Theorem cdlemg2kq 40972

Description: cdlemg2k 40971 with 𝑃 and 𝑄 swapped. TODO: FIX COMMENT. (Contributed by NM, 15-May-2013.)

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)
cdlemg2j.m	⊢ ∧ = (meet‘𝐾)
cdlemg2j.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)

Assertion

Ref	Expression
cdlemg2kq	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → ((𝐹‘𝑃) ∨ (𝐹‘𝑄)) = ((𝐹‘𝑄) ∨ 𝑈))

Proof of Theorem cdlemg2kq

Step	Hyp	Ref	Expression
1		simp1 1137	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simp2r 1202	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
3		simp2l 1201	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
4		simp3 1139	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ 𝑇)
5		cdlemg2inv.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
6		cdlemg2inv.t	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
7		cdlemg2j.l	. . . 4 ⊢ ≤ = (le‘𝐾)
8		cdlemg2j.j	. . . 4 ⊢ ∨ = (join‘𝐾)
9		cdlemg2j.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
10		cdlemg2j.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
11		eqid 2737	. . . 4 ⊢ ((𝑄 ∨ 𝑃) ∧ 𝑊) = ((𝑄 ∨ 𝑃) ∧ 𝑊)
12	5, 6, 7, 8, 9, 10, 11	cdlemg2k 40971	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) = ((𝐹‘𝑄) ∨ ((𝑄 ∨ 𝑃) ∧ 𝑊)))
13	1, 2, 3, 4, 12	syl121anc 1378	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) = ((𝐹‘𝑄) ∨ ((𝑄 ∨ 𝑃) ∧ 𝑊)))
14		simp1l 1199	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → 𝐾 ∈ HL)
15		simp2ll 1242	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → 𝑃 ∈ 𝐴)
16	7, 9, 5, 6	ltrnat 40510	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝐹‘𝑃) ∈ 𝐴)
17	1, 4, 15, 16	syl3anc 1374	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → (𝐹‘𝑃) ∈ 𝐴)
18		simp2rl 1244	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → 𝑄 ∈ 𝐴)
19	7, 9, 5, 6	ltrnat 40510	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐴) → (𝐹‘𝑄) ∈ 𝐴)
20	1, 4, 18, 19	syl3anc 1374	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → (𝐹‘𝑄) ∈ 𝐴)
21	8, 9	hlatjcom 39738	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ (𝐹‘𝑄) ∈ 𝐴) → ((𝐹‘𝑃) ∨ (𝐹‘𝑄)) = ((𝐹‘𝑄) ∨ (𝐹‘𝑃)))
22	14, 17, 20, 21	syl3anc 1374	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → ((𝐹‘𝑃) ∨ (𝐹‘𝑄)) = ((𝐹‘𝑄) ∨ (𝐹‘𝑃)))
23		cdlemg2j.u	. . . 4 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
24	8, 9	hlatjcom 39738	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
25	14, 15, 18, 24	syl3anc 1374	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
26	25	oveq1d 7383	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → ((𝑃 ∨ 𝑄) ∧ 𝑊) = ((𝑄 ∨ 𝑃) ∧ 𝑊))
27	23, 26	eqtrid 2784	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → 𝑈 = ((𝑄 ∨ 𝑃) ∧ 𝑊))
28	27	oveq2d 7384	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → ((𝐹‘𝑄) ∨ 𝑈) = ((𝐹‘𝑄) ∨ ((𝑄 ∨ 𝑃) ∧ 𝑊)))
29	13, 22, 28	3eqtr4d 2782	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → ((𝐹‘𝑃) ∨ (𝐹‘𝑄)) = ((𝐹‘𝑄) ∨ 𝑈))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   class class class wbr 5100  ‘cfv 6500  (class class class)co 7368  lecple 17196  joincjn 18246  meetcmee 18247  Atomscatm 39633  HLchlt 39720  LHypclh 40354  LTrncltrn 40471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-riotaBAD 39323

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-undef 8225  df-map 8777  df-proset 18229  df-poset 18248  df-plt 18263  df-lub 18279  df-glb 18280  df-join 18281  df-meet 18282  df-p0 18358  df-p1 18359  df-lat 18367  df-clat 18434  df-oposet 39546  df-ol 39548  df-oml 39549  df-covers 39636  df-ats 39637  df-atl 39668  df-cvlat 39692  df-hlat 39721  df-llines 39868  df-lplanes 39869  df-lvols 39870  df-lines 39871  df-psubsp 39873  df-pmap 39874  df-padd 40166  df-lhyp 40358  df-laut 40359  df-ldil 40474  df-ltrn 40475  df-trl 40529

This theorem is referenced by:  cdlemg18b  41049  cdlemg18c  41050