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Theorem ispsubcl2N 34251
 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 2610 . . 3 (Atoms‘𝐾) = (Atoms‘𝐾)
2 eqid 2610 . . 3 (⊥𝑃𝐾) = (⊥𝑃𝐾)
3 pmapsubcl.c . . 3 𝐶 = (PSubCl‘𝐾)
41, 2, 3ispsubclN 34241 . 2 (𝐾 ∈ HL → (𝑋𝐶 ↔ (𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋)))
5 hlop 33667 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ OP)
65adantr 480 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾)) → 𝐾 ∈ OP)
7 hlclat 33663 . . . . . . . . . 10 (𝐾 ∈ HL → 𝐾 ∈ CLat)
87adantr 480 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾)) → 𝐾 ∈ CLat)
91, 2polssatN 34212 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾)) → ((⊥𝑃𝐾)‘𝑋) ⊆ (Atoms‘𝐾))
10 pmapsubcl.b . . . . . . . . . . 11 𝐵 = (Base‘𝐾)
1110, 1atssbase 33595 . . . . . . . . . 10 (Atoms‘𝐾) ⊆ 𝐵
129, 11syl6ss 3580 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾)) → ((⊥𝑃𝐾)‘𝑋) ⊆ 𝐵)
13 eqid 2610 . . . . . . . . . 10 (lub‘𝐾) = (lub‘𝐾)
1410, 13clatlubcl 16935 . . . . . . . . 9 ((𝐾 ∈ CLat ∧ ((⊥𝑃𝐾)‘𝑋) ⊆ 𝐵) → ((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋)) ∈ 𝐵)
158, 12, 14syl2anc 691 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾)) → ((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋)) ∈ 𝐵)
16 eqid 2610 . . . . . . . . 9 (oc‘𝐾) = (oc‘𝐾)
1710, 16opoccl 33499 . . . . . . . 8 ((𝐾 ∈ OP ∧ ((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋)) ∈ 𝐵) → ((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))) ∈ 𝐵)
186, 15, 17syl2anc 691 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾)) → ((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))) ∈ 𝐵)
1918ex 449 . . . . . 6 (𝐾 ∈ HL → (𝑋 ⊆ (Atoms‘𝐾) → ((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))) ∈ 𝐵))
2019adantrd 483 . . . . 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 34210 . . . . . . . . 9 ((𝐾 ∈ HL ∧ ((⊥𝑃𝐾)‘𝑋) ⊆ (Atoms‘𝐾)) → ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋)))))
239, 22syldan 486 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾)) → ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋)))))
2423ex 449 . . . . . . 7 (𝐾 ∈ HL → (𝑋 ⊆ (Atoms‘𝐾) → ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))))))
25 eqeq1 2614 . . . . . . . 8 (((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋 → (((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋)))) ↔ 𝑋 = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))))))
2625biimpcd 238 . . . . . . 7 (((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋)))) → (((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋𝑋 = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))))))
2724, 26syl6 34 . . . . . 6 (𝐾 ∈ HL → (𝑋 ⊆ (Atoms‘𝐾) → (((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋𝑋 = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋)))))))
2827impd 446 . . . . 5 (𝐾 ∈ HL → ((𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋) → 𝑋 = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))))))
2920, 28jcad 554 . . . 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 6103 . . . . . 6 (𝑦 = ((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))) → (𝑀𝑦) = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋)))))
3130eqeq2d 2620 . . . . 5 (𝑦 = ((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))) → (𝑋 = (𝑀𝑦) ↔ 𝑋 = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))))))
3231rspcev 3282 . . . 4 ((((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))) ∈ 𝐵𝑋 = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))))) → ∃𝑦𝐵 𝑋 = (𝑀𝑦))
3329, 32syl6 34 . . 3 (𝐾 ∈ HL → ((𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋) → ∃𝑦𝐵 𝑋 = (𝑀𝑦)))
3410, 1, 21pmapssat 34063 . . . . 5 ((𝐾 ∈ HL ∧ 𝑦𝐵) → (𝑀𝑦) ⊆ (Atoms‘𝐾))
3510, 21, 22polpmapN 34217 . . . . 5 ((𝐾 ∈ HL ∧ 𝑦𝐵) → ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘(𝑀𝑦))) = (𝑀𝑦))
36 sseq1 3589 . . . . . . 7 (𝑋 = (𝑀𝑦) → (𝑋 ⊆ (Atoms‘𝐾) ↔ (𝑀𝑦) ⊆ (Atoms‘𝐾)))
37 fveq2 6103 . . . . . . . . 9 (𝑋 = (𝑀𝑦) → ((⊥𝑃𝐾)‘𝑋) = ((⊥𝑃𝐾)‘(𝑀𝑦)))
3837fveq2d 6107 . . . . . . . 8 (𝑋 = (𝑀𝑦) → ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘(𝑀𝑦))))
39 id 22 . . . . . . . 8 (𝑋 = (𝑀𝑦) → 𝑋 = (𝑀𝑦))
4038, 39eqeq12d 2625 . . . . . . 7 (𝑋 = (𝑀𝑦) → (((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋 ↔ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘(𝑀𝑦))) = (𝑀𝑦)))
4136, 40anbi12d 743 . . . . . 6 (𝑋 = (𝑀𝑦) → ((𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋) ↔ ((𝑀𝑦) ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘(𝑀𝑦))) = (𝑀𝑦))))
4241biimprcd 239 . . . . 5 (((𝑀𝑦) ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘(𝑀𝑦))) = (𝑀𝑦)) → (𝑋 = (𝑀𝑦) → (𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋)))
4334, 35, 42syl2anc 691 . . . 4 ((𝐾 ∈ HL ∧ 𝑦𝐵) → (𝑋 = (𝑀𝑦) → (𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋)))
4443rexlimdva 3013 . . 3 (𝐾 ∈ HL → (∃𝑦𝐵 𝑋 = (𝑀𝑦) → (𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋)))
4533, 44impbid 201 . 2 (𝐾 ∈ HL → ((𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋) ↔ ∃𝑦𝐵 𝑋 = (𝑀𝑦)))
464, 45bitrd 267 1 (𝐾 ∈ HL → (𝑋𝐶 ↔ ∃𝑦𝐵 𝑋 = (𝑀𝑦)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897   ⊆ wss 3540  ‘cfv 5804  Basecbs 15695  occoc 15776  lubclub 16765  CLatccla 16930  OPcops 33477  Atomscatm 33568  HLchlt 33655  pmapcpmap 33801  ⊥𝑃cpolN 34206  PSubClcpscN 34238 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-riotaBAD 33257 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-undef 7286  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-p1 16863  df-lat 16869  df-clat 16931  df-oposet 33481  df-ol 33483  df-oml 33484  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656  df-psubsp 33807  df-pmap 33808  df-polarityN 34207  df-psubclN 34239 This theorem is referenced by: (None)
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