Theorem cdlemg2l 41187

Description: TODO: FIX COMMENT. (Contributed by NM, 23-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
cdlemg2l	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) = ((𝐹‘(𝐺‘𝑃)) ∨ 𝑈))

Proof of Theorem cdlemg2l

Step	Hyp	Ref	Expression
1		cdlemg2inv.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
2		cdlemg2inv.t	. . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
3		cdlemg2j.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
4		cdlemg2j.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
5		cdlemg2j.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
6		cdlemg2j.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
7		cdlemg2j.u	. . . . 5 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
8	1, 2, 3, 4, 5, 6, 7	cdlemg2k 41185	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐺 ∈ 𝑇) → ((𝐺‘𝑃) ∨ (𝐺‘𝑄)) = ((𝐺‘𝑃) ∨ 𝑈))
9	8	3adant3l 1193	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → ((𝐺‘𝑃) ∨ (𝐺‘𝑄)) = ((𝐺‘𝑃) ∨ 𝑈))
10	9	fveq2d 6865	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → (𝐹‘((𝐺‘𝑃) ∨ (𝐺‘𝑄))) = (𝐹‘((𝐺‘𝑃) ∨ 𝑈)))
11		simp1 1148	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
12		simp3l 1214	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → 𝐹 ∈ 𝑇)
13		simp3r 1215	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → 𝐺 ∈ 𝑇)
14		simp2l 1212	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
15	3, 5, 1, 2	ltrnel 40723	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐺‘𝑃) ∈ 𝐴 ∧ ¬ (𝐺‘𝑃) ≤ 𝑊))
16	11, 13, 14, 15	syl3anc 1389	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → ((𝐺‘𝑃) ∈ 𝐴 ∧ ¬ (𝐺‘𝑃) ≤ 𝑊))
17	16	simpld 498	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → (𝐺‘𝑃) ∈ 𝐴)
18		eqid 2761	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
19	18, 5	atbase 39873	. . . 4 ⊢ ((𝐺‘𝑃) ∈ 𝐴 → (𝐺‘𝑃) ∈ (Base‘𝐾))
20	17, 19	syl 17	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → (𝐺‘𝑃) ∈ (Base‘𝐾))
21		simp2r 1213	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
22	3, 5, 1, 2	ltrnel 40723	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ((𝐺‘𝑄) ∈ 𝐴 ∧ ¬ (𝐺‘𝑄) ≤ 𝑊))
23	11, 13, 21, 22	syl3anc 1389	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → ((𝐺‘𝑄) ∈ 𝐴 ∧ ¬ (𝐺‘𝑄) ≤ 𝑊))
24	23	simpld 498	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → (𝐺‘𝑄) ∈ 𝐴)
25	18, 5	atbase 39873	. . . 4 ⊢ ((𝐺‘𝑄) ∈ 𝐴 → (𝐺‘𝑄) ∈ (Base‘𝐾))
26	24, 25	syl 17	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → (𝐺‘𝑄) ∈ (Base‘𝐾))
27	18, 4, 1, 2	ltrnj 40716	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝐺‘𝑃) ∈ (Base‘𝐾) ∧ (𝐺‘𝑄) ∈ (Base‘𝐾))) → (𝐹‘((𝐺‘𝑃) ∨ (𝐺‘𝑄))) = ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))))
28	11, 12, 20, 26, 27	syl112anc 1392	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → (𝐹‘((𝐺‘𝑃) ∨ (𝐺‘𝑄))) = ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))))
29	1, 2, 3, 4, 5, 6, 7	cdlemg2fv2 41184	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ ((𝐺‘𝑃) ∈ 𝐴 ∧ ¬ (𝐺‘𝑃) ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → (𝐹‘((𝐺‘𝑃) ∨ 𝑈)) = ((𝐹‘(𝐺‘𝑃)) ∨ 𝑈))
30	11, 14, 21, 16, 12, 29	syl131anc 1401	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → (𝐹‘((𝐺‘𝑃) ∨ 𝑈)) = ((𝐹‘(𝐺‘𝑃)) ∨ 𝑈))
31	10, 28, 30	3eqtr3d 2804	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) = ((𝐹‘(𝐺‘𝑃)) ∨ 𝑈))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141   class class class wbr 5097  ‘cfv 6515  (class class class)co 7390  Basecbs 17235  lecple 17283  joincjn 18333  meetcmee 18334  Atomscatm 39847  HLchlt 39934  LHypclh 40568  LTrncltrn 40685

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7712  ax-riotaBAD 39537

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-iin 4949  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7964  df-2nd 7965  df-undef 8246  df-map 8803  df-proset 18316  df-poset 18335  df-plt 18350  df-lub 18366  df-glb 18367  df-join 18368  df-meet 18369  df-p0 18445  df-p1 18446  df-lat 18454  df-clat 18521  df-oposet 39760  df-ol 39762  df-oml 39763  df-covers 39850  df-ats 39851  df-atl 39882  df-cvlat 39906  df-hlat 39935  df-llines 40082  df-lplanes 40083  df-lvols 40084  df-lines 40085  df-psubsp 40087  df-pmap 40088  df-padd 40380  df-lhyp 40572  df-laut 40573  df-ldil 40688  df-ltrn 40689  df-trl 40743

This theorem is referenced by:  cdlemg9a  41216  cdlemg9b  41217  cdlemg12b  41228