Theorem cdlemg2klem 39770

Description: cdleme42keg 39661 with simpler hypotheses. TODO: FIX COMMENT. (Contributed by NM, 22-Apr-2013.)

Hypotheses

Ref	Expression
cdlemg2.b	⊢ 𝐵 = (Base‘𝐾)
cdlemg2.l	⊢ ≤ = (le‘𝐾)
cdlemg2.j	⊢ ∨ = (join‘𝐾)
cdlemg2.m	⊢ ∧ = (meet‘𝐾)
cdlemg2.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemg2.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemg2.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemg2ex.u	⊢ 𝑈 = ((𝑝 ∨ 𝑞) ∧ 𝑊)
cdlemg2ex.d	⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑞 ∨ ((𝑝 ∨ 𝑡) ∧ 𝑊)))
cdlemg2ex.e	⊢ 𝐸 = ((𝑝 ∨ 𝑞) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊)))
cdlemg2ex.g	⊢ 𝐺 = (𝑥 ∈ 𝐵 ↦ if((𝑝 ≠ 𝑞 ∧ ¬ 𝑥 ≤ 𝑊), (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 ≤ (𝑝 ∨ 𝑞), (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑝 ∨ 𝑞)) → 𝑦 = 𝐸)), ⦋𝑠 / 𝑡⦌𝐷) ∨ (𝑥 ∧ 𝑊)))), 𝑥))
cdlemg2klem.v	⊢ 𝑉 = ((𝑃 ∨ 𝑄) ∧ 𝑊)

Assertion

Ref	Expression
cdlemg2klem	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → ((𝐹‘𝑃) ∨ (𝐹‘𝑄)) = ((𝐹‘𝑃) ∨ 𝑉))

Distinct variable groups:   𝑡,𝑠,𝑥,𝑦,𝑧,𝐴   𝐵,𝑠,𝑡,𝑥,𝑦,𝑧   𝐷,𝑠,𝑥,𝑦,𝑧   𝑥,𝐸,𝑦,𝑧   𝐻,𝑠,𝑡,𝑥,𝑦,𝑧   ∨ ,𝑠,𝑡,𝑥,𝑦,𝑧   𝐾,𝑠,𝑡,𝑥,𝑦,𝑧   ≤ ,𝑠,𝑡,𝑥,𝑦,𝑧   ∧ ,𝑠,𝑡,𝑥,𝑦,𝑧   𝑃,𝑠,𝑡,𝑥,𝑦,𝑧   𝑄,𝑠,𝑡,𝑥,𝑦,𝑧   𝑈,𝑠,𝑡,𝑥,𝑦,𝑧   𝑊,𝑠,𝑡,𝑥,𝑦,𝑧   𝑞,𝑝,𝐴   𝐹,𝑝,𝑞   𝐻,𝑝,𝑞   𝐾,𝑝,𝑞   ≤ ,𝑝,𝑞   𝑇,𝑝,𝑞   𝑊,𝑝,𝑞,𝑠,𝑡,𝑥,𝑦,𝑧   ∨ ,𝑝,𝑞   𝑃,𝑝,𝑞   𝑄,𝑝,𝑞   𝐵,𝑝,𝑞   ∧ ,𝑝,𝑞   𝑉,𝑝,𝑞,𝑠,𝑡,𝑥,𝑧

Allowed substitution hints:   𝐷(𝑡,𝑞,𝑝)   𝑇(𝑥,𝑦,𝑧,𝑡,𝑠)   𝑈(𝑞,𝑝)   𝐸(𝑡,𝑠,𝑞,𝑝)   𝐹(𝑥,𝑦,𝑧,𝑡,𝑠)   𝐺(𝑥,𝑦,𝑧,𝑡,𝑠,𝑞,𝑝)   𝑉(𝑦)

Proof of Theorem cdlemg2klem

Step	Hyp	Ref	Expression
1		cdlemg2.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
2		cdlemg2.l	. . 3 ⊢ ≤ = (le‘𝐾)
3		cdlemg2.j	. . 3 ⊢ ∨ = (join‘𝐾)
4		cdlemg2.m	. . 3 ⊢ ∧ = (meet‘𝐾)
5		cdlemg2.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
6		cdlemg2.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
7		cdlemg2.t	. . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
8		cdlemg2ex.u	. . 3 ⊢ 𝑈 = ((𝑝 ∨ 𝑞) ∧ 𝑊)
9		cdlemg2ex.d	. . 3 ⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑞 ∨ ((𝑝 ∨ 𝑡) ∧ 𝑊)))
10		cdlemg2ex.e	. . 3 ⊢ 𝐸 = ((𝑝 ∨ 𝑞) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊)))
11		cdlemg2ex.g	. . 3 ⊢ 𝐺 = (𝑥 ∈ 𝐵 ↦ if((𝑝 ≠ 𝑞 ∧ ¬ 𝑥 ≤ 𝑊), (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 ≤ (𝑝 ∨ 𝑞), (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑝 ∨ 𝑞)) → 𝑦 = 𝐸)), ⦋𝑠 / 𝑡⦌𝐷) ∨ (𝑥 ∧ 𝑊)))), 𝑥))
12		fveq1 6890	. . . . 5 ⊢ (𝐹 = 𝐺 → (𝐹‘𝑃) = (𝐺‘𝑃))
13		fveq1 6890	. . . . 5 ⊢ (𝐹 = 𝐺 → (𝐹‘𝑄) = (𝐺‘𝑄))
14	12, 13	oveq12d 7430	. . . 4 ⊢ (𝐹 = 𝐺 → ((𝐹‘𝑃) ∨ (𝐹‘𝑄)) = ((𝐺‘𝑃) ∨ (𝐺‘𝑄)))
15	12	oveq1d 7427	. . . 4 ⊢ (𝐹 = 𝐺 → ((𝐹‘𝑃) ∨ 𝑉) = ((𝐺‘𝑃) ∨ 𝑉))
16	14, 15	eqeq12d 2747	. . 3 ⊢ (𝐹 = 𝐺 → (((𝐹‘𝑃) ∨ (𝐹‘𝑄)) = ((𝐹‘𝑃) ∨ 𝑉) ↔ ((𝐺‘𝑃) ∨ (𝐺‘𝑄)) = ((𝐺‘𝑃) ∨ 𝑉)))
17		vex 3477	. . . . 5 ⊢ 𝑠 ∈ V
18		eqid 2731	. . . . . 6 ⊢ ((𝑠 ∨ 𝑈) ∧ (𝑞 ∨ ((𝑝 ∨ 𝑠) ∧ 𝑊))) = ((𝑠 ∨ 𝑈) ∧ (𝑞 ∨ ((𝑝 ∨ 𝑠) ∧ 𝑊)))
19	9, 18	cdleme31sc 39559	. . . . 5 ⊢ (𝑠 ∈ V → ⦋𝑠 / 𝑡⦌𝐷 = ((𝑠 ∨ 𝑈) ∧ (𝑞 ∨ ((𝑝 ∨ 𝑠) ∧ 𝑊))))
20	17, 19	ax-mp 5	. . . 4 ⊢ ⦋𝑠 / 𝑡⦌𝐷 = ((𝑠 ∨ 𝑈) ∧ (𝑞 ∨ ((𝑝 ∨ 𝑠) ∧ 𝑊)))
21		eqid 2731	. . . 4 ⊢ (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑝 ∨ 𝑞)) → 𝑦 = 𝐸)) = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑝 ∨ 𝑞)) → 𝑦 = 𝐸))
22		eqid 2731	. . . 4 ⊢ if(𝑠 ≤ (𝑝 ∨ 𝑞), (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑝 ∨ 𝑞)) → 𝑦 = 𝐸)), ⦋𝑠 / 𝑡⦌𝐷) = if(𝑠 ≤ (𝑝 ∨ 𝑞), (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑝 ∨ 𝑞)) → 𝑦 = 𝐸)), ⦋𝑠 / 𝑡⦌𝐷)
23		eqid 2731	. . . 4 ⊢ (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 ≤ (𝑝 ∨ 𝑞), (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑝 ∨ 𝑞)) → 𝑦 = 𝐸)), ⦋𝑠 / 𝑡⦌𝐷) ∨ (𝑥 ∧ 𝑊)))) = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 ≤ (𝑝 ∨ 𝑞), (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑝 ∨ 𝑞)) → 𝑦 = 𝐸)), ⦋𝑠 / 𝑡⦌𝐷) ∨ (𝑥 ∧ 𝑊))))
24		cdlemg2klem.v	. . . 4 ⊢ 𝑉 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
25	1, 2, 3, 4, 5, 6, 8, 20, 9, 10, 21, 22, 23, 11, 24	cdleme42keg 39661	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → ((𝐺‘𝑃) ∨ (𝐺‘𝑄)) = ((𝐺‘𝑃) ∨ 𝑉))
26	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 25	cdlemg2ce 39767	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → ((𝐹‘𝑃) ∨ (𝐹‘𝑄)) = ((𝐹‘𝑃) ∨ 𝑉))
27	26	3com23 1125	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → ((𝐹‘𝑃) ∨ (𝐹‘𝑄)) = ((𝐹‘𝑃) ∨ 𝑉))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   ≠ wne 2939  ∀wral 3060  Vcvv 3473  ⦋csb 3893  ifcif 4528   class class class wbr 5148   ↦ cmpt 5231  ‘cfv 6543  ℩crio 7367  (class class class)co 7412  Basecbs 17149  lecple 17209  joincjn 18269  meetcmee 18270  Atomscatm 38437  HLchlt 38524  LHypclh 39159  LTrncltrn 39276

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-riotaBAD 38127

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7979  df-2nd 7980  df-undef 8262  df-map 8826  df-proset 18253  df-poset 18271  df-plt 18288  df-lub 18304  df-glb 18305  df-join 18306  df-meet 18307  df-p0 18383  df-p1 18384  df-lat 18390  df-clat 18457  df-oposet 38350  df-ol 38352  df-oml 38353  df-covers 38440  df-ats 38441  df-atl 38472  df-cvlat 38496  df-hlat 38525  df-llines 38673  df-lplanes 38674  df-lvols 38675  df-lines 38676  df-psubsp 38678  df-pmap 38679  df-padd 38971  df-lhyp 39163  df-laut 39164  df-ldil 39279  df-ltrn 39280  df-trl 39334

This theorem is referenced by:  cdlemg2k  39776