Theorem cdlemg8 40740

Description: TODO: FIX COMMENT. (Contributed by NM, 29-Apr-2013.)

Hypotheses

Ref	Expression
cdlemg8.l	⊢ ≤ = (le‘𝐾)
cdlemg8.j	⊢ ∨ = (join‘𝐾)
cdlemg8.m	⊢ ∧ = (meet‘𝐾)
cdlemg8.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemg8.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemg8.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)

Assertion

Ref	Expression
cdlemg8	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) = (𝑃 ∨ 𝑄))) → ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ 𝑊) = ((𝑄 ∨ (𝐹‘(𝐺‘𝑄))) ∧ 𝑊))

Proof of Theorem cdlemg8

Step	Hyp	Ref	Expression
1		simpl1 1192	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) = (𝑃 ∨ 𝑄))) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simpl21 1252	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) = (𝑃 ∨ 𝑄))) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
3		simpl22 1253	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) = (𝑃 ∨ 𝑄))) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
4		simpl23 1254	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) = (𝑃 ∨ 𝑄))) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃) → 𝐹 ∈ 𝑇)
5		simpl3l 1229	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) = (𝑃 ∨ 𝑄))) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃) → 𝐺 ∈ 𝑇)
6		simpr 484	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) = (𝑃 ∨ 𝑄))) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃) → (𝐹‘(𝐺‘𝑃)) = 𝑃)
7		cdlemg8.l	. . . 4 ⊢ ≤ = (le‘𝐾)
8		cdlemg8.j	. . . 4 ⊢ ∨ = (join‘𝐾)
9		cdlemg8.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
10		cdlemg8.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
11		cdlemg8.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
12		cdlemg8.t	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
13	7, 8, 9, 10, 11, 12	cdlemg8a 40736	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) → ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ 𝑊) = ((𝑄 ∨ (𝐹‘(𝐺‘𝑄))) ∧ 𝑊))
14	1, 2, 3, 4, 5, 6, 13	syl123anc 1389	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) = (𝑃 ∨ 𝑄))) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃) → ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ 𝑊) = ((𝑄 ∨ (𝐹‘(𝐺‘𝑄))) ∧ 𝑊))
15		simpl1 1192	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) = (𝑃 ∨ 𝑄))) ∧ (𝐹‘(𝐺‘𝑃)) ≠ 𝑃) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
16		simpl2 1193	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) = (𝑃 ∨ 𝑄))) ∧ (𝐹‘(𝐺‘𝑃)) ≠ 𝑃) → ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇))
17		simpl3l 1229	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) = (𝑃 ∨ 𝑄))) ∧ (𝐹‘(𝐺‘𝑃)) ≠ 𝑃) → 𝐺 ∈ 𝑇)
18		simpl3r 1230	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) = (𝑃 ∨ 𝑄))) ∧ (𝐹‘(𝐺‘𝑃)) ≠ 𝑃) → ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) = (𝑃 ∨ 𝑄))
19		simpr 484	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) = (𝑃 ∨ 𝑄))) ∧ (𝐹‘(𝐺‘𝑃)) ≠ 𝑃) → (𝐹‘(𝐺‘𝑃)) ≠ 𝑃)
20	7, 8, 9, 10, 11, 12	cdlemg8d 40739	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) = (𝑃 ∨ 𝑄) ∧ (𝐹‘(𝐺‘𝑃)) ≠ 𝑃)) → ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ 𝑊) = ((𝑄 ∨ (𝐹‘(𝐺‘𝑄))) ∧ 𝑊))
21	15, 16, 17, 18, 19, 20	syl113anc 1384	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) = (𝑃 ∨ 𝑄))) ∧ (𝐹‘(𝐺‘𝑃)) ≠ 𝑃) → ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ 𝑊) = ((𝑄 ∨ (𝐹‘(𝐺‘𝑄))) ∧ 𝑊))
22	14, 21	pm2.61dane 3017	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) = (𝑃 ∨ 𝑄))) → ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ 𝑊) = ((𝑄 ∨ (𝐹‘(𝐺‘𝑄))) ∧ 𝑊))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2930   class class class wbr 5095  ‘cfv 6489  (class class class)co 7355  lecple 17178  joincjn 18227  meetcmee 18228  Atomscatm 39372  HLchlt 39459  LHypclh 40093  LTrncltrn 40210

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-riotaBAD 39062

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-iin 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-1st 7930  df-2nd 7931  df-undef 8212  df-map 8761  df-proset 18210  df-poset 18229  df-plt 18244  df-lub 18260  df-glb 18261  df-join 18262  df-meet 18263  df-p0 18339  df-p1 18340  df-lat 18348  df-clat 18415  df-oposet 39285  df-ol 39287  df-oml 39288  df-covers 39375  df-ats 39376  df-atl 39407  df-cvlat 39431  df-hlat 39460  df-llines 39607  df-lplanes 39608  df-lvols 39609  df-lines 39610  df-psubsp 39612  df-pmap 39613  df-padd 39905  df-lhyp 40097  df-laut 40098  df-ldil 40213  df-ltrn 40214  df-trl 40268

This theorem is referenced by:  cdlemg15  40765  cdlemg16z  40768  cdlemg37  40798