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Theorem dalem25 36953
Description: Lemma for dath 36991. 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 36905 . . 3 (𝜑𝐶𝑆)
653ad2ant1 1130 . 2 ((𝜑𝑌 = 𝑍𝜓) → 𝐶𝑆)
7 dalem.ps . . . . . . . . . . 11 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
87dalemclccjdd 36943 . . . . . . . . . 10 (𝜓𝐶 (𝑐 𝑑))
983ad2ant3 1132 . . . . . . . . 9 ((𝜑𝑌 = 𝑍𝜓) → 𝐶 (𝑐 𝑑))
109adantr 484 . . . . . . . 8 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → 𝐶 (𝑐 𝑑))
11 simpr 488 . . . . . . . . . 10 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → 𝑐 = 𝐺)
12 dalem23.g . . . . . . . . . . . . 13 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
131dalemkelat 36879 . . . . . . . . . . . . . . 15 (𝜑𝐾 ∈ Lat)
14133ad2ant1 1130 . . . . . . . . . . . . . 14 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ Lat)
151dalemkehl 36878 . . . . . . . . . . . . . . . 16 (𝜑𝐾 ∈ HL)
16153ad2ant1 1130 . . . . . . . . . . . . . . 15 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
177dalemccea 36938 . . . . . . . . . . . . . . . 16 (𝜓𝑐𝐴)
18173ad2ant3 1132 . . . . . . . . . . . . . . 15 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝐴)
191dalempea 36881 . . . . . . . . . . . . . . . 16 (𝜑𝑃𝐴)
20193ad2ant1 1130 . . . . . . . . . . . . . . 15 ((𝜑𝑌 = 𝑍𝜓) → 𝑃𝐴)
21 eqid 2822 . . . . . . . . . . . . . . . 16 (Base‘𝐾) = (Base‘𝐾)
2221, 3, 4hlatjcl 36622 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ 𝑐𝐴𝑃𝐴) → (𝑐 𝑃) ∈ (Base‘𝐾))
2316, 18, 20, 22syl3anc 1368 . . . . . . . . . . . . . 14 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑃) ∈ (Base‘𝐾))
247dalemddea 36939 . . . . . . . . . . . . . . . 16 (𝜓𝑑𝐴)
25243ad2ant3 1132 . . . . . . . . . . . . . . 15 ((𝜑𝑌 = 𝑍𝜓) → 𝑑𝐴)
261dalemsea 36884 . . . . . . . . . . . . . . . 16 (𝜑𝑆𝐴)
27263ad2ant1 1130 . . . . . . . . . . . . . . 15 ((𝜑𝑌 = 𝑍𝜓) → 𝑆𝐴)
2821, 3, 4hlatjcl 36622 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ 𝑑𝐴𝑆𝐴) → (𝑑 𝑆) ∈ (Base‘𝐾))
2916, 25, 27, 28syl3anc 1368 . . . . . . . . . . . . . 14 ((𝜑𝑌 = 𝑍𝜓) → (𝑑 𝑆) ∈ (Base‘𝐾))
30 dalem23.m . . . . . . . . . . . . . . 15 = (meet‘𝐾)
3121, 2, 30latmle2 17678 . . . . . . . . . . . . . 14 ((𝐾 ∈ Lat ∧ (𝑐 𝑃) ∈ (Base‘𝐾) ∧ (𝑑 𝑆) ∈ (Base‘𝐾)) → ((𝑐 𝑃) (𝑑 𝑆)) (𝑑 𝑆))
3214, 23, 29, 31syl3anc 1368 . . . . . . . . . . . . 13 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 𝑃) (𝑑 𝑆)) (𝑑 𝑆))
3312, 32eqbrtrid 5077 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑍𝜓) → 𝐺 (𝑑 𝑆))
343, 4hlatjcom 36623 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑑𝐴𝑆𝐴) → (𝑑 𝑆) = (𝑆 𝑑))
3516, 25, 27, 34syl3anc 1368 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑍𝜓) → (𝑑 𝑆) = (𝑆 𝑑))
3633, 35breqtrd 5068 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → 𝐺 (𝑆 𝑑))
3736adantr 484 . . . . . . . . . 10 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → 𝐺 (𝑆 𝑑))
3811, 37eqbrtrd 5064 . . . . . . . . 9 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → 𝑐 (𝑆 𝑑))
392, 3, 4hlatlej2 36631 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑑𝐴) → 𝑑 (𝑆 𝑑))
4016, 27, 25, 39syl3anc 1368 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → 𝑑 (𝑆 𝑑))
4140adantr 484 . . . . . . . . 9 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → 𝑑 (𝑆 𝑑))
427, 4dalemcceb 36944 . . . . . . . . . . . 12 (𝜓𝑐 ∈ (Base‘𝐾))
43423ad2ant3 1132 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 ∈ (Base‘𝐾))
4421, 4atbase 36544 . . . . . . . . . . . . 13 (𝑑𝐴𝑑 ∈ (Base‘𝐾))
4524, 44syl 17 . . . . . . . . . . . 12 (𝜓𝑑 ∈ (Base‘𝐾))
46453ad2ant3 1132 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → 𝑑 ∈ (Base‘𝐾))
4721, 3, 4hlatjcl 36622 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑑𝐴) → (𝑆 𝑑) ∈ (Base‘𝐾))
4816, 27, 25, 47syl3anc 1368 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → (𝑆 𝑑) ∈ (Base‘𝐾))
4921, 2, 3latjle12 17663 . . . . . . . . . . 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 1369 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 (𝑆 𝑑) ∧ 𝑑 (𝑆 𝑑)) ↔ (𝑐 𝑑) (𝑆 𝑑)))
5150adantr 484 . . . . . . . . 9 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → ((𝑐 (𝑆 𝑑) ∧ 𝑑 (𝑆 𝑑)) ↔ (𝑐 𝑑) (𝑆 𝑑)))
5238, 41, 51mpbi2and 711 . . . . . . . 8 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → (𝑐 𝑑) (𝑆 𝑑))
531, 4dalemceb 36893 . . . . . . . . . . 11 (𝜑𝐶 ∈ (Base‘𝐾))
54533ad2ant1 1130 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → 𝐶 ∈ (Base‘𝐾))
5521, 3, 4hlatjcl 36622 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑐𝐴𝑑𝐴) → (𝑐 𝑑) ∈ (Base‘𝐾))
5616, 18, 25, 55syl3anc 1368 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑑) ∈ (Base‘𝐾))
5721, 2lattr 17657 . . . . . . . . . 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 1369 . . . . . . . . 9 ((𝜑𝑌 = 𝑍𝜓) → ((𝐶 (𝑐 𝑑) ∧ (𝑐 𝑑) (𝑆 𝑑)) → 𝐶 (𝑆 𝑑)))
5958adantr 484 . . . . . . . 8 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → ((𝐶 (𝑐 𝑑) ∧ (𝑐 𝑑) (𝑆 𝑑)) → 𝐶 (𝑆 𝑑)))
6010, 52, 59mp2and 698 . . . . . . 7 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → 𝐶 (𝑆 𝑑))
61 dalem23.o . . . . . . . . . . 11 𝑂 = (LPlanes‘𝐾)
621, 61dalemyeb 36904 . . . . . . . . . 10 (𝜑𝑌 ∈ (Base‘𝐾))
63623ad2ant1 1130 . . . . . . . . 9 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 ∈ (Base‘𝐾))
6421, 2, 30latmlem1 17682 . . . . . . . . 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 1369 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → (𝐶 (𝑆 𝑑) → (𝐶 𝑌) ((𝑆 𝑑) 𝑌)))
6665adantr 484 . . . . . . 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 36935 . . . . . . . . 9 ((𝜑𝑌 = 𝑍) → 𝐶 𝑌)
71703adant3 1129 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝐶 𝑌)
7221, 2, 30latleeqm1 17680 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝐶 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (𝐶 𝑌 ↔ (𝐶 𝑌) = 𝐶))
7314, 54, 63, 72syl3anc 1368 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → (𝐶 𝑌 ↔ (𝐶 𝑌) = 𝐶))
7471, 73mpbid 235 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → (𝐶 𝑌) = 𝐶)
7574adantr 484 . . . . . 6 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → (𝐶 𝑌) = 𝐶)
761, 2, 3, 4, 69dalemsly 36910 . . . . . . . . 9 ((𝜑𝑌 = 𝑍) → 𝑆 𝑌)
77763adant3 1129 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝑆 𝑌)
787dalem-ddly 36941 . . . . . . . . 9 (𝜓 → ¬ 𝑑 𝑌)
79783ad2ant3 1132 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑑 𝑌)
8021, 2, 3, 30, 42atjm 36700 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑆𝐴𝑑𝐴𝑌 ∈ (Base‘𝐾)) ∧ (𝑆 𝑌 ∧ ¬ 𝑑 𝑌)) → ((𝑆 𝑑) 𝑌) = 𝑆)
8116, 27, 25, 63, 77, 79, 80syl132anc 1385 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → ((𝑆 𝑑) 𝑌) = 𝑆)
8281adantr 484 . . . . . 6 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → ((𝑆 𝑑) 𝑌) = 𝑆)
8367, 75, 823brtr3d 5073 . . . . 5 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → 𝐶 𝑆)
84 hlatl 36615 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
8515, 84syl 17 . . . . . . . 8 (𝜑𝐾 ∈ AtLat)
861, 2, 3, 4, 61, 68dalemcea 36915 . . . . . . . 8 (𝜑𝐶𝐴)
872, 4atcmp 36566 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝐶𝐴𝑆𝐴) → (𝐶 𝑆𝐶 = 𝑆))
8885, 86, 26, 87syl3anc 1368 . . . . . . 7 (𝜑 → (𝐶 𝑆𝐶 = 𝑆))
89883ad2ant1 1130 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → (𝐶 𝑆𝐶 = 𝑆))
9089adantr 484 . . . . 5 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → (𝐶 𝑆𝐶 = 𝑆))
9183, 90mpbid 235 . . . 4 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → 𝐶 = 𝑆)
9291ex 416 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 = 𝐺𝐶 = 𝑆))
9392necon3d 3032 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝐶𝑆𝑐𝐺))
946, 93mpd 15 1 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝐺)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2114  wne 3011   class class class wbr 5042  cfv 6334  (class class class)co 7140  Basecbs 16474  lecple 16563  joincjn 17545  meetcmee 17546  Latclat 17646  Atomscatm 36518  AtLatcal 36519  HLchlt 36605  LPlanesclpl 36747
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-proset 17529  df-poset 17547  df-plt 17559  df-lub 17575  df-glb 17576  df-join 17577  df-meet 17578  df-p0 17640  df-lat 17647  df-clat 17709  df-oposet 36431  df-ol 36433  df-oml 36434  df-covers 36521  df-ats 36522  df-atl 36553  df-cvlat 36577  df-hlat 36606  df-llines 36753  df-lplanes 36754
This theorem is referenced by:  dalem28  36955  dalem31N  36958
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