Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme32f Structured version   Visualization version   GIF version

Theorem cdleme32f 34548
Description: Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 20-Feb-2013.)
Hypotheses
Ref Expression
cdleme32.b 𝐵 = (Base‘𝐾)
cdleme32.l = (le‘𝐾)
cdleme32.j = (join‘𝐾)
cdleme32.m = (meet‘𝐾)
cdleme32.a 𝐴 = (Atoms‘𝐾)
cdleme32.h 𝐻 = (LHyp‘𝐾)
cdleme32.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdleme32.c 𝐶 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
cdleme32.d 𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
cdleme32.e 𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))
cdleme32.i 𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸))
cdleme32.n 𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐶)
cdleme32.o 𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))
cdleme32.f 𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))
Assertion
Ref Expression
cdleme32f ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) → (𝐹𝑋) (𝐹𝑌))
Distinct variable groups:   𝑡,𝑠,𝑥,𝑦,𝑧,𝐴   𝐵,𝑠,𝑡,𝑥,𝑦,𝑧   𝑦,𝐶   𝐷,𝑠,𝑦,𝑧   𝑦,𝐸   𝐻,𝑠,𝑡   ,𝑠,𝑡,𝑥,𝑦,𝑧   𝐾,𝑠,𝑡   ,𝑠,𝑡,𝑥,𝑦,𝑧   ,𝑠,𝑡,𝑥,𝑦,𝑧   𝑥,𝑁,𝑧   𝑃,𝑠,𝑡,𝑥,𝑦,𝑧   𝑄,𝑠,𝑡,𝑥,𝑦,𝑧   𝑈,𝑠,𝑡,𝑥,𝑦,𝑧   𝑊,𝑠,𝑡,𝑥,𝑦,𝑧   𝑋,𝑠,𝑡,𝑥,𝑧   𝑦,𝐻   𝑦,𝐾   𝑦,𝑌   𝑧,𝐻   𝑧,𝐾   𝑌,𝑠,𝑡,𝑥,𝑧
Allowed substitution hints:   𝐶(𝑥,𝑧,𝑡,𝑠)   𝐷(𝑥,𝑡)   𝐸(𝑥,𝑧,𝑡,𝑠)   𝐹(𝑥,𝑦,𝑧,𝑡,𝑠)   𝐻(𝑥)   𝐼(𝑥,𝑦,𝑧,𝑡,𝑠)   𝐾(𝑥)   𝑁(𝑦,𝑡,𝑠)   𝑂(𝑥,𝑦,𝑧,𝑡,𝑠)   𝑋(𝑦)

Proof of Theorem cdleme32f
StepHypRef Expression
1 simp11 1083 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simp21r 1171 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) → 𝑌𝐵)
3 simp23r 1175 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) → ¬ 𝑌 𝑊)
4 cdleme32.b . . . 4 𝐵 = (Base‘𝐾)
5 cdleme32.l . . . 4 = (le‘𝐾)
6 cdleme32.j . . . 4 = (join‘𝐾)
7 cdleme32.m . . . 4 = (meet‘𝐾)
8 cdleme32.a . . . 4 𝐴 = (Atoms‘𝐾)
9 cdleme32.h . . . 4 𝐻 = (LHyp‘𝐾)
104, 5, 6, 7, 8, 9lhpmcvr2 34124 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) → ∃𝑠𝐴𝑠 𝑊 ∧ (𝑠 (𝑌 𝑊)) = 𝑌))
111, 2, 3, 10syl12anc 1315 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) → ∃𝑠𝐴𝑠 𝑊 ∧ (𝑠 (𝑌 𝑊)) = 𝑌))
12 nfv 1829 . . 3 𝑠(((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌)
13 cdleme32.f . . . . . 6 𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))
14 nfcv 2750 . . . . . . 7 𝑠𝐵
15 nfv 1829 . . . . . . . 8 𝑠(𝑃𝑄 ∧ ¬ 𝑥 𝑊)
16 cdleme32.o . . . . . . . . 9 𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))
17 nfra1 2924 . . . . . . . . . 10 𝑠𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊)))
1817, 14nfriota 6498 . . . . . . . . 9 𝑠(𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))
1916, 18nfcxfr 2748 . . . . . . . 8 𝑠𝑂
20 nfcv 2750 . . . . . . . 8 𝑠𝑥
2115, 19, 20nfif 4064 . . . . . . 7 𝑠if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥)
2214, 21nfmpt 4668 . . . . . 6 𝑠(𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))
2313, 22nfcxfr 2748 . . . . 5 𝑠𝐹
24 nfcv 2750 . . . . 5 𝑠𝑋
2523, 24nffv 6095 . . . 4 𝑠(𝐹𝑋)
26 nfcv 2750 . . . 4 𝑠
27 nfcv 2750 . . . . 5 𝑠𝑌
2823, 27nffv 6095 . . . 4 𝑠(𝐹𝑌)
2925, 26, 28nfbr 4623 . . 3 𝑠(𝐹𝑋) (𝐹𝑌)
30 simpl1 1056 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊 ∧ (𝑠 (𝑌 𝑊)) = 𝑌))) → ((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)))
31 simpl2 1057 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊 ∧ (𝑠 (𝑌 𝑊)) = 𝑌))) → ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)))
32 simprl 789 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊 ∧ (𝑠 (𝑌 𝑊)) = 𝑌))) → 𝑠𝐴)
33 simprrl 799 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊 ∧ (𝑠 (𝑌 𝑊)) = 𝑌))) → ¬ 𝑠 𝑊)
3432, 33jca 552 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊 ∧ (𝑠 (𝑌 𝑊)) = 𝑌))) → (𝑠𝐴 ∧ ¬ 𝑠 𝑊))
35 simprrr 800 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊 ∧ (𝑠 (𝑌 𝑊)) = 𝑌))) → (𝑠 (𝑌 𝑊)) = 𝑌)
36 simpl3 1058 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊 ∧ (𝑠 (𝑌 𝑊)) = 𝑌))) → 𝑋 𝑌)
37 cdleme32.u . . . . . 6 𝑈 = ((𝑃 𝑄) 𝑊)
38 cdleme32.c . . . . . 6 𝐶 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
39 cdleme32.d . . . . . 6 𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
40 cdleme32.e . . . . . 6 𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))
41 cdleme32.i . . . . . 6 𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸))
42 cdleme32.n . . . . . 6 𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐶)
434, 5, 6, 7, 8, 9, 37, 38, 39, 40, 41, 42, 16, 13cdleme32e 34547 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝐹𝑋) (𝐹𝑌))
4430, 31, 34, 35, 36, 43syl113anc 1329 . . . 4 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊 ∧ (𝑠 (𝑌 𝑊)) = 𝑌))) → (𝐹𝑋) (𝐹𝑌))
4544exp32 628 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) → (𝑠𝐴 → ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑌 𝑊)) = 𝑌) → (𝐹𝑋) (𝐹𝑌))))
4612, 29, 45rexlimd 3007 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) → (∃𝑠𝐴𝑠 𝑊 ∧ (𝑠 (𝑌 𝑊)) = 𝑌) → (𝐹𝑋) (𝐹𝑌)))
4711, 46mpd 15 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) → (𝐹𝑋) (𝐹𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1976  wne 2779  wral 2895  wrex 2896  ifcif 4035   class class class wbr 4577  cmpt 4637  cfv 5790  crio 6488  (class class class)co 6527  Basecbs 15641  lecple 15721  joincjn 16713  meetcmee 16714  Atomscatm 33364  HLchlt 33451  LHypclh 34084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-riotaBAD 33053
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-iin 4452  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7036  df-2nd 7037  df-undef 7263  df-preset 16697  df-poset 16715  df-plt 16727  df-lub 16743  df-glb 16744  df-join 16745  df-meet 16746  df-p0 16808  df-p1 16809  df-lat 16815  df-clat 16877  df-oposet 33277  df-ol 33279  df-oml 33280  df-covers 33367  df-ats 33368  df-atl 33399  df-cvlat 33423  df-hlat 33452  df-llines 33598  df-lplanes 33599  df-lvols 33600  df-lines 33601  df-psubsp 33603  df-pmap 33604  df-padd 33896  df-lhyp 34088
This theorem is referenced by:  cdleme32le  34549
  Copyright terms: Public domain W3C validator