Theorem cdlemc3 40315

Description: Part of proof of Lemma C in [Crawley] p. 113. (Contributed by NM, 26-May-2012.)

Hypotheses

Ref	Expression
cdlemc3.l	⊢ ≤ = (le‘𝐾)
cdlemc3.j	⊢ ∨ = (join‘𝐾)
cdlemc3.m	⊢ ∧ = (meet‘𝐾)
cdlemc3.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemc3.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemc3.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemc3.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)

Assertion

Ref	Expression
cdlemc3	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → ((𝐹‘𝑃) ≤ (𝑄 ∨ (𝑅‘𝐹)) → 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))))

Proof of Theorem cdlemc3

Step	Hyp	Ref	Expression
1		simpll 766	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → 𝐾 ∈ HL)
2		simpl 482	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
3		simpr1 1195	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → 𝐹 ∈ 𝑇)
4		simpr2l 1233	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → 𝑃 ∈ 𝐴)
5		cdlemc3.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
6		cdlemc3.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
7		cdlemc3.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
8		cdlemc3.t	. . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
9	5, 6, 7, 8	ltrnat 40262	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝐹‘𝑃) ∈ 𝐴)
10	2, 3, 4, 9	syl3anc 1373	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → (𝐹‘𝑃) ∈ 𝐴)
11		simpr3l 1235	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → 𝑄 ∈ 𝐴)
12		eqid 2733	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
13		cdlemc3.r	. . . . 5 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
14	12, 7, 8, 13	trlcl 40286	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) ∈ (Base‘𝐾))
15	3, 14	syldan 591	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → (𝑅‘𝐹) ∈ (Base‘𝐾))
16	5, 6, 7, 8	ltrnel 40261	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊))
17	16	3adant3r3 1185	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊))
18	5, 6, 7, 8, 13	trlnle 40308	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊)) → ¬ (𝐹‘𝑃) ≤ (𝑅‘𝐹))
19	2, 3, 17, 18	syl3anc 1373	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → ¬ (𝐹‘𝑃) ≤ (𝑅‘𝐹))
20		cdlemc3.j	. . . 4 ⊢ ∨ = (join‘𝐾)
21	12, 5, 20, 6	hlexch2 39505	. . 3 ⊢ ((𝐾 ∈ HL ∧ ((𝐹‘𝑃) ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑅‘𝐹) ∈ (Base‘𝐾)) ∧ ¬ (𝐹‘𝑃) ≤ (𝑅‘𝐹)) → ((𝐹‘𝑃) ≤ (𝑄 ∨ (𝑅‘𝐹)) → 𝑄 ≤ ((𝐹‘𝑃) ∨ (𝑅‘𝐹))))
22	1, 10, 11, 15, 19, 21	syl131anc 1385	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → ((𝐹‘𝑃) ≤ (𝑄 ∨ (𝑅‘𝐹)) → 𝑄 ≤ ((𝐹‘𝑃) ∨ (𝑅‘𝐹))))
23	5, 20, 6, 7, 8, 13	trljat2 40289	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∨ (𝑅‘𝐹)) = (𝑃 ∨ (𝐹‘𝑃)))
24	23	3adant3r3 1185	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → ((𝐹‘𝑃) ∨ (𝑅‘𝐹)) = (𝑃 ∨ (𝐹‘𝑃)))
25	24	breq2d 5107	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → (𝑄 ≤ ((𝐹‘𝑃) ∨ (𝑅‘𝐹)) ↔ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))))
26	22, 25	sylibd 239	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → ((𝐹‘𝑃) ≤ (𝑄 ∨ (𝑅‘𝐹)) → 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   class class class wbr 5095  ‘cfv 6488  (class class class)co 7354  Basecbs 17124  lecple 17172  joincjn 18221  meetcmee 18222  Atomscatm 39385  HLchlt 39472  LHypclh 40106  LTrncltrn 40223  trLctrl 40280

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 7676

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  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 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  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 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-riota 7311  df-ov 7357  df-oprab 7358  df-mpo 7359  df-1st 7929  df-2nd 7930  df-map 8760  df-proset 18204  df-poset 18223  df-plt 18238  df-lub 18254  df-glb 18255  df-join 18256  df-meet 18257  df-p0 18333  df-p1 18334  df-lat 18342  df-clat 18409  df-oposet 39298  df-ol 39300  df-oml 39301  df-covers 39388  df-ats 39389  df-atl 39420  df-cvlat 39444  df-hlat 39473  df-psubsp 39625  df-pmap 39626  df-padd 39918  df-lhyp 40110  df-laut 40111  df-ldil 40226  df-ltrn 40227  df-trl 40281

This theorem is referenced by:  cdlemc4  40316