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Theorem dalem25 36716
Description: Lemma for dath 36754. 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 36668 . . 3 (𝜑𝐶𝑆)
653ad2ant1 1125 . 2 ((𝜑𝑌 = 𝑍𝜓) → 𝐶𝑆)
7 dalem.ps . . . . . . . . . . 11 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
87dalemclccjdd 36706 . . . . . . . . . 10 (𝜓𝐶 (𝑐 𝑑))
983ad2ant3 1127 . . . . . . . . 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 36642 . . . . . . . . . . . . . . 15 (𝜑𝐾 ∈ Lat)
14133ad2ant1 1125 . . . . . . . . . . . . . 14 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ Lat)
151dalemkehl 36641 . . . . . . . . . . . . . . . 16 (𝜑𝐾 ∈ HL)
16153ad2ant1 1125 . . . . . . . . . . . . . . 15 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
177dalemccea 36701 . . . . . . . . . . . . . . . 16 (𝜓𝑐𝐴)
18173ad2ant3 1127 . . . . . . . . . . . . . . 15 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝐴)
191dalempea 36644 . . . . . . . . . . . . . . . 16 (𝜑𝑃𝐴)
20193ad2ant1 1125 . . . . . . . . . . . . . . 15 ((𝜑𝑌 = 𝑍𝜓) → 𝑃𝐴)
21 eqid 2821 . . . . . . . . . . . . . . . 16 (Base‘𝐾) = (Base‘𝐾)
2221, 3, 4hlatjcl 36385 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ 𝑐𝐴𝑃𝐴) → (𝑐 𝑃) ∈ (Base‘𝐾))
2316, 18, 20, 22syl3anc 1363 . . . . . . . . . . . . . 14 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑃) ∈ (Base‘𝐾))
247dalemddea 36702 . . . . . . . . . . . . . . . 16 (𝜓𝑑𝐴)
25243ad2ant3 1127 . . . . . . . . . . . . . . 15 ((𝜑𝑌 = 𝑍𝜓) → 𝑑𝐴)
261dalemsea 36647 . . . . . . . . . . . . . . . 16 (𝜑𝑆𝐴)
27263ad2ant1 1125 . . . . . . . . . . . . . . 15 ((𝜑𝑌 = 𝑍𝜓) → 𝑆𝐴)
2821, 3, 4hlatjcl 36385 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ 𝑑𝐴𝑆𝐴) → (𝑑 𝑆) ∈ (Base‘𝐾))
2916, 25, 27, 28syl3anc 1363 . . . . . . . . . . . . . 14 ((𝜑𝑌 = 𝑍𝜓) → (𝑑 𝑆) ∈ (Base‘𝐾))
30 dalem23.m . . . . . . . . . . . . . . 15 = (meet‘𝐾)
3121, 2, 30latmle2 17677 . . . . . . . . . . . . . 14 ((𝐾 ∈ Lat ∧ (𝑐 𝑃) ∈ (Base‘𝐾) ∧ (𝑑 𝑆) ∈ (Base‘𝐾)) → ((𝑐 𝑃) (𝑑 𝑆)) (𝑑 𝑆))
3214, 23, 29, 31syl3anc 1363 . . . . . . . . . . . . 13 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 𝑃) (𝑑 𝑆)) (𝑑 𝑆))
3312, 32eqbrtrid 5093 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑍𝜓) → 𝐺 (𝑑 𝑆))
343, 4hlatjcom 36386 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑑𝐴𝑆𝐴) → (𝑑 𝑆) = (𝑆 𝑑))
3516, 25, 27, 34syl3anc 1363 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑍𝜓) → (𝑑 𝑆) = (𝑆 𝑑))
3633, 35breqtrd 5084 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → 𝐺 (𝑆 𝑑))
3736adantr 481 . . . . . . . . . 10 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → 𝐺 (𝑆 𝑑))
3811, 37eqbrtrd 5080 . . . . . . . . 9 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → 𝑐 (𝑆 𝑑))
392, 3, 4hlatlej2 36394 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑑𝐴) → 𝑑 (𝑆 𝑑))
4016, 27, 25, 39syl3anc 1363 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → 𝑑 (𝑆 𝑑))
4140adantr 481 . . . . . . . . 9 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → 𝑑 (𝑆 𝑑))
427, 4dalemcceb 36707 . . . . . . . . . . . 12 (𝜓𝑐 ∈ (Base‘𝐾))
43423ad2ant3 1127 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 ∈ (Base‘𝐾))
4421, 4atbase 36307 . . . . . . . . . . . . 13 (𝑑𝐴𝑑 ∈ (Base‘𝐾))
4524, 44syl 17 . . . . . . . . . . . 12 (𝜓𝑑 ∈ (Base‘𝐾))
46453ad2ant3 1127 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → 𝑑 ∈ (Base‘𝐾))
4721, 3, 4hlatjcl 36385 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑑𝐴) → (𝑆 𝑑) ∈ (Base‘𝐾))
4816, 27, 25, 47syl3anc 1363 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → (𝑆 𝑑) ∈ (Base‘𝐾))
4921, 2, 3latjle12 17662 . . . . . . . . . . 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 1364 . . . . . . . . . 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 708 . . . . . . . 8 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → (𝑐 𝑑) (𝑆 𝑑))
531, 4dalemceb 36656 . . . . . . . . . . 11 (𝜑𝐶 ∈ (Base‘𝐾))
54533ad2ant1 1125 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → 𝐶 ∈ (Base‘𝐾))
5521, 3, 4hlatjcl 36385 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑐𝐴𝑑𝐴) → (𝑐 𝑑) ∈ (Base‘𝐾))
5616, 18, 25, 55syl3anc 1363 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑑) ∈ (Base‘𝐾))
5721, 2lattr 17656 . . . . . . . . . 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 1364 . . . . . . . . 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 695 . . . . . . 7 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → 𝐶 (𝑆 𝑑))
61 dalem23.o . . . . . . . . . . 11 𝑂 = (LPlanes‘𝐾)
621, 61dalemyeb 36667 . . . . . . . . . 10 (𝜑𝑌 ∈ (Base‘𝐾))
63623ad2ant1 1125 . . . . . . . . 9 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 ∈ (Base‘𝐾))
6421, 2, 30latmlem1 17681 . . . . . . . . 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 1364 . . . . . . . 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 36698 . . . . . . . . 9 ((𝜑𝑌 = 𝑍) → 𝐶 𝑌)
71703adant3 1124 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝐶 𝑌)
7221, 2, 30latleeqm1 17679 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝐶 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (𝐶 𝑌 ↔ (𝐶 𝑌) = 𝐶))
7314, 54, 63, 72syl3anc 1363 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → (𝐶 𝑌 ↔ (𝐶 𝑌) = 𝐶))
7471, 73mpbid 233 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → (𝐶 𝑌) = 𝐶)
7574adantr 481 . . . . . 6 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → (𝐶 𝑌) = 𝐶)
761, 2, 3, 4, 69dalemsly 36673 . . . . . . . . 9 ((𝜑𝑌 = 𝑍) → 𝑆 𝑌)
77763adant3 1124 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝑆 𝑌)
787dalem-ddly 36704 . . . . . . . . 9 (𝜓 → ¬ 𝑑 𝑌)
79783ad2ant3 1127 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑑 𝑌)
8021, 2, 3, 30, 42atjm 36463 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑆𝐴𝑑𝐴𝑌 ∈ (Base‘𝐾)) ∧ (𝑆 𝑌 ∧ ¬ 𝑑 𝑌)) → ((𝑆 𝑑) 𝑌) = 𝑆)
8116, 27, 25, 63, 77, 79, 80syl132anc 1380 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → ((𝑆 𝑑) 𝑌) = 𝑆)
8281adantr 481 . . . . . 6 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → ((𝑆 𝑑) 𝑌) = 𝑆)
8367, 75, 823brtr3d 5089 . . . . 5 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → 𝐶 𝑆)
84 hlatl 36378 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
8515, 84syl 17 . . . . . . . 8 (𝜑𝐾 ∈ AtLat)
861, 2, 3, 4, 61, 68dalemcea 36678 . . . . . . . 8 (𝜑𝐶𝐴)
872, 4atcmp 36329 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝐶𝐴𝑆𝐴) → (𝐶 𝑆𝐶 = 𝑆))
8885, 86, 26, 87syl3anc 1363 . . . . . . 7 (𝜑 → (𝐶 𝑆𝐶 = 𝑆))
89883ad2ant1 1125 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → (𝐶 𝑆𝐶 = 𝑆))
9089adantr 481 . . . . 5 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → (𝐶 𝑆𝐶 = 𝑆))
9183, 90mpbid 233 . . . 4 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → 𝐶 = 𝑆)
9291ex 413 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 = 𝐺𝐶 = 𝑆))
9392necon3d 3037 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝐶𝑆𝑐𝐺))
946, 93mpd 15 1 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝐺)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wne 3016   class class class wbr 5058  cfv 6349  (class class class)co 7145  Basecbs 16473  lecple 16562  joincjn 17544  meetcmee 17545  Latclat 17645  Atomscatm 36281  AtLatcal 36282  HLchlt 36368  LPlanesclpl 36510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  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-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4833  df-iun 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 7103  df-ov 7148  df-oprab 7149  df-proset 17528  df-poset 17546  df-plt 17558  df-lub 17574  df-glb 17575  df-join 17576  df-meet 17577  df-p0 17639  df-lat 17646  df-clat 17708  df-oposet 36194  df-ol 36196  df-oml 36197  df-covers 36284  df-ats 36285  df-atl 36316  df-cvlat 36340  df-hlat 36369  df-llines 36516  df-lplanes 36517
This theorem is referenced by:  dalem28  36718  dalem31N  36721
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