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Theorem cdlemg6c 37750
 Description: TODO: FIX COMMENT. (Contributed by NM, 27-Apr-2013.)
Hypotheses
Ref Expression
cdlemg4.l = (le‘𝐾)
cdlemg4.a 𝐴 = (Atoms‘𝐾)
cdlemg4.h 𝐻 = (LHyp‘𝐾)
cdlemg4.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemg4.r 𝑅 = ((trL‘𝐾)‘𝑊)
cdlemg4.j = (join‘𝐾)
cdlemg4b.v 𝑉 = (𝑅𝐺)
Assertion
Ref Expression
cdlemg6c (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → (((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ ¬ 𝑟 (𝑃 𝑉)) → (𝐹‘(𝐺𝑄)) = 𝑄))
Distinct variable groups:   𝐴,𝑟   𝐹,𝑟   𝐺,𝑟   𝐻,𝑟   ,𝑟   𝐾,𝑟   ,𝑟   𝑃,𝑟   𝑄,𝑟   𝑇,𝑟   𝑉,𝑟   𝑊,𝑟
Allowed substitution hint:   𝑅(𝑟)

Proof of Theorem cdlemg6c
StepHypRef Expression
1 simpl1 1187 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ ¬ 𝑟 (𝑃 𝑉))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simprl 769 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ ¬ 𝑟 (𝑃 𝑉))) → (𝑟𝐴 ∧ ¬ 𝑟 𝑊))
3 simpl22 1248 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ ¬ 𝑟 (𝑃 𝑉))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
4 simpl23 1249 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ ¬ 𝑟 (𝑃 𝑉))) → 𝐹𝑇)
5 simpl31 1250 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ ¬ 𝑟 (𝑃 𝑉))) → 𝐺𝑇)
6 simprr 771 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ ¬ 𝑟 (𝑃 𝑉))) → ¬ 𝑟 (𝑃 𝑉))
7 simpl1l 1220 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ ¬ 𝑟 (𝑃 𝑉))) → 𝐾 ∈ HL)
8 simp22l 1288 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → 𝑄𝐴)
98adantr 483 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ ¬ 𝑟 (𝑃 𝑉))) → 𝑄𝐴)
10 simprll 777 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ ¬ 𝑟 (𝑃 𝑉))) → 𝑟𝐴)
11 cdlemg4b.v . . . . . . 7 𝑉 = (𝑅𝐺)
12 eqid 2821 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
13 cdlemg4.h . . . . . . . . 9 𝐻 = (LHyp‘𝐾)
14 cdlemg4.t . . . . . . . . 9 𝑇 = ((LTrn‘𝐾)‘𝑊)
15 cdlemg4.r . . . . . . . . 9 𝑅 = ((trL‘𝐾)‘𝑊)
1612, 13, 14, 15trlcl 37294 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇) → (𝑅𝐺) ∈ (Base‘𝐾))
171, 5, 16syl2anc 586 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ ¬ 𝑟 (𝑃 𝑉))) → (𝑅𝐺) ∈ (Base‘𝐾))
1811, 17eqeltrid 2917 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ ¬ 𝑟 (𝑃 𝑉))) → 𝑉 ∈ (Base‘𝐾))
19 simp22r 1289 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → ¬ 𝑄 𝑊)
2019adantr 483 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ ¬ 𝑟 (𝑃 𝑉))) → ¬ 𝑄 𝑊)
21 cdlemg4.l . . . . . . . . . . 11 = (le‘𝐾)
2221, 13, 14, 15trlle 37314 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇) → (𝑅𝐺) 𝑊)
231, 5, 22syl2anc 586 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ ¬ 𝑟 (𝑃 𝑉))) → (𝑅𝐺) 𝑊)
2411, 23eqbrtrid 5093 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ ¬ 𝑟 (𝑃 𝑉))) → 𝑉 𝑊)
25 simp1l 1193 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → 𝐾 ∈ HL)
2625hllatd 36494 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → 𝐾 ∈ Lat)
2726adantr 483 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ ¬ 𝑟 (𝑃 𝑉))) → 𝐾 ∈ Lat)
28 cdlemg4.a . . . . . . . . . . . 12 𝐴 = (Atoms‘𝐾)
2912, 28atbase 36419 . . . . . . . . . . 11 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
308, 29syl 17 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → 𝑄 ∈ (Base‘𝐾))
3130adantr 483 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ ¬ 𝑟 (𝑃 𝑉))) → 𝑄 ∈ (Base‘𝐾))
32 simp1r 1194 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → 𝑊𝐻)
3312, 13lhpbase 37128 . . . . . . . . . . 11 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
3432, 33syl 17 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → 𝑊 ∈ (Base‘𝐾))
3534adantr 483 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ ¬ 𝑟 (𝑃 𝑉))) → 𝑊 ∈ (Base‘𝐾))
3612, 21lattr 17660 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑄 𝑉𝑉 𝑊) → 𝑄 𝑊))
3727, 31, 18, 35, 36syl13anc 1368 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ ¬ 𝑟 (𝑃 𝑉))) → ((𝑄 𝑉𝑉 𝑊) → 𝑄 𝑊))
3824, 37mpan2d 692 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ ¬ 𝑟 (𝑃 𝑉))) → (𝑄 𝑉𝑄 𝑊))
3920, 38mtod 200 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ ¬ 𝑟 (𝑃 𝑉))) → ¬ 𝑄 𝑉)
40 cdlemg4.j . . . . . . 7 = (join‘𝐾)
4112, 21, 40, 28hlexch2 36513 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑟𝐴𝑉 ∈ (Base‘𝐾)) ∧ ¬ 𝑄 𝑉) → (𝑄 (𝑟 𝑉) → 𝑟 (𝑄 𝑉)))
427, 9, 10, 18, 39, 41syl131anc 1379 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ ¬ 𝑟 (𝑃 𝑉))) → (𝑄 (𝑟 𝑉) → 𝑟 (𝑄 𝑉)))
43 simpl32 1251 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ ¬ 𝑟 (𝑃 𝑉))) → 𝑄 (𝑃 𝑉))
44 simp21l 1286 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → 𝑃𝐴)
4544adantr 483 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ ¬ 𝑟 (𝑃 𝑉))) → 𝑃𝐴)
4612, 28atbase 36419 . . . . . . . . 9 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
4745, 46syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ ¬ 𝑟 (𝑃 𝑉))) → 𝑃 ∈ (Base‘𝐾))
4812, 21, 40latlej2 17665 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾)) → 𝑉 (𝑃 𝑉))
4927, 47, 18, 48syl3anc 1367 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ ¬ 𝑟 (𝑃 𝑉))) → 𝑉 (𝑃 𝑉))
5012, 40latjcl 17655 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾)) → (𝑃 𝑉) ∈ (Base‘𝐾))
5127, 47, 18, 50syl3anc 1367 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ ¬ 𝑟 (𝑃 𝑉))) → (𝑃 𝑉) ∈ (Base‘𝐾))
5212, 21, 40latjle12 17666 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾) ∧ (𝑃 𝑉) ∈ (Base‘𝐾))) → ((𝑄 (𝑃 𝑉) ∧ 𝑉 (𝑃 𝑉)) ↔ (𝑄 𝑉) (𝑃 𝑉)))
5327, 31, 18, 51, 52syl13anc 1368 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ ¬ 𝑟 (𝑃 𝑉))) → ((𝑄 (𝑃 𝑉) ∧ 𝑉 (𝑃 𝑉)) ↔ (𝑄 𝑉) (𝑃 𝑉)))
5443, 49, 53mpbi2and 710 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ ¬ 𝑟 (𝑃 𝑉))) → (𝑄 𝑉) (𝑃 𝑉))
5512, 28atbase 36419 . . . . . . . 8 (𝑟𝐴𝑟 ∈ (Base‘𝐾))
5610, 55syl 17 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ ¬ 𝑟 (𝑃 𝑉))) → 𝑟 ∈ (Base‘𝐾))
5712, 40latjcl 17655 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾)) → (𝑄 𝑉) ∈ (Base‘𝐾))
5827, 31, 18, 57syl3anc 1367 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ ¬ 𝑟 (𝑃 𝑉))) → (𝑄 𝑉) ∈ (Base‘𝐾))
5912, 21lattr 17660 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑟 ∈ (Base‘𝐾) ∧ (𝑄 𝑉) ∈ (Base‘𝐾) ∧ (𝑃 𝑉) ∈ (Base‘𝐾))) → ((𝑟 (𝑄 𝑉) ∧ (𝑄 𝑉) (𝑃 𝑉)) → 𝑟 (𝑃 𝑉)))
6027, 56, 58, 51, 59syl13anc 1368 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ ¬ 𝑟 (𝑃 𝑉))) → ((𝑟 (𝑄 𝑉) ∧ (𝑄 𝑉) (𝑃 𝑉)) → 𝑟 (𝑃 𝑉)))
6154, 60mpan2d 692 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ ¬ 𝑟 (𝑃 𝑉))) → (𝑟 (𝑄 𝑉) → 𝑟 (𝑃 𝑉)))
6242, 61syld 47 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ ¬ 𝑟 (𝑃 𝑉))) → (𝑄 (𝑟 𝑉) → 𝑟 (𝑃 𝑉)))
636, 62mtod 200 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ ¬ 𝑟 (𝑃 𝑉))) → ¬ 𝑄 (𝑟 𝑉))
64 simpl21 1247 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ ¬ 𝑟 (𝑃 𝑉))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
65 simpl33 1252 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ ¬ 𝑟 (𝑃 𝑉))) → (𝐹‘(𝐺𝑃)) = 𝑃)
6621, 28, 13, 14, 15, 40, 11cdlemg6a 37748 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ¬ 𝑟 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → (𝐹‘(𝐺𝑟)) = 𝑟)
671, 64, 2, 4, 5, 6, 65, 66syl133anc 1389 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ ¬ 𝑟 (𝑃 𝑉))) → (𝐹‘(𝐺𝑟)) = 𝑟)
6821, 28, 13, 14, 15, 40, 11cdlemg6b 37749 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ¬ 𝑄 (𝑟 𝑉) ∧ (𝐹‘(𝐺𝑟)) = 𝑟)) → (𝐹‘(𝐺𝑄)) = 𝑄)
691, 2, 3, 4, 5, 63, 67, 68syl133anc 1389 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) ∧ ((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ ¬ 𝑟 (𝑃 𝑉))) → (𝐹‘(𝐺𝑄)) = 𝑄)
7069ex 415 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → (((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ ¬ 𝑟 (𝑃 𝑉)) → (𝐹‘(𝐺𝑄)) = 𝑄))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110   class class class wbr 5058  ‘cfv 6349  (class class class)co 7150  Basecbs 16477  lecple 16566  joincjn 17548  Latclat 17649  Atomscatm 36393  HLchlt 36480  LHypclh 37114  LTrncltrn 37231  trLctrl 37288 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-riotaBAD 36083 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-iin 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7683  df-2nd 7684  df-undef 7933  df-map 8402  df-proset 17532  df-poset 17550  df-plt 17562  df-lub 17578  df-glb 17579  df-join 17580  df-meet 17581  df-p0 17643  df-p1 17644  df-lat 17650  df-clat 17712  df-oposet 36306  df-ol 36308  df-oml 36309  df-covers 36396  df-ats 36397  df-atl 36428  df-cvlat 36452  df-hlat 36481  df-llines 36628  df-lplanes 36629  df-lvols 36630  df-lines 36631  df-psubsp 36633  df-pmap 36634  df-padd 36926  df-lhyp 37118  df-laut 37119  df-ldil 37234  df-ltrn 37235  df-trl 37289 This theorem is referenced by:  cdlemg6d  37751
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