Theorem cdlemg2jOLDN 40504

Description: TODO: Replace this with ltrnj 40038. (Contributed by NM, 22-Apr-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cdlemg2inv.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemg2inv.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemg2j.l	⊢ ≤ = (le‘𝐾)
cdlemg2j.j	⊢ ∨ = (join‘𝐾)
cdlemg2j.a	⊢ 𝐴 = (Atoms‘𝐾)

Assertion

Ref	Expression
cdlemg2jOLDN	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → (𝐹‘(𝑃 ∨ 𝑄)) = ((𝐹‘𝑃) ∨ (𝐹‘𝑄)))

Proof of Theorem cdlemg2jOLDN

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

Step	Hyp	Ref	Expression
1		eqid 2734	. 2 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		cdlemg2j.l	. 2 ⊢ ≤ = (le‘𝐾)
3		cdlemg2j.j	. 2 ⊢ ∨ = (join‘𝐾)
4		eqid 2734	. 2 ⊢ (meet‘𝐾) = (meet‘𝐾)
5		cdlemg2j.a	. 2 ⊢ 𝐴 = (Atoms‘𝐾)
6		cdlemg2inv.h	. 2 ⊢ 𝐻 = (LHyp‘𝐾)
7		cdlemg2inv.t	. 2 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
8		eqid 2734	. 2 ⊢ ((𝑝 ∨ 𝑞)(meet‘𝐾)𝑊) = ((𝑝 ∨ 𝑞)(meet‘𝐾)𝑊)
9		eqid 2734	. 2 ⊢ ((𝑡 ∨ ((𝑝 ∨ 𝑞)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑞 ∨ ((𝑝 ∨ 𝑡)(meet‘𝐾)𝑊))) = ((𝑡 ∨ ((𝑝 ∨ 𝑞)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑞 ∨ ((𝑝 ∨ 𝑡)(meet‘𝐾)𝑊)))
10		eqid 2734	. 2 ⊢ ((𝑝 ∨ 𝑞)(meet‘𝐾)(((𝑡 ∨ ((𝑝 ∨ 𝑞)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑞 ∨ ((𝑝 ∨ 𝑡)(meet‘𝐾)𝑊))) ∨ ((𝑠 ∨ 𝑡)(meet‘𝐾)𝑊))) = ((𝑝 ∨ 𝑞)(meet‘𝐾)(((𝑡 ∨ ((𝑝 ∨ 𝑞)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑞 ∨ ((𝑝 ∨ 𝑡)(meet‘𝐾)𝑊))) ∨ ((𝑠 ∨ 𝑡)(meet‘𝐾)𝑊)))
11		eqid 2734	. 2 ⊢ (𝑥 ∈ (Base‘𝐾) ↦ if((𝑝 ≠ 𝑞 ∧ ¬ 𝑥 ≤ 𝑊), (℩𝑧 ∈ (Base‘𝐾)∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥(meet‘𝐾)𝑊)) = 𝑥) → 𝑧 = (if(𝑠 ≤ (𝑝 ∨ 𝑞), (℩𝑦 ∈ (Base‘𝐾)∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑝 ∨ 𝑞)) → 𝑦 = ((𝑝 ∨ 𝑞)(meet‘𝐾)(((𝑡 ∨ ((𝑝 ∨ 𝑞)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑞 ∨ ((𝑝 ∨ 𝑡)(meet‘𝐾)𝑊))) ∨ ((𝑠 ∨ 𝑡)(meet‘𝐾)𝑊))))), ⦋𝑠 / 𝑡⦌((𝑡 ∨ ((𝑝 ∨ 𝑞)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑞 ∨ ((𝑝 ∨ 𝑡)(meet‘𝐾)𝑊)))) ∨ (𝑥(meet‘𝐾)𝑊)))), 𝑥)) = (𝑥 ∈ (Base‘𝐾) ↦ if((𝑝 ≠ 𝑞 ∧ ¬ 𝑥 ≤ 𝑊), (℩𝑧 ∈ (Base‘𝐾)∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥(meet‘𝐾)𝑊)) = 𝑥) → 𝑧 = (if(𝑠 ≤ (𝑝 ∨ 𝑞), (℩𝑦 ∈ (Base‘𝐾)∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑝 ∨ 𝑞)) → 𝑦 = ((𝑝 ∨ 𝑞)(meet‘𝐾)(((𝑡 ∨ ((𝑝 ∨ 𝑞)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑞 ∨ ((𝑝 ∨ 𝑡)(meet‘𝐾)𝑊))) ∨ ((𝑠 ∨ 𝑡)(meet‘𝐾)𝑊))))), ⦋𝑠 / 𝑡⦌((𝑡 ∨ ((𝑝 ∨ 𝑞)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑞 ∨ ((𝑝 ∨ 𝑡)(meet‘𝐾)𝑊)))) ∨ (𝑥(meet‘𝐾)𝑊)))), 𝑥))
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	cdlemg2jlemOLDN 40499	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → (𝐹‘(𝑃 ∨ 𝑄)) = ((𝐹‘𝑃) ∨ (𝐹‘𝑄)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2103   ≠ wne 2942  ∀wral 3063  ⦋csb 3915  ifcif 4548   class class class wbr 5169   ↦ cmpt 5252  ‘cfv 6572  ℩crio 7400  (class class class)co 7445  Basecbs 17253  lecple 17313  joincjn 18376  meetcmee 18377  Atomscatm 39168  HLchlt 39255  LHypclh 39890  LTrncltrn 40007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-rep 5306  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766  ax-riotaBAD 38858

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-rmo 3383  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5021  df-iin 5022  df-br 5170  df-opab 5232  df-mpt 5253  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-riota 7401  df-ov 7448  df-oprab 7449  df-mpo 7450  df-1st 8026  df-2nd 8027  df-undef 8310  df-map 8882  df-proset 18360  df-poset 18378  df-plt 18395  df-lub 18411  df-glb 18412  df-join 18413  df-meet 18414  df-p0 18490  df-p1 18491  df-lat 18497  df-clat 18564  df-oposet 39081  df-ol 39083  df-oml 39084  df-covers 39171  df-ats 39172  df-atl 39203  df-cvlat 39227  df-hlat 39256  df-llines 39404  df-lplanes 39405  df-lvols 39406  df-lines 39407  df-psubsp 39409  df-pmap 39410  df-padd 39702  df-lhyp 39894  df-laut 39895  df-ldil 40010  df-ltrn 40011  df-trl 40065

This theorem is referenced by: (None)