Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ispsubcl2N Structured version   Visualization version   GIF version

Theorem ispsubcl2N 35754
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 2760 . . 3 (Atoms‘𝐾) = (Atoms‘𝐾)
2 eqid 2760 . . 3 (⊥𝑃𝐾) = (⊥𝑃𝐾)
3 pmapsubcl.c . . 3 𝐶 = (PSubCl‘𝐾)
41, 2, 3ispsubclN 35744 . 2 (𝐾 ∈ HL → (𝑋𝐶 ↔ (𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋)))
5 hlop 35170 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ OP)
65adantr 472 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾)) → 𝐾 ∈ OP)
7 hlclat 35166 . . . . . . . . . 10 (𝐾 ∈ HL → 𝐾 ∈ CLat)
87adantr 472 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾)) → 𝐾 ∈ CLat)
91, 2polssatN 35715 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾)) → ((⊥𝑃𝐾)‘𝑋) ⊆ (Atoms‘𝐾))
10 pmapsubcl.b . . . . . . . . . . 11 𝐵 = (Base‘𝐾)
1110, 1atssbase 35098 . . . . . . . . . 10 (Atoms‘𝐾) ⊆ 𝐵
129, 11syl6ss 3756 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾)) → ((⊥𝑃𝐾)‘𝑋) ⊆ 𝐵)
13 eqid 2760 . . . . . . . . . 10 (lub‘𝐾) = (lub‘𝐾)
1410, 13clatlubcl 17333 . . . . . . . . 9 ((𝐾 ∈ CLat ∧ ((⊥𝑃𝐾)‘𝑋) ⊆ 𝐵) → ((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋)) ∈ 𝐵)
158, 12, 14syl2anc 696 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾)) → ((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋)) ∈ 𝐵)
16 eqid 2760 . . . . . . . . 9 (oc‘𝐾) = (oc‘𝐾)
1710, 16opoccl 35002 . . . . . . . 8 ((𝐾 ∈ OP ∧ ((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋)) ∈ 𝐵) → ((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))) ∈ 𝐵)
186, 15, 17syl2anc 696 . . . . . . 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 485 . . . . 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 35713 . . . . . . . . 9 ((𝐾 ∈ HL ∧ ((⊥𝑃𝐾)‘𝑋) ⊆ (Atoms‘𝐾)) → ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋)))))
239, 22syldan 488 . . . . . . . 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 2764 . . . . . . . 8 (((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋 → (((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋)))) ↔ 𝑋 = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))))))
2625biimpcd 239 . . . . . . 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 446 . . . . 5 (𝐾 ∈ HL → ((𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋) → 𝑋 = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))))))
2920, 28jcad 556 . . . 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 6353 . . . . . 6 (𝑦 = ((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))) → (𝑀𝑦) = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋)))))
3130eqeq2d 2770 . . . . 5 (𝑦 = ((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))) → (𝑋 = (𝑀𝑦) ↔ 𝑋 = (𝑀‘((oc‘𝐾)‘((lub‘𝐾)‘((⊥𝑃𝐾)‘𝑋))))))
3231rspcev 3449 . . . 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 35566 . . . . 5 ((𝐾 ∈ HL ∧ 𝑦𝐵) → (𝑀𝑦) ⊆ (Atoms‘𝐾))
3510, 21, 22polpmapN 35720 . . . . 5 ((𝐾 ∈ HL ∧ 𝑦𝐵) → ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘(𝑀𝑦))) = (𝑀𝑦))
36 sseq1 3767 . . . . . . 7 (𝑋 = (𝑀𝑦) → (𝑋 ⊆ (Atoms‘𝐾) ↔ (𝑀𝑦) ⊆ (Atoms‘𝐾)))
37 fveq2 6353 . . . . . . . . 9 (𝑋 = (𝑀𝑦) → ((⊥𝑃𝐾)‘𝑋) = ((⊥𝑃𝐾)‘(𝑀𝑦)))
3837fveq2d 6357 . . . . . . . 8 (𝑋 = (𝑀𝑦) → ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘(𝑀𝑦))))
39 id 22 . . . . . . . 8 (𝑋 = (𝑀𝑦) → 𝑋 = (𝑀𝑦))
4038, 39eqeq12d 2775 . . . . . . 7 (𝑋 = (𝑀𝑦) → (((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋 ↔ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘(𝑀𝑦))) = (𝑀𝑦)))
4136, 40anbi12d 749 . . . . . 6 (𝑋 = (𝑀𝑦) → ((𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋) ↔ ((𝑀𝑦) ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘(𝑀𝑦))) = (𝑀𝑦))))
4241biimprcd 240 . . . . 5 (((𝑀𝑦) ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘(𝑀𝑦))) = (𝑀𝑦)) → (𝑋 = (𝑀𝑦) → (𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋)))
4334, 35, 42syl2anc 696 . . . 4 ((𝐾 ∈ HL ∧ 𝑦𝐵) → (𝑋 = (𝑀𝑦) → (𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋)))
4443rexlimdva 3169 . . 3 (𝐾 ∈ HL → (∃𝑦𝐵 𝑋 = (𝑀𝑦) → (𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋)))
4533, 44impbid 202 . 2 (𝐾 ∈ HL → ((𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋) ↔ ∃𝑦𝐵 𝑋 = (𝑀𝑦)))
464, 45bitrd 268 1 (𝐾 ∈ HL → (𝑋𝐶 ↔ ∃𝑦𝐵 𝑋 = (𝑀𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wrex 3051  wss 3715  cfv 6049  Basecbs 16079  occoc 16171  lubclub 17163  CLatccla 17328  OPcops 34980  Atomscatm 35071  HLchlt 35158  pmapcpmap 35304  𝑃cpolN 35709  PSubClcpscN 35741
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-riotaBAD 34760
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-undef 7569  df-preset 17149  df-poset 17167  df-plt 17179  df-lub 17195  df-glb 17196  df-join 17197  df-meet 17198  df-p0 17260  df-p1 17261  df-lat 17267  df-clat 17329  df-oposet 34984  df-ol 34986  df-oml 34987  df-covers 35074  df-ats 35075  df-atl 35106  df-cvlat 35130  df-hlat 35159  df-psubsp 35310  df-pmap 35311  df-polarityN 35710  df-psubclN 35742
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator