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Theorem dalem25 39397
Description: Lemma for dath 39435. 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 39349 . . 3 (𝜑𝐶𝑆)
653ad2ant1 1130 . 2 ((𝜑𝑌 = 𝑍𝜓) → 𝐶𝑆)
7 dalem.ps . . . . . . . . . . 11 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
87dalemclccjdd 39387 . . . . . . . . . 10 (𝜓𝐶 (𝑐 𝑑))
983ad2ant3 1132 . . . . . . . . 9 ((𝜑𝑌 = 𝑍𝜓) → 𝐶 (𝑐 𝑑))
109adantr 479 . . . . . . . 8 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → 𝐶 (𝑐 𝑑))
11 simpr 483 . . . . . . . . . 10 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → 𝑐 = 𝐺)
12 dalem23.g . . . . . . . . . . . . 13 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
131dalemkelat 39323 . . . . . . . . . . . . . . 15 (𝜑𝐾 ∈ Lat)
14133ad2ant1 1130 . . . . . . . . . . . . . 14 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ Lat)
151dalemkehl 39322 . . . . . . . . . . . . . . . 16 (𝜑𝐾 ∈ HL)
16153ad2ant1 1130 . . . . . . . . . . . . . . 15 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
177dalemccea 39382 . . . . . . . . . . . . . . . 16 (𝜓𝑐𝐴)
18173ad2ant3 1132 . . . . . . . . . . . . . . 15 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝐴)
191dalempea 39325 . . . . . . . . . . . . . . . 16 (𝜑𝑃𝐴)
20193ad2ant1 1130 . . . . . . . . . . . . . . 15 ((𝜑𝑌 = 𝑍𝜓) → 𝑃𝐴)
21 eqid 2726 . . . . . . . . . . . . . . . 16 (Base‘𝐾) = (Base‘𝐾)
2221, 3, 4hlatjcl 39065 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ 𝑐𝐴𝑃𝐴) → (𝑐 𝑃) ∈ (Base‘𝐾))
2316, 18, 20, 22syl3anc 1368 . . . . . . . . . . . . . 14 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑃) ∈ (Base‘𝐾))
247dalemddea 39383 . . . . . . . . . . . . . . . 16 (𝜓𝑑𝐴)
25243ad2ant3 1132 . . . . . . . . . . . . . . 15 ((𝜑𝑌 = 𝑍𝜓) → 𝑑𝐴)
261dalemsea 39328 . . . . . . . . . . . . . . . 16 (𝜑𝑆𝐴)
27263ad2ant1 1130 . . . . . . . . . . . . . . 15 ((𝜑𝑌 = 𝑍𝜓) → 𝑆𝐴)
2821, 3, 4hlatjcl 39065 . . . . . . . . . . . . . . 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 18490 . . . . . . . . . . . . . 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 5188 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑍𝜓) → 𝐺 (𝑑 𝑆))
343, 4hlatjcom 39066 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑑𝐴𝑆𝐴) → (𝑑 𝑆) = (𝑆 𝑑))
3516, 25, 27, 34syl3anc 1368 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑍𝜓) → (𝑑 𝑆) = (𝑆 𝑑))
3633, 35breqtrd 5179 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → 𝐺 (𝑆 𝑑))
3736adantr 479 . . . . . . . . . 10 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → 𝐺 (𝑆 𝑑))
3811, 37eqbrtrd 5175 . . . . . . . . 9 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → 𝑐 (𝑆 𝑑))
392, 3, 4hlatlej2 39074 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑑𝐴) → 𝑑 (𝑆 𝑑))
4016, 27, 25, 39syl3anc 1368 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → 𝑑 (𝑆 𝑑))
4140adantr 479 . . . . . . . . 9 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → 𝑑 (𝑆 𝑑))
427, 4dalemcceb 39388 . . . . . . . . . . . 12 (𝜓𝑐 ∈ (Base‘𝐾))
43423ad2ant3 1132 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 ∈ (Base‘𝐾))
4421, 4atbase 38987 . . . . . . . . . . . . 13 (𝑑𝐴𝑑 ∈ (Base‘𝐾))
4524, 44syl 17 . . . . . . . . . . . 12 (𝜓𝑑 ∈ (Base‘𝐾))
46453ad2ant3 1132 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → 𝑑 ∈ (Base‘𝐾))
4721, 3, 4hlatjcl 39065 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑑𝐴) → (𝑆 𝑑) ∈ (Base‘𝐾))
4816, 27, 25, 47syl3anc 1368 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → (𝑆 𝑑) ∈ (Base‘𝐾))
4921, 2, 3latjle12 18475 . . . . . . . . . . 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 479 . . . . . . . . 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 39337 . . . . . . . . . . 11 (𝜑𝐶 ∈ (Base‘𝐾))
54533ad2ant1 1130 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → 𝐶 ∈ (Base‘𝐾))
5521, 3, 4hlatjcl 39065 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑐𝐴𝑑𝐴) → (𝑐 𝑑) ∈ (Base‘𝐾))
5616, 18, 25, 55syl3anc 1368 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑑) ∈ (Base‘𝐾))
5721, 2lattr 18469 . . . . . . . . . 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 479 . . . . . . . 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 39348 . . . . . . . . . 10 (𝜑𝑌 ∈ (Base‘𝐾))
63623ad2ant1 1130 . . . . . . . . 9 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 ∈ (Base‘𝐾))
6421, 2, 30latmlem1 18494 . . . . . . . . 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 479 . . . . . . 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 39379 . . . . . . . . 9 ((𝜑𝑌 = 𝑍) → 𝐶 𝑌)
71703adant3 1129 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝐶 𝑌)
7221, 2, 30latleeqm1 18492 . . . . . . . . 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 231 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → (𝐶 𝑌) = 𝐶)
7574adantr 479 . . . . . 6 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → (𝐶 𝑌) = 𝐶)
761, 2, 3, 4, 69dalemsly 39354 . . . . . . . . 9 ((𝜑𝑌 = 𝑍) → 𝑆 𝑌)
77763adant3 1129 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝑆 𝑌)
787dalem-ddly 39385 . . . . . . . . 9 (𝜓 → ¬ 𝑑 𝑌)
79783ad2ant3 1132 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑑 𝑌)
8021, 2, 3, 30, 42atjm 39144 . . . . . . . 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 479 . . . . . 6 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → ((𝑆 𝑑) 𝑌) = 𝑆)
8367, 75, 823brtr3d 5184 . . . . 5 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → 𝐶 𝑆)
84 hlatl 39058 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
8515, 84syl 17 . . . . . . . 8 (𝜑𝐾 ∈ AtLat)
861, 2, 3, 4, 61, 68dalemcea 39359 . . . . . . . 8 (𝜑𝐶𝐴)
872, 4atcmp 39009 . . . . . . . 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 479 . . . . 5 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → (𝐶 𝑆𝐶 = 𝑆))
9183, 90mpbid 231 . . . 4 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → 𝐶 = 𝑆)
9291ex 411 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 = 𝐺𝐶 = 𝑆))
9392necon3d 2951 . 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 394  w3a 1084   = wceq 1534  wcel 2099  wne 2930   class class class wbr 5153  cfv 6554  (class class class)co 7424  Basecbs 17213  lecple 17273  joincjn 18336  meetcmee 18337  Latclat 18456  Atomscatm 38961  AtLatcal 38962  HLchlt 39048  LPlanesclpl 39191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-proset 18320  df-poset 18338  df-plt 18355  df-lub 18371  df-glb 18372  df-join 18373  df-meet 18374  df-p0 18450  df-lat 18457  df-clat 18524  df-oposet 38874  df-ol 38876  df-oml 38877  df-covers 38964  df-ats 38965  df-atl 38996  df-cvlat 39020  df-hlat 39049  df-llines 39197  df-lplanes 39198
This theorem is referenced by:  dalem28  39399  dalem31N  39402
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