Theorem ltrniotafvawN 38154

Description: Version of cdleme46fvaw 38077 with simpler hypotheses. TODO: Fix comment. (Contributed by NM, 18-Apr-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ltrniotaval.l	⊢ ≤ = (le‘𝐾)
ltrniotaval.a	⊢ 𝐴 = (Atoms‘𝐾)
ltrniotaval.h	⊢ 𝐻 = (LHyp‘𝐾)
ltrniotaval.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
ltrniotaval.f	⊢ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄)

Assertion

Ref	Expression
ltrniotafvawN	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → ((𝐹‘𝑅) ∈ 𝐴 ∧ ¬ (𝐹‘𝑅) ≤ 𝑊))

Distinct variable groups:   𝐴,𝑓   𝑓,𝐻   𝑓,𝐾   ≤ ,𝑓   𝑃,𝑓   𝑄,𝑓   𝑇,𝑓   𝑓,𝑊

Allowed substitution hints:   𝑅(𝑓)   𝐹(𝑓)

Proof of Theorem ltrniotafvawN

Dummy variables 𝑠 𝑡 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2758	. 2 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		ltrniotaval.l	. 2 ⊢ ≤ = (le‘𝐾)
3		eqid 2758	. 2 ⊢ (join‘𝐾) = (join‘𝐾)
4		eqid 2758	. 2 ⊢ (meet‘𝐾) = (meet‘𝐾)
5		ltrniotaval.a	. 2 ⊢ 𝐴 = (Atoms‘𝐾)
6		ltrniotaval.h	. 2 ⊢ 𝐻 = (LHyp‘𝐾)
7		eqid 2758	. 2 ⊢ ((𝑃(join‘𝐾)𝑄)(meet‘𝐾)𝑊) = ((𝑃(join‘𝐾)𝑄)(meet‘𝐾)𝑊)
8		eqid 2758	. 2 ⊢ ((𝑡(join‘𝐾)((𝑃(join‘𝐾)𝑄)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄(join‘𝐾)((𝑃(join‘𝐾)𝑡)(meet‘𝐾)𝑊))) = ((𝑡(join‘𝐾)((𝑃(join‘𝐾)𝑄)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄(join‘𝐾)((𝑃(join‘𝐾)𝑡)(meet‘𝐾)𝑊)))
9		eqid 2758	. 2 ⊢ ((𝑃(join‘𝐾)𝑄)(meet‘𝐾)(((𝑡(join‘𝐾)((𝑃(join‘𝐾)𝑄)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄(join‘𝐾)((𝑃(join‘𝐾)𝑡)(meet‘𝐾)𝑊)))(join‘𝐾)((𝑠(join‘𝐾)𝑡)(meet‘𝐾)𝑊))) = ((𝑃(join‘𝐾)𝑄)(meet‘𝐾)(((𝑡(join‘𝐾)((𝑃(join‘𝐾)𝑄)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄(join‘𝐾)((𝑃(join‘𝐾)𝑡)(meet‘𝐾)𝑊)))(join‘𝐾)((𝑠(join‘𝐾)𝑡)(meet‘𝐾)𝑊)))
10		eqid 2758	. 2 ⊢ (𝑥 ∈ (Base‘𝐾) ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), (℩𝑧 ∈ (Base‘𝐾)∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠(join‘𝐾)(𝑥(meet‘𝐾)𝑊)) = 𝑥) → 𝑧 = (if(𝑠 ≤ (𝑃(join‘𝐾)𝑄), (℩𝑦 ∈ (Base‘𝐾)∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃(join‘𝐾)𝑄)) → 𝑦 = ((𝑃(join‘𝐾)𝑄)(meet‘𝐾)(((𝑡(join‘𝐾)((𝑃(join‘𝐾)𝑄)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄(join‘𝐾)((𝑃(join‘𝐾)𝑡)(meet‘𝐾)𝑊)))(join‘𝐾)((𝑠(join‘𝐾)𝑡)(meet‘𝐾)𝑊))))), ⦋𝑠 / 𝑡⦌((𝑡(join‘𝐾)((𝑃(join‘𝐾)𝑄)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄(join‘𝐾)((𝑃(join‘𝐾)𝑡)(meet‘𝐾)𝑊))))(join‘𝐾)(𝑥(meet‘𝐾)𝑊)))), 𝑥)) = (𝑥 ∈ (Base‘𝐾) ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), (℩𝑧 ∈ (Base‘𝐾)∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠(join‘𝐾)(𝑥(meet‘𝐾)𝑊)) = 𝑥) → 𝑧 = (if(𝑠 ≤ (𝑃(join‘𝐾)𝑄), (℩𝑦 ∈ (Base‘𝐾)∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃(join‘𝐾)𝑄)) → 𝑦 = ((𝑃(join‘𝐾)𝑄)(meet‘𝐾)(((𝑡(join‘𝐾)((𝑃(join‘𝐾)𝑄)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄(join‘𝐾)((𝑃(join‘𝐾)𝑡)(meet‘𝐾)𝑊)))(join‘𝐾)((𝑠(join‘𝐾)𝑡)(meet‘𝐾)𝑊))))), ⦋𝑠 / 𝑡⦌((𝑡(join‘𝐾)((𝑃(join‘𝐾)𝑄)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄(join‘𝐾)((𝑃(join‘𝐾)𝑡)(meet‘𝐾)𝑊))))(join‘𝐾)(𝑥(meet‘𝐾)𝑊)))), 𝑥))
11		ltrniotaval.t	. 2 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
12		ltrniotaval.f	. 2 ⊢ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄)
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	cdlemg1fvawlemN 38149	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → ((𝐹‘𝑅) ∈ 𝐴 ∧ ¬ (𝐹‘𝑅) ≤ 𝑊))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  ∀wral 3070  ⦋csb 3805  ifcif 4420   class class class wbr 5032   ↦ cmpt 5112  ‘cfv 6335  ℩crio 7107  (class class class)co 7150  Basecbs 16541  lecple 16630  joincjn 17620  meetcmee 17621  Atomscatm 36839  HLchlt 36926  LHypclh 37560  LTrncltrn 37677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-riotaBAD 36529

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-iin 4886  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7693  df-2nd 7694  df-undef 7949  df-map 8418  df-proset 17604  df-poset 17622  df-plt 17634  df-lub 17650  df-glb 17651  df-join 17652  df-meet 17653  df-p0 17715  df-p1 17716  df-lat 17722  df-clat 17784  df-oposet 36752  df-ol 36754  df-oml 36755  df-covers 36842  df-ats 36843  df-atl 36874  df-cvlat 36898  df-hlat 36927  df-llines 37074  df-lplanes 37075  df-lvols 37076  df-lines 37077  df-psubsp 37079  df-pmap 37080  df-padd 37372  df-lhyp 37564  df-laut 37565  df-ldil 37680  df-ltrn 37681  df-trl 37735

This theorem is referenced by: (None)