Theorem cdlemg16 37797

Description: Part of proof of Lemma G of [Crawley] p. 116; 2nd line p. 117, which says that (our) cdlemg10 37781 "implies (2)" (of p. 116). No details are provided by the authors, so there may be a shorter proof; but ours requires the 14 lemmas, one using Desargues's law dalaw 37026, in order to make this inference. This final step eliminates the (𝑅‘𝐹) ≠ (𝑅‘𝐺) condition from cdlemg12 37790. TODO: FIX COMMENT. TODO: should we also eliminate 𝑃 ≠ 𝑄 here (or earlier)? Do it if we don't need to add it in for something else later. (Contributed by NM, 6-May-2013.)

Hypotheses

Ref	Expression
cdlemg12.l	⊢ ≤ = (le‘𝐾)
cdlemg12.j	⊢ ∨ = (join‘𝐾)
cdlemg12.m	⊢ ∧ = (meet‘𝐾)
cdlemg12.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemg12.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemg12.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemg12b.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)

Assertion

Ref	Expression
cdlemg16	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄))) → ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ 𝑊) = ((𝑄 ∨ (𝐹‘(𝐺‘𝑄))) ∧ 𝑊))

Proof of Theorem cdlemg16

Step	Hyp	Ref	Expression
1		simpl1 1187	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄))) ∧ (𝑅‘𝐹) = (𝑅‘𝐺)) → ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)))
2		simpl21 1247	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄))) ∧ (𝑅‘𝐹) = (𝑅‘𝐺)) → 𝐹 ∈ 𝑇)
3		simpl22 1248	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄))) ∧ (𝑅‘𝐹) = (𝑅‘𝐺)) → 𝐺 ∈ 𝑇)
4		simpr 487	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄))) ∧ (𝑅‘𝐹) = (𝑅‘𝐺)) → (𝑅‘𝐹) = (𝑅‘𝐺))
5		cdlemg12.l	. . . 4 ⊢ ≤ = (le‘𝐾)
6		cdlemg12.j	. . . 4 ⊢ ∨ = (join‘𝐾)
7		cdlemg12.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
8		cdlemg12.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
9		cdlemg12.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
10		cdlemg12.t	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
11		cdlemg12b.r	. . . 4 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
12	5, 6, 7, 8, 9, 10, 11	cdlemg15 37796	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝐺)) → ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ 𝑊) = ((𝑄 ∨ (𝐹‘(𝐺‘𝑄))) ∧ 𝑊))
13	1, 2, 3, 4, 12	syl121anc 1371	. 2 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄))) ∧ (𝑅‘𝐹) = (𝑅‘𝐺)) → ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ 𝑊) = ((𝑄 ∨ (𝐹‘(𝐺‘𝑄))) ∧ 𝑊))
14		simpl1 1187	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄))) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺)) → ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)))
15		simpl2 1188	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄))) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺)) → (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄))
16		simpl31 1250	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄))) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺)) → ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))
17		simpl32 1251	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄))) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺)) → ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))
18	16, 17	jca 514	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄))) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺)) → (¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄)))
19		simpr 487	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄))) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺)) → (𝑅‘𝐹) ≠ (𝑅‘𝐺))
20		simpl33 1252	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄))) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺)) → ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄))
21	5, 6, 7, 8, 9, 10, 11	cdlemg12 37790	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄))) → ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ 𝑊) = ((𝑄 ∨ (𝐹‘(𝐺‘𝑄))) ∧ 𝑊))
22	14, 15, 18, 19, 20, 21	syl113anc 1378	. 2 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄))) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺)) → ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ 𝑊) = ((𝑄 ∨ (𝐹‘(𝐺‘𝑄))) ∧ 𝑊))
23	13, 22	pm2.61dane 3107	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ((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 398   ∧ w3a 1083   = wceq 1536   ∈ wcel 2113   ≠ wne 3019   class class class wbr 5069  ‘cfv 6358  (class class class)co 7159  lecple 16575  joincjn 17557  meetcmee 17558  Atomscatm 36403  HLchlt 36490  LHypclh 37124  LTrncltrn 37241  trLctrl 37298

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-riotaBAD 36093

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rmo 3149  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-iun 4924  df-iin 4925  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7117  df-ov 7162  df-oprab 7163  df-mpo 7164  df-1st 7692  df-2nd 7693  df-undef 7942  df-map 8411  df-proset 17541  df-poset 17559  df-plt 17571  df-lub 17587  df-glb 17588  df-join 17589  df-meet 17590  df-p0 17652  df-p1 17653  df-lat 17659  df-clat 17721  df-oposet 36316  df-ol 36318  df-oml 36319  df-covers 36406  df-ats 36407  df-atl 36438  df-cvlat 36462  df-hlat 36491  df-llines 36638  df-lplanes 36639  df-lvols 36640  df-lines 36641  df-psubsp 36643  df-pmap 36644  df-padd 36936  df-lhyp 37128  df-laut 37129  df-ldil 37244  df-ltrn 37245  df-trl 37299

This theorem is referenced by:  cdlemg16z  37799