))) = (𝑀𝑦))
35 sseq1 3946 . . . . . . 7 (𝑋 = (𝑀𝑦) → (𝑋 ⊆ (Atoms‘𝐾) ↔ (𝑀𝑦) ⊆ (Atoms‘𝐾)))
36 2fveq3 6779 . . . . . . . 8 (𝑋 = (𝑀𝑦) → ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘(𝑀𝑦))))
37 id 22 . . . . . . . 8 (𝑋 = (𝑀𝑦) → 𝑋 = (𝑀𝑦))
3836, 37eqeq12d 2754 . . . . . . 7 (𝑋 = (𝑀𝑦) → (((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋 ↔ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘(𝑀𝑦))) = (𝑀𝑦)))
3935, 38anbi12d 631 . . . . . 6 (𝑋 = (𝑀𝑦) → ((𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋) ↔ ((𝑀𝑦) ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘(𝑀𝑦))) = (𝑀𝑦))))
4039biimprcd 249 . . . . 5 (((𝑀𝑦) ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘(𝑀𝑦))) = (𝑀𝑦)) → (𝑋 = (𝑀𝑦) → (𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋)))
4133, 34, 40syl2anc 584 . . . 4 ((𝐾 ∈ HL ∧ 𝑦𝐵) → (𝑋 = (𝑀𝑦) → (𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋)))
4241rexlimdva 3213 . . 3 (𝐾 ∈ HL → (∃𝑦𝐵 𝑋 = (𝑀𝑦) → (𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋)))
4332, 42impbid 211 . 2 (𝐾 ∈ HL → ((𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋) ↔ ∃𝑦𝐵 𝑋 = (𝑀𝑦)))
444, 43bitrd 278 1 (𝐾 ∈ HL → (𝑋𝐶 ↔ ∃𝑦𝐵 𝑋 = (𝑀𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106  wrex 3065  wss 3887  cfv 6433  Basecbs 16912  occoc 16970  lubclub 18027  CLatccla 18216  OPcops 37186  Atomscatm 37277  HLchlt 37364  pmapcpmap 37511  𝑃cpolN 37916  PSubClcpscN 37948
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-riotaBAD 36967
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-undef 8089  df-proset 18013  df-poset 18031  df-plt 18048  df-lub 18064  df-glb 18065  df-join 18066  df-meet 18067  df-p0 18143  df-p1 18144  df-lat 18150  df-clat 18217  df-oposet 37190  df-ol 37192  df-oml 37193  df-covers 37280  df-ats 37281  df-atl 37312  df-cvlat 37336  df-hlat 37365  df-psubsp 37517  df-pmap 37518  df-polarityN 37917  df-psubclN 37949
This theorem is referenced by: (None)
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