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Theorem dalem25 38161
Description: Lemma for dath 38199. Show that the dummy center of perspectivity 𝑐 is different from auxiliary atom 𝐺. (Contributed by NM, 3-Aug-2012.)
Hypotheses
Ref Expression
dalem.ph (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))
dalem.l = (le‘𝐾)
dalem.j = (join‘𝐾)
dalem.a 𝐴 = (Atoms‘𝐾)
dalem.ps (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
dalem23.m = (meet‘𝐾)
dalem23.o 𝑂 = (LPlanes‘𝐾)
dalem23.y 𝑌 = ((𝑃 𝑄) 𝑅)
dalem23.z 𝑍 = ((𝑆 𝑇) 𝑈)
dalem23.g 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
Assertion
Ref Expression
dalem25 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝐺)

Proof of Theorem dalem25
StepHypRef Expression
1 dalem.ph . . . 4 (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))
2 dalem.l . . . 4 = (le‘𝐾)
3 dalem.j . . . 4 = (join‘𝐾)
4 dalem.a . . . 4 𝐴 = (Atoms‘𝐾)
51, 2, 3, 4dalemcnes 38113 . . 3 (𝜑𝐶𝑆)
653ad2ant1 1133 . 2 ((𝜑𝑌 = 𝑍𝜓) → 𝐶𝑆)
7 dalem.ps . . . . . . . . . . 11 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
87dalemclccjdd 38151 . . . . . . . . . 10 (𝜓𝐶 (𝑐 𝑑))
983ad2ant3 1135 . . . . . . . . 9 ((𝜑𝑌 = 𝑍𝜓) → 𝐶 (𝑐 𝑑))
109adantr 481 . . . . . . . 8 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → 𝐶 (𝑐 𝑑))
11 simpr 485 . . . . . . . . . 10 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → 𝑐 = 𝐺)
12 dalem23.g . . . . . . . . . . . . 13 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
131dalemkelat 38087 . . . . . . . . . . . . . . 15 (𝜑𝐾 ∈ Lat)
14133ad2ant1 1133 . . . . . . . . . . . . . 14 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ Lat)
151dalemkehl 38086 . . . . . . . . . . . . . . . 16 (𝜑𝐾 ∈ HL)
16153ad2ant1 1133 . . . . . . . . . . . . . . 15 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
177dalemccea 38146 . . . . . . . . . . . . . . . 16 (𝜓𝑐𝐴)
18173ad2ant3 1135 . . . . . . . . . . . . . . 15 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝐴)
191dalempea 38089 . . . . . . . . . . . . . . . 16 (𝜑𝑃𝐴)
20193ad2ant1 1133 . . . . . . . . . . . . . . 15 ((𝜑𝑌 = 𝑍𝜓) → 𝑃𝐴)
21 eqid 2736 . . . . . . . . . . . . . . . 16 (Base‘𝐾) = (Base‘𝐾)
2221, 3, 4hlatjcl 37829 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ 𝑐𝐴𝑃𝐴) → (𝑐 𝑃) ∈ (Base‘𝐾))
2316, 18, 20, 22syl3anc 1371 . . . . . . . . . . . . . 14 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑃) ∈ (Base‘𝐾))
247dalemddea 38147 . . . . . . . . . . . . . . . 16 (𝜓𝑑𝐴)
25243ad2ant3 1135 . . . . . . . . . . . . . . 15 ((𝜑𝑌 = 𝑍𝜓) → 𝑑𝐴)
261dalemsea 38092 . . . . . . . . . . . . . . . 16 (𝜑𝑆𝐴)
27263ad2ant1 1133 . . . . . . . . . . . . . . 15 ((𝜑𝑌 = 𝑍𝜓) → 𝑆𝐴)
2821, 3, 4hlatjcl 37829 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ 𝑑𝐴𝑆𝐴) → (𝑑 𝑆) ∈ (Base‘𝐾))
2916, 25, 27, 28syl3anc 1371 . . . . . . . . . . . . . 14 ((𝜑𝑌 = 𝑍𝜓) → (𝑑 𝑆) ∈ (Base‘𝐾))
30 dalem23.m . . . . . . . . . . . . . . 15 = (meet‘𝐾)
3121, 2, 30latmle2 18354 . . . . . . . . . . . . . 14 ((𝐾 ∈ Lat ∧ (𝑐 𝑃) ∈ (Base‘𝐾) ∧ (𝑑 𝑆) ∈ (Base‘𝐾)) → ((𝑐 𝑃) (𝑑 𝑆)) (𝑑 𝑆))
3214, 23, 29, 31syl3anc 1371 . . . . . . . . . . . . 13 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 𝑃) (𝑑 𝑆)) (𝑑 𝑆))
3312, 32eqbrtrid 5140 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑍𝜓) → 𝐺 (𝑑 𝑆))
343, 4hlatjcom 37830 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑑𝐴𝑆𝐴) → (𝑑 𝑆) = (𝑆 𝑑))
3516, 25, 27, 34syl3anc 1371 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑍𝜓) → (𝑑 𝑆) = (𝑆 𝑑))
3633, 35breqtrd 5131 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → 𝐺 (𝑆 𝑑))
3736adantr 481 . . . . . . . . . 10 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → 𝐺 (𝑆 𝑑))
3811, 37eqbrtrd 5127 . . . . . . . . 9 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → 𝑐 (𝑆 𝑑))
392, 3, 4hlatlej2 37838 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑑𝐴) → 𝑑 (𝑆 𝑑))
4016, 27, 25, 39syl3anc 1371 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → 𝑑 (𝑆 𝑑))
4140adantr 481 . . . . . . . . 9 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → 𝑑 (𝑆 𝑑))
427, 4dalemcceb 38152 . . . . . . . . . . . 12 (𝜓𝑐 ∈ (Base‘𝐾))
43423ad2ant3 1135 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 ∈ (Base‘𝐾))
4421, 4atbase 37751 . . . . . . . . . . . . 13 (𝑑𝐴𝑑 ∈ (Base‘𝐾))
4524, 44syl 17 . . . . . . . . . . . 12 (𝜓𝑑 ∈ (Base‘𝐾))
46453ad2ant3 1135 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → 𝑑 ∈ (Base‘𝐾))
4721, 3, 4hlatjcl 37829 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑑𝐴) → (𝑆 𝑑) ∈ (Base‘𝐾))
4816, 27, 25, 47syl3anc 1371 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → (𝑆 𝑑) ∈ (Base‘𝐾))
4921, 2, 3latjle12 18339 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ (𝑐 ∈ (Base‘𝐾) ∧ 𝑑 ∈ (Base‘𝐾) ∧ (𝑆 𝑑) ∈ (Base‘𝐾))) → ((𝑐 (𝑆 𝑑) ∧ 𝑑 (𝑆 𝑑)) ↔ (𝑐 𝑑) (𝑆 𝑑)))
5014, 43, 46, 48, 49syl13anc 1372 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 (𝑆 𝑑) ∧ 𝑑 (𝑆 𝑑)) ↔ (𝑐 𝑑) (𝑆 𝑑)))
5150adantr 481 . . . . . . . . 9 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → ((𝑐 (𝑆 𝑑) ∧ 𝑑 (𝑆 𝑑)) ↔ (𝑐 𝑑) (𝑆 𝑑)))
5238, 41, 51mpbi2and 710 . . . . . . . 8 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → (𝑐 𝑑) (𝑆 𝑑))
531, 4dalemceb 38101 . . . . . . . . . . 11 (𝜑𝐶 ∈ (Base‘𝐾))
54533ad2ant1 1133 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → 𝐶 ∈ (Base‘𝐾))
5521, 3, 4hlatjcl 37829 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑐𝐴𝑑𝐴) → (𝑐 𝑑) ∈ (Base‘𝐾))
5616, 18, 25, 55syl3anc 1371 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑑) ∈ (Base‘𝐾))
5721, 2lattr 18333 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝐶 ∈ (Base‘𝐾) ∧ (𝑐 𝑑) ∈ (Base‘𝐾) ∧ (𝑆 𝑑) ∈ (Base‘𝐾))) → ((𝐶 (𝑐 𝑑) ∧ (𝑐 𝑑) (𝑆 𝑑)) → 𝐶 (𝑆 𝑑)))
5814, 54, 56, 48, 57syl13anc 1372 . . . . . . . . 9 ((𝜑𝑌 = 𝑍𝜓) → ((𝐶 (𝑐 𝑑) ∧ (𝑐 𝑑) (𝑆 𝑑)) → 𝐶 (𝑆 𝑑)))
5958adantr 481 . . . . . . . 8 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → ((𝐶 (𝑐 𝑑) ∧ (𝑐 𝑑) (𝑆 𝑑)) → 𝐶 (𝑆 𝑑)))
6010, 52, 59mp2and 697 . . . . . . 7 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → 𝐶 (𝑆 𝑑))
61 dalem23.o . . . . . . . . . . 11 𝑂 = (LPlanes‘𝐾)
621, 61dalemyeb 38112 . . . . . . . . . 10 (𝜑𝑌 ∈ (Base‘𝐾))
63623ad2ant1 1133 . . . . . . . . 9 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 ∈ (Base‘𝐾))
6421, 2, 30latmlem1 18358 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝐶 ∈ (Base‘𝐾) ∧ (𝑆 𝑑) ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾))) → (𝐶 (𝑆 𝑑) → (𝐶 𝑌) ((𝑆 𝑑) 𝑌)))
6514, 54, 48, 63, 64syl13anc 1372 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → (𝐶 (𝑆 𝑑) → (𝐶 𝑌) ((𝑆 𝑑) 𝑌)))
6665adantr 481 . . . . . . 7 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → (𝐶 (𝑆 𝑑) → (𝐶 𝑌) ((𝑆 𝑑) 𝑌)))
6760, 66mpd 15 . . . . . 6 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → (𝐶 𝑌) ((𝑆 𝑑) 𝑌))
68 dalem23.y . . . . . . . . . 10 𝑌 = ((𝑃 𝑄) 𝑅)
69 dalem23.z . . . . . . . . . 10 𝑍 = ((𝑆 𝑇) 𝑈)
701, 2, 3, 4, 61, 68, 69dalem17 38143 . . . . . . . . 9 ((𝜑𝑌 = 𝑍) → 𝐶 𝑌)
71703adant3 1132 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝐶 𝑌)
7221, 2, 30latleeqm1 18356 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝐶 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (𝐶 𝑌 ↔ (𝐶 𝑌) = 𝐶))
7314, 54, 63, 72syl3anc 1371 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → (𝐶 𝑌 ↔ (𝐶 𝑌) = 𝐶))
7471, 73mpbid 231 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → (𝐶 𝑌) = 𝐶)
7574adantr 481 . . . . . 6 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → (𝐶 𝑌) = 𝐶)
761, 2, 3, 4, 69dalemsly 38118 . . . . . . . . 9 ((𝜑𝑌 = 𝑍) → 𝑆 𝑌)
77763adant3 1132 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝑆 𝑌)
787dalem-ddly 38149 . . . . . . . . 9 (𝜓 → ¬ 𝑑 𝑌)
79783ad2ant3 1135 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑑 𝑌)
8021, 2, 3, 30, 42atjm 37908 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑆𝐴𝑑𝐴𝑌 ∈ (Base‘𝐾)) ∧ (𝑆 𝑌 ∧ ¬ 𝑑 𝑌)) → ((𝑆 𝑑) 𝑌) = 𝑆)
8116, 27, 25, 63, 77, 79, 80syl132anc 1388 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → ((𝑆 𝑑) 𝑌) = 𝑆)
8281adantr 481 . . . . . 6 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → ((𝑆 𝑑) 𝑌) = 𝑆)
8367, 75, 823brtr3d 5136 . . . . 5 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → 𝐶 𝑆)
84 hlatl 37822 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
8515, 84syl 17 . . . . . . . 8 (𝜑𝐾 ∈ AtLat)
861, 2, 3, 4, 61, 68dalemcea 38123 . . . . . . . 8 (𝜑𝐶𝐴)
872, 4atcmp 37773 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝐶𝐴𝑆𝐴) → (𝐶 𝑆𝐶 = 𝑆))
8885, 86, 26, 87syl3anc 1371 . . . . . . 7 (𝜑 → (𝐶 𝑆𝐶 = 𝑆))
89883ad2ant1 1133 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → (𝐶 𝑆𝐶 = 𝑆))
9089adantr 481 . . . . 5 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → (𝐶 𝑆𝐶 = 𝑆))
9183, 90mpbid 231 . . . 4 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → 𝐶 = 𝑆)
9291ex 413 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 = 𝐺𝐶 = 𝑆))
9392necon3d 2964 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝐶𝑆𝑐𝐺))
946, 93mpd 15 1 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝐺)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943   class class class wbr 5105  cfv 6496  (class class class)co 7357  Basecbs 17083  lecple 17140  joincjn 18200  meetcmee 18201  Latclat 18320  Atomscatm 37725  AtLatcal 37726  HLchlt 37812  LPlanesclpl 37955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-proset 18184  df-poset 18202  df-plt 18219  df-lub 18235  df-glb 18236  df-join 18237  df-meet 18238  df-p0 18314  df-lat 18321  df-clat 18388  df-oposet 37638  df-ol 37640  df-oml 37641  df-covers 37728  df-ats 37729  df-atl 37760  df-cvlat 37784  df-hlat 37813  df-llines 37961  df-lplanes 37962
This theorem is referenced by:  dalem28  38163  dalem31N  38166
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