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Theorem ispsubcl2N 35771
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 2774 . . 3 (Atoms‘𝐾) = (Atoms‘𝐾)
2 eqid 2774 . . 3 (⊥𝑃𝐾) = (⊥𝑃𝐾)
3 pmapsubcl.c . . 3 𝐶 = (PSubCl‘𝐾)
41, 2, 3ispsubclN 35761 . 2 (𝐾 ∈ HL → (𝑋𝐶 ↔ (𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋)))
5 hlop 35186 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ OP)
65adantr 467 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾)) → 𝐾 ∈ OP)
7 hlclat 35182 . . . . . . . . . 10 (𝐾 ∈ HL → 𝐾 ∈ CLat)
87adantr 467 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾)) → 𝐾 ∈ CLat)
91, 2polssatN 35732 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾)) → ((⊥𝑃𝐾)‘𝑋) ⊆ (Atoms‘𝐾))
10 pmapsubcl.b . . . . . . . . . . 11 𝐵 = (Base‘𝐾)
1110, 1atssbase 35114 . . . . . . . . . 10 (Atoms‘𝐾) ⊆ 𝐵
129, 11syl6ss 3770 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾)) → ((⊥𝑃𝐾)‘𝑋) ⊆ 𝐵)
13 eqid 2774 . . . . . . . . . 10 (lub‘𝐾) = (lub‘𝐾)
1410, 13clatlubcl 17340 . . . . . . . . 9 ((𝐾 ∈ CLat ∧ ((⊥𝑃𝐾)‘𝑋) ⊆ 𝐵) → ((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋)) ∈ 𝐵)
158, 12, 14syl2anc 574 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾)) → ((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋)) ∈ 𝐵)
16 eqid 2774 . . . . . . . . 9 (oc‘𝐾) = (oc‘𝐾)
1710, 16opoccl 35018 . . . . . . . 8 ((𝐾 ∈ OP ∧ ((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋)) ∈ 𝐵) → ((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))) ∈ 𝐵)
186, 15, 17syl2anc 574 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾)) → ((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))) ∈ 𝐵)
1918ex 398 . . . . . 6 (𝐾 ∈ HL → (𝑋 ⊆ (Atoms‘𝐾) → ((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))) ∈ 𝐵))
2019adantrd 480 . . . . 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 35730 . . . . . . . . 9 ((𝐾 ∈ HL ∧ ((⊥𝑃𝐾)‘𝑋) ⊆ (Atoms‘𝐾)) → ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋)))))
239, 22syldan 580 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾)) → ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋)))))
2423ex 398 . . . . . . 7 (𝐾 ∈ HL → (𝑋 ⊆ (Atoms‘𝐾) → ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))))))
25 eqeq1 2778 . . . . . . . 8 (((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋 → (((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋)))) ↔ 𝑋 = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))))))
2625biimpcd 240 . . . . . . 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 397 . . . . 5 (𝐾 ∈ HL → ((𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋) → 𝑋 = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))))))
2920, 28jcad 503 . . . 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 6348 . . . . . 6 (𝑦 = ((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))) → (𝑀𝑦) = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋)))))
3130eqeq2d 2784 . . . . 5 (𝑦 = ((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))) → (𝑋 = (𝑀𝑦) ↔ 𝑋 = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))))))
3231rspcev 3465 . . . 4 ((((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))) ∈ 𝐵𝑋 = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))))) → ∃𝑦𝐵 𝑋 = (𝑀𝑦))
3329, 32syl6 35 . . 3 (𝐾 ∈ HL → ((𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋) → ∃𝑦𝐵 𝑋 = (𝑀𝑦)))
3410, 1, 21pmapssat 35583 . . . . 5 ((𝐾 ∈ HL ∧ 𝑦𝐵) → (𝑀𝑦) ⊆ (Atoms‘𝐾))
3510, 21, 22polpmapN 35737 . . . . 5 ((𝐾 ∈ HL ∧ 𝑦𝐵) → ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘(𝑀𝑦))) = (𝑀𝑦))
36 sseq1 3782 . . . . . . 7 (𝑋 = (𝑀𝑦) → (𝑋 ⊆ (Atoms‘𝐾) ↔ (𝑀𝑦) ⊆ (Atoms‘𝐾)))
37 2fveq3 6353 . . . . . . . 8 (𝑋 = (𝑀𝑦) → ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘(𝑀𝑦))))
38 id 22 . . . . . . . 8 (𝑋 = (𝑀𝑦) → 𝑋 = (𝑀𝑦))
3937, 38eqeq12d 2789 . . . . . . 7 (𝑋 = (𝑀𝑦) → (((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋 ↔ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘(𝑀𝑦))) = (𝑀𝑦)))
4036, 39anbi12d 617 . . . . . 6 (𝑋 = (𝑀𝑦) → ((𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋) ↔ ((𝑀𝑦) ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘(𝑀𝑦))) = (𝑀𝑦))))
4140biimprcd 241 . . . . 5 (((𝑀𝑦) ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘(𝑀𝑦))) = (𝑀𝑦)) → (𝑋 = (𝑀𝑦) → (𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋)))
4234, 35, 41syl2anc 574 . . . 4 ((𝐾 ∈ HL ∧ 𝑦𝐵) → (𝑋 = (𝑀𝑦) → (𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋)))
4342rexlimdva 3183 . . 3 (𝐾 ∈ HL → (∃𝑦𝐵 𝑋 = (𝑀𝑦) → (𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋)))
4433, 43impbid 203 . 2 (𝐾 ∈ HL → ((𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋) ↔ ∃𝑦𝐵 𝑋 = (𝑀𝑦)))
454, 44bitrd 269 1 (𝐾 ∈ HL → (𝑋𝐶 ↔ ∃𝑦𝐵 𝑋 = (𝑀𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 383   = wceq 1634  wcel 2148  wrex 3065  wss 3729  cfv 6042  Basecbs 16084  occoc 16177  lubclub 17170  CLatccla 17335  OPcops 34996  Atomscatm 35087  HLchlt 35174  pmapcpmap 35321  𝑃cpolN 35726  PSubClcpscN 35758
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-8 2150  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754  ax-rep 4917  ax-sep 4928  ax-nul 4936  ax-pow 4988  ax-pr 5048  ax-un 7117  ax-riotaBAD 34776
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3an 1100  df-tru 1637  df-ex 1856  df-nf 1861  df-sb 2053  df-eu 2625  df-mo 2626  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ne 2947  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3071  df-rmo 3072  df-rab 3073  df-v 3357  df-sbc 3594  df-csb 3689  df-dif 3732  df-un 3734  df-in 3736  df-ss 3743  df-nul 4074  df-if 4236  df-pw 4309  df-sn 4327  df-pr 4329  df-op 4333  df-uni 4586  df-iun 4667  df-iin 4668  df-br 4798  df-opab 4860  df-mpt 4877  df-id 5171  df-xp 5269  df-rel 5270  df-cnv 5271  df-co 5272  df-dm 5273  df-rn 5274  df-res 5275  df-ima 5276  df-iota 6005  df-fun 6044  df-fn 6045  df-f 6046  df-f1 6047  df-fo 6048  df-f1o 6049  df-fv 6050  df-riota 6773  df-ov 6815  df-oprab 6816  df-undef 7572  df-preset 17156  df-poset 17174  df-plt 17186  df-lub 17202  df-glb 17203  df-join 17204  df-meet 17205  df-p0 17267  df-p1 17268  df-lat 17274  df-clat 17336  df-oposet 35000  df-ol 35002  df-oml 35003  df-covers 35090  df-ats 35091  df-atl 35122  df-cvlat 35146  df-hlat 35175  df-psubsp 35327  df-pmap 35328  df-polarityN 35727  df-psubclN 35759
This theorem is referenced by: (None)
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