Theorem cdlemg17dN 40682

Description: TODO: fix comment. (Contributed by NM, 9-May-2013.) (New usage is discouraged.)

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
cdlemg17dN	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ ((𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)) ∧ (𝐺‘𝑃) ≠ 𝑃)) → (𝑅‘𝐺) = ((𝑃 ∨ 𝑄) ∧ 𝑊))

Distinct variable groups:   𝐴,𝑟   𝐺,𝑟   ∨ ,𝑟   ≤ ,𝑟   𝑃,𝑟   𝑄,𝑟   𝑊,𝑟

Allowed substitution hints:   𝑅(𝑟)   𝑇(𝑟)   𝐻(𝑟)   𝐾(𝑟)   ∧ (𝑟)

Proof of Theorem cdlemg17dN

Step	Hyp	Ref	Expression
1		simp1 1136	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ ((𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)) ∧ (𝐺‘𝑃) ≠ 𝑃)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐺 ∈ 𝑇))
2		simp21 1207	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ ((𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)) ∧ (𝐺‘𝑃) ≠ 𝑃)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
3		simpl1 1192	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐾 ∈ HL)
4		simpl2 1193	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑊 ∈ 𝐻)
5		simpl3 1194	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐺 ∈ 𝑇)
6		simpr 484	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
7		cdlemg12.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
8		cdlemg12.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
9		cdlemg12.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
10		cdlemg12.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
11		cdlemg12.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
12		cdlemg12.t	. . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
13		cdlemg12b.r	. . . . 5 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
14	7, 8, 9, 10, 11, 12, 13	trlval2 40182	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐺) = ((𝑃 ∨ (𝐺‘𝑃)) ∧ 𝑊))
15	3, 4, 5, 6, 14	syl211anc 1378	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐺) = ((𝑃 ∨ (𝐺‘𝑃)) ∧ 𝑊))
16	1, 2, 15	syl2anc 584	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ ((𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)) ∧ (𝐺‘𝑃) ≠ 𝑃)) → (𝑅‘𝐺) = ((𝑃 ∨ (𝐺‘𝑃)) ∧ 𝑊))
17		simp11 1204	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ ((𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)) ∧ (𝐺‘𝑃) ≠ 𝑃)) → 𝐾 ∈ HL)
18		simp12 1205	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ ((𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)) ∧ (𝐺‘𝑃) ≠ 𝑃)) → 𝑊 ∈ 𝐻)
19	17, 18	jca 511	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ ((𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)) ∧ (𝐺‘𝑃) ≠ 𝑃)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
20		simp22 1208	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ ((𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)) ∧ (𝐺‘𝑃) ≠ 𝑃)) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
21		simp13 1206	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ ((𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)) ∧ (𝐺‘𝑃) ≠ 𝑃)) → 𝐺 ∈ 𝑇)
22		simp23 1209	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ ((𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)) ∧ (𝐺‘𝑃) ≠ 𝑃)) → 𝑃 ≠ 𝑄)
23		simp33 1212	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ ((𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)) ∧ (𝐺‘𝑃) ≠ 𝑃)) → (𝐺‘𝑃) ≠ 𝑃)
24		simp31 1210	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ ((𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)) ∧ (𝐺‘𝑃) ≠ 𝑃)) → (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))
25		simp32 1211	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ ((𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)) ∧ (𝐺‘𝑃) ≠ 𝑃)) → ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))
26	7, 8, 9, 10, 11, 12, 13	cdlemg17b 40681	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (𝐺‘𝑃) = 𝑄)
27	19, 2, 20, 21, 22, 23, 24, 25, 26	syl323anc 1402	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ ((𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)) ∧ (𝐺‘𝑃) ≠ 𝑃)) → (𝐺‘𝑃) = 𝑄)
28	27	oveq2d 7421	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ ((𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)) ∧ (𝐺‘𝑃) ≠ 𝑃)) → (𝑃 ∨ (𝐺‘𝑃)) = (𝑃 ∨ 𝑄))
29	28	oveq1d 7420	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ ((𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)) ∧ (𝐺‘𝑃) ≠ 𝑃)) → ((𝑃 ∨ (𝐺‘𝑃)) ∧ 𝑊) = ((𝑃 ∨ 𝑄) ∧ 𝑊))
30	16, 29	eqtrd 2770	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ ((𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)) ∧ (𝐺‘𝑃) ≠ 𝑃)) → (𝑅‘𝐺) = ((𝑃 ∨ 𝑄) ∧ 𝑊))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108   ≠ wne 2932  ∃wrex 3060   class class class wbr 5119  ‘cfv 6531  (class class class)co 7405  lecple 17278  joincjn 18323  meetcmee 18324  Atomscatm 39281  HLchlt 39368  LHypclh 40003  LTrncltrn 40120  trLctrl 40177

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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-iin 4970  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7988  df-2nd 7989  df-map 8842  df-proset 18306  df-poset 18325  df-plt 18340  df-lub 18356  df-glb 18357  df-join 18358  df-meet 18359  df-p0 18435  df-p1 18436  df-lat 18442  df-clat 18509  df-oposet 39194  df-ol 39196  df-oml 39197  df-covers 39284  df-ats 39285  df-atl 39316  df-cvlat 39340  df-hlat 39369  df-psubsp 39522  df-pmap 39523  df-padd 39815  df-lhyp 40007  df-laut 40008  df-ldil 40123  df-ltrn 40124  df-trl 40178

This theorem is referenced by: (None)