Theorem cdlemg2jOLDN 40557

Description: TODO: Replace this with ltrnj 40091. (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 2740	. 2 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		cdlemg2j.l	. 2 ⊢ ≤ = (le‘𝐾)
3		cdlemg2j.j	. 2 ⊢ ∨ = (join‘𝐾)
4		eqid 2740	. 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 2740	. 2 ⊢ ((𝑝 ∨ 𝑞)(meet‘𝐾)𝑊) = ((𝑝 ∨ 𝑞)(meet‘𝐾)𝑊)
9		eqid 2740	. 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 2740	. 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 2740	. 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 40552	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 2108   ≠ wne 2946  ∀wral 3067  ⦋csb 3921  ifcif 4548   class class class wbr 5166   ↦ cmpt 5249  ‘cfv 6575  ℩crio 7405  (class class class)co 7450  Basecbs 17260  lecple 17320  joincjn 18383  meetcmee 18384  Atomscatm 39221  HLchlt 39308  LHypclh 39943  LTrncltrn 40060

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 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7772  ax-riotaBAD 38911

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 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6527  df-fun 6577  df-fn 6578  df-f 6579  df-f1 6580  df-fo 6581  df-f1o 6582  df-fv 6583  df-riota 7406  df-ov 7453  df-oprab 7454  df-mpo 7455  df-1st 8032  df-2nd 8033  df-undef 8316  df-map 8888  df-proset 18367  df-poset 18385  df-plt 18402  df-lub 18418  df-glb 18419  df-join 18420  df-meet 18421  df-p0 18497  df-p1 18498  df-lat 18504  df-clat 18571  df-oposet 39134  df-ol 39136  df-oml 39137  df-covers 39224  df-ats 39225  df-atl 39256  df-cvlat 39280  df-hlat 39309  df-llines 39457  df-lplanes 39458  df-lvols 39459  df-lines 39460  df-psubsp 39462  df-pmap 39463  df-padd 39755  df-lhyp 39947  df-laut 39948  df-ldil 40063  df-ltrn 40064  df-trl 40118

This theorem is referenced by: (None)