Theorem cdlemg4e 36419

Description: TODO: FIX COMMENT. (Contributed by NM, 25-Apr-2013.)

Hypotheses

Ref	Expression
cdlemg4.l	⊢ ≤ = (le‘𝐾)
cdlemg4.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemg4.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemg4.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemg4.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
cdlemg4.j	⊢ ∨ = (join‘𝐾)
cdlemg4b.v	⊢ 𝑉 = (𝑅‘𝐺)
cdlemg4.m	⊢ ∧ = (meet‘𝐾)

Assertion

Ref	Expression
cdlemg4e	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) → (𝐹‘(𝐺‘𝑄)) = (((𝐺‘𝑄) ∨ (𝑅‘𝐹)) ∧ ((𝐹‘(𝐺‘𝑃)) ∨ (((𝐺‘𝑃) ∨ (𝐺‘𝑄)) ∧ 𝑊))))

Proof of Theorem cdlemg4e

Step	Hyp	Ref	Expression
1		simp1 1130	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simp23 1250	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) → 𝐹 ∈ 𝑇)
3		simp31 1251	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) → 𝐺 ∈ 𝑇)
4		simp21 1248	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
5		cdlemg4.l	. . . 4 ⊢ ≤ = (le‘𝐾)
6		cdlemg4.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
7		cdlemg4.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
8		cdlemg4.t	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
9	5, 6, 7, 8	ltrnel 35943	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐺‘𝑃) ∈ 𝐴 ∧ ¬ (𝐺‘𝑃) ≤ 𝑊))
10	1, 3, 4, 9	syl3anc 1476	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) → ((𝐺‘𝑃) ∈ 𝐴 ∧ ¬ (𝐺‘𝑃) ≤ 𝑊))
11		simp22 1249	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
12	5, 6, 7, 8	ltrnel 35943	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ((𝐺‘𝑄) ∈ 𝐴 ∧ ¬ (𝐺‘𝑄) ≤ 𝑊))
13	1, 3, 11, 12	syl3anc 1476	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) → ((𝐺‘𝑄) ∈ 𝐴 ∧ ¬ (𝐺‘𝑄) ≤ 𝑊))
14		cdlemg4.r	. . 3 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
15		cdlemg4.j	. . 3 ⊢ ∨ = (join‘𝐾)
16		cdlemg4b.v	. . 3 ⊢ 𝑉 = (𝑅‘𝐺)
17	5, 6, 7, 8, 14, 15, 16	cdlemg4d 36418	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) → ¬ (𝐺‘𝑄) ≤ ((𝐺‘𝑃) ∨ (𝐹‘(𝐺‘𝑃))))
18		cdlemg4.m	. . 3 ⊢ ∧ = (meet‘𝐾)
19	5, 15, 18, 6, 7, 8, 14	cdlemc 36002	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ ((𝐺‘𝑃) ∈ 𝐴 ∧ ¬ (𝐺‘𝑃) ≤ 𝑊) ∧ ((𝐺‘𝑄) ∈ 𝐴 ∧ ¬ (𝐺‘𝑄) ≤ 𝑊)) ∧ ¬ (𝐺‘𝑄) ≤ ((𝐺‘𝑃) ∨ (𝐹‘(𝐺‘𝑃)))) → (𝐹‘(𝐺‘𝑄)) = (((𝐺‘𝑄) ∨ (𝑅‘𝐹)) ∧ ((𝐹‘(𝐺‘𝑃)) ∨ (((𝐺‘𝑃) ∨ (𝐺‘𝑄)) ∧ 𝑊))))
20	1, 2, 10, 13, 17, 19	syl131anc 1489	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) → (𝐹‘(𝐺‘𝑄)) = (((𝐺‘𝑄) ∨ (𝑅‘𝐹)) ∧ ((𝐹‘(𝐺‘𝑃)) ∨ (((𝐺‘𝑃) ∨ (𝐺‘𝑄)) ∧ 𝑊))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 382   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145   class class class wbr 4786  ‘cfv 6031  (class class class)co 6792  lecple 16155  joincjn 17151  meetcmee 17152  Atomscatm 35068  HLchlt 35155  LHypclh 35788  LTrncltrn 35905  trLctrl 35963

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7095

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-iin 4657  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-1st 7314  df-2nd 7315  df-map 8010  df-preset 17135  df-poset 17153  df-plt 17165  df-lub 17181  df-glb 17182  df-join 17183  df-meet 17184  df-p0 17246  df-p1 17247  df-lat 17253  df-clat 17315  df-oposet 34981  df-ol 34983  df-oml 34984  df-covers 35071  df-ats 35072  df-atl 35103  df-cvlat 35127  df-hlat 35156  df-llines 35302  df-psubsp 35307  df-pmap 35308  df-padd 35600  df-lhyp 35792  df-laut 35793  df-ldil 35908  df-ltrn 35909  df-trl 35964

This theorem is referenced by:  cdlemg4f  36420