Theorem ltrniotacl 40544

Description: Version of cdleme50ltrn 40522 with simpler hypotheses. TODO: Fix comment. (Contributed by NM, 17-Apr-2013.)

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
ltrniotacl	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐹 ∈ 𝑇)

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

Allowed substitution hint:   𝐹(𝑓)

Proof of Theorem ltrniotacl

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

Step	Hyp	Ref	Expression
1		eqid 2735	. 2 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		ltrniotaval.l	. 2 ⊢ ≤ = (le‘𝐾)
3		eqid 2735	. 2 ⊢ (join‘𝐾) = (join‘𝐾)
4		eqid 2735	. 2 ⊢ (meet‘𝐾) = (meet‘𝐾)
5		ltrniotaval.a	. 2 ⊢ 𝐴 = (Atoms‘𝐾)
6		ltrniotaval.h	. 2 ⊢ 𝐻 = (LHyp‘𝐾)
7		eqid 2735	. 2 ⊢ ((𝑃(join‘𝐾)𝑄)(meet‘𝐾)𝑊) = ((𝑃(join‘𝐾)𝑄)(meet‘𝐾)𝑊)
8		eqid 2735	. 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 2735	. 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 2735	. 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	cdlemg1ltrnlem 40539	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐹 ∈ 𝑇)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108   ≠ wne 2932  ∀wral 3051  ⦋csb 3874  ifcif 4500   class class class wbr 5119   ↦ cmpt 5201  ‘cfv 6530  ℩crio 7359  (class class class)co 7403  Basecbs 17226  lecple 17276  joincjn 18321  meetcmee 18322  Atomscatm 39227  HLchlt 39314  LHypclh 39949  LTrncltrn 40066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727  ax-riotaBAD 38917

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-iin 4970  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-riota 7360  df-ov 7406  df-oprab 7407  df-mpo 7408  df-1st 7986  df-2nd 7987  df-undef 8270  df-map 8840  df-proset 18304  df-poset 18323  df-plt 18338  df-lub 18354  df-glb 18355  df-join 18356  df-meet 18357  df-p0 18433  df-p1 18434  df-lat 18440  df-clat 18507  df-oposet 39140  df-ol 39142  df-oml 39143  df-covers 39230  df-ats 39231  df-atl 39262  df-cvlat 39286  df-hlat 39315  df-llines 39463  df-lplanes 39464  df-lvols 39465  df-lines 39466  df-psubsp 39468  df-pmap 39469  df-padd 39761  df-lhyp 39953  df-laut 39954  df-ldil 40069  df-ltrn 40070  df-trl 40124

This theorem is referenced by:  ltrniotacnvval  40547  ltrniotaidvalN  40548  ltrniotavalbN  40549  cdlemg1ci2  40551  cdlemki  40806  cdlemkj  40828  cdlemm10N  41083  dicssdvh  41151  dicvaddcl  41155  dicvscacl  41156  dicn0  41157  diclspsn  41159  cdlemn2  41160  cdlemn2a  41161  cdlemn3  41162  cdlemn4  41163  cdlemn4a  41164  cdlemn6  41167  cdlemn8  41169  cdlemn9  41170  cdlemn11a  41172  dihordlem7b  41180  dihopelvalcpre  41213  dih1  41251  dihmeetlem1N  41255  dihglblem5apreN  41256  dihglbcpreN  41265  dihmeetlem4preN  41271  dihmeetlem13N  41284  dih1dimatlem0  41293  dihatlat  41299  dihatexv  41303  dihjatcclem3  41385  dihjatcclem4  41386