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Theorem ispsubcl2N 34047
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 2609 . . 3 (Atoms‘𝐾) = (Atoms‘𝐾)
2 eqid 2609 . . 3 (⊥𝑃𝐾) = (⊥𝑃𝐾)
3 pmapsubcl.c . . 3 𝐶 = (PSubCl‘𝐾)
41, 2, 3ispsubclN 34037 . 2 (𝐾 ∈ HL → (𝑋𝐶 ↔ (𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋)))
5 hlop 33463 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ OP)
65adantr 479 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾)) → 𝐾 ∈ OP)
7 hlclat 33459 . . . . . . . . . 10 (𝐾 ∈ HL → 𝐾 ∈ CLat)
87adantr 479 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾)) → 𝐾 ∈ CLat)
91, 2polssatN 34008 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾)) → ((⊥𝑃𝐾)‘𝑋) ⊆ (Atoms‘𝐾))
10 pmapsubcl.b . . . . . . . . . . 11 𝐵 = (Base‘𝐾)
1110, 1atssbase 33391 . . . . . . . . . 10 (Atoms‘𝐾) ⊆ 𝐵
129, 11syl6ss 3579 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾)) → ((⊥𝑃𝐾)‘𝑋) ⊆ 𝐵)
13 eqid 2609 . . . . . . . . . 10 (lub‘𝐾) = (lub‘𝐾)
1410, 13clatlubcl 16881 . . . . . . . . 9 ((𝐾 ∈ CLat ∧ ((⊥𝑃𝐾)‘𝑋) ⊆ 𝐵) → ((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋)) ∈ 𝐵)
158, 12, 14syl2anc 690 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾)) → ((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋)) ∈ 𝐵)
16 eqid 2609 . . . . . . . . 9 (oc‘𝐾) = (oc‘𝐾)
1710, 16opoccl 33295 . . . . . . . 8 ((𝐾 ∈ OP ∧ ((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋)) ∈ 𝐵) → ((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))) ∈ 𝐵)
186, 15, 17syl2anc 690 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾)) → ((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))) ∈ 𝐵)
1918ex 448 . . . . . 6 (𝐾 ∈ HL → (𝑋 ⊆ (Atoms‘𝐾) → ((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))) ∈ 𝐵))
2019adantrd 482 . . . . 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 34006 . . . . . . . . 9 ((𝐾 ∈ HL ∧ ((⊥𝑃𝐾)‘𝑋) ⊆ (Atoms‘𝐾)) → ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋)))))
239, 22syldan 485 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾)) → ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋)))))
2423ex 448 . . . . . . 7 (𝐾 ∈ HL → (𝑋 ⊆ (Atoms‘𝐾) → ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))))))
25 eqeq1 2613 . . . . . . . 8 (((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋 → (((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋)))) ↔ 𝑋 = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))))))
2625biimpcd 237 . . . . . . 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 445 . . . . 5 (𝐾 ∈ HL → ((𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋) → 𝑋 = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))))))
2920, 28jcad 553 . . . 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 6088 . . . . . 6 (𝑦 = ((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))) → (𝑀𝑦) = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋)))))
3130eqeq2d 2619 . . . . 5 (𝑦 = ((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))) → (𝑋 = (𝑀𝑦) ↔ 𝑋 = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))))))
3231rspcev 3281 . . . 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 33859 . . . . 5 ((𝐾 ∈ HL ∧ 𝑦𝐵) → (𝑀𝑦) ⊆ (Atoms‘𝐾))
3510, 21, 22polpmapN 34013 . . . . 5 ((𝐾 ∈ HL ∧ 𝑦𝐵) → ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘(𝑀𝑦))) = (𝑀𝑦))
36 sseq1 3588 . . . . . . 7 (𝑋 = (𝑀𝑦) → (𝑋 ⊆ (Atoms‘𝐾) ↔ (𝑀𝑦) ⊆ (Atoms‘𝐾)))
37 fveq2 6088 . . . . . . . . 9 (𝑋 = (𝑀𝑦) → ((⊥𝑃𝐾)‘𝑋) = ((⊥𝑃𝐾)‘(𝑀𝑦)))
3837fveq2d 6092 . . . . . . . 8 (𝑋 = (𝑀𝑦) → ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘(𝑀𝑦))))
39 id 22 . . . . . . . 8 (𝑋 = (𝑀𝑦) → 𝑋 = (𝑀𝑦))
4038, 39eqeq12d 2624 . . . . . . 7 (𝑋 = (𝑀𝑦) → (((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋 ↔ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘(𝑀𝑦))) = (𝑀𝑦)))
4136, 40anbi12d 742 . . . . . 6 (𝑋 = (𝑀𝑦) → ((𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋) ↔ ((𝑀𝑦) ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘(𝑀𝑦))) = (𝑀𝑦))))
4241biimprcd 238 . . . . 5 (((𝑀𝑦) ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘(𝑀𝑦))) = (𝑀𝑦)) → (𝑋 = (𝑀𝑦) → (𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋)))
4334, 35, 42syl2anc 690 . . . 4 ((𝐾 ∈ HL ∧ 𝑦𝐵) → (𝑋 = (𝑀𝑦) → (𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋)))
4443rexlimdva 3012 . . 3 (𝐾 ∈ HL → (∃𝑦𝐵 𝑋 = (𝑀𝑦) → (𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋)))
4533, 44impbid 200 . 2 (𝐾 ∈ HL → ((𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋) ↔ ∃𝑦𝐵 𝑋 = (𝑀𝑦)))
464, 45bitrd 266 1 (𝐾 ∈ HL → (𝑋𝐶 ↔ ∃𝑦𝐵 𝑋 = (𝑀𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wrex 2896  wss 3539  cfv 5790  Basecbs 15641  occoc 15722  lubclub 16711  CLatccla 16876  OPcops 33273  Atomscatm 33364  HLchlt 33451  pmapcpmap 33597  𝑃cpolN 34002  PSubClcpscN 34034
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-riotaBAD 33053
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-iin 4452  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-undef 7263  df-preset 16697  df-poset 16715  df-plt 16727  df-lub 16743  df-glb 16744  df-join 16745  df-meet 16746  df-p0 16808  df-p1 16809  df-lat 16815  df-clat 16877  df-oposet 33277  df-ol 33279  df-oml 33280  df-covers 33367  df-ats 33368  df-atl 33399  df-cvlat 33423  df-hlat 33452  df-psubsp 33603  df-pmap 33604  df-polarityN 34003  df-psubclN 34035
This theorem is referenced by: (None)
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