Theorem cdlemg2m 41705

Description: TODO: FIX COMMENT. (Contributed by NM, 25-Apr-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
cdlemg2m	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → (((𝐹‘𝑃) ∨ (𝐹‘𝑄)) ∧ 𝑊) = 𝑈)

Proof of Theorem cdlemg2m

Step	Hyp	Ref	Expression
1		cdlemg2inv.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
2		cdlemg2inv.t	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
3		cdlemg2j.l	. . . 4 ⊢ ≤ = (le‘𝐾)
4		cdlemg2j.j	. . . 4 ⊢ ∨ = (join‘𝐾)
5		cdlemg2j.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
6		cdlemg2j.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
7		cdlemg2j.u	. . . 4 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
8	1, 2, 3, 4, 5, 6, 7	cdlemg2k 41702	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → ((𝐹‘𝑃) ∨ (𝐹‘𝑄)) = ((𝐹‘𝑃) ∨ 𝑈))
9	8	oveq1d 7435	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → (((𝐹‘𝑃) ∨ (𝐹‘𝑄)) ∧ 𝑊) = (((𝐹‘𝑃) ∨ 𝑈) ∧ 𝑊))
10		simp1 1154	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
11		simp3 1156	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ 𝑇)
12		simp2l 1218	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
13		eqid 2760	. . . . . 6 ⊢ (0.‘𝐾) = (0.‘𝐾)
14	3, 6, 13, 5, 1, 2	ltrnmw 41252	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∧ 𝑊) = (0.‘𝐾))
15	10, 11, 12, 14	syl3anc 1398	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → ((𝐹‘𝑃) ∧ 𝑊) = (0.‘𝐾))
16	15	oveq1d 7435	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → (((𝐹‘𝑃) ∧ 𝑊) ∨ 𝑈) = ((0.‘𝐾) ∨ 𝑈))
17		simp1l 1216	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → 𝐾 ∈ HL)
18	3, 5, 1, 2	ltrnel 41240	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊))
19	10, 11, 12, 18	syl3anc 1398	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊))
20	19	simpld 500	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → (𝐹‘𝑃) ∈ 𝐴)
21		simp1r 1217	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → 𝑊 ∈ 𝐻)
22		simp2ll 1259	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → 𝑃 ∈ 𝐴)
23		simp2rl 1261	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → 𝑄 ∈ 𝐴)
24		eqid 2760	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
25	3, 4, 6, 5, 1, 7, 24	cdleme0aa 41311	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑈 ∈ (Base‘𝐾))
26	17, 21, 22, 23, 25	syl211anc 1403	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → 𝑈 ∈ (Base‘𝐾))
27	24, 1	lhpbase 41099	. . . . 5 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
28	21, 27	syl 18	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → 𝑊 ∈ (Base‘𝐾))
29	17	hllatd 40465	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → 𝐾 ∈ Lat)
30	24, 4, 5	hlatjcl 40468	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
31	17, 22, 23, 30	syl3anc 1398	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
32	24, 3, 6	latmle2 18640	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ 𝑊)
33	29, 31, 28, 32	syl3anc 1398	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ 𝑊)
34	7, 33	eqbrtrid 5139	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → 𝑈 ≤ 𝑊)
35	24, 3, 4, 6, 5	atmod4i2 40968	. . . 4 ⊢ ((𝐾 ∈ HL ∧ ((𝐹‘𝑃) ∈ 𝐴 ∧ 𝑈 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑈 ≤ 𝑊) → (((𝐹‘𝑃) ∧ 𝑊) ∨ 𝑈) = (((𝐹‘𝑃) ∨ 𝑈) ∧ 𝑊))
36	17, 20, 26, 28, 34, 35	syl131anc 1410	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → (((𝐹‘𝑃) ∧ 𝑊) ∨ 𝑈) = (((𝐹‘𝑃) ∨ 𝑈) ∧ 𝑊))
37		hlol 40462	. . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
38	17, 37	syl 18	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → 𝐾 ∈ OL)
39	24, 4, 13	olj02 40327	. . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑈 ∈ (Base‘𝐾)) → ((0.‘𝐾) ∨ 𝑈) = 𝑈)
40	38, 26, 39	syl2anc 596	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → ((0.‘𝐾) ∨ 𝑈) = 𝑈)
41	16, 36, 40	3eqtr3d 2803	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → (((𝐹‘𝑃) ∨ 𝑈) ∧ 𝑊) = 𝑈)
42	9, 41	eqtrd 2795	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → (((𝐹‘𝑃) ∨ (𝐹‘𝑄)) ∧ 𝑊) = 𝑈)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   class class class wbr 5102  ‘cfv 6538  (class class class)co 7420  Basecbs 17388  lecple 17436  joincjn 18486  meetcmee 18487  0.cp0 18596  Latclat 18606  OLcol 40275  Atomscatm 40364  HLchlt 40451  LHypclh 41085  LTrncltrn 41202

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-riotaBAD 40054

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-undef 8291  df-map 8850  df-proset 18469  df-poset 18488  df-plt 18503  df-lub 18519  df-glb 18520  df-join 18521  df-meet 18522  df-p0 18598  df-p1 18599  df-lat 18607  df-clat 18674  df-oposet 40277  df-ol 40279  df-oml 40280  df-covers 40367  df-ats 40368  df-atl 40399  df-cvlat 40423  df-hlat 40452  df-llines 40599  df-lplanes 40600  df-lvols 40601  df-lines 40602  df-psubsp 40604  df-pmap 40605  df-padd 40897  df-lhyp 41089  df-laut 41090  df-ldil 41205  df-ltrn 41206  df-trl 41260

This theorem is used by:  cdlemg4f  41716  cdlemg10b  41736