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Theorem dalem25 35487
Description: Lemma for dath 35525. 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 35439 . . 3 (𝜑𝐶𝑆)
653ad2ant1 1128 . 2 ((𝜑𝑌 = 𝑍𝜓) → 𝐶𝑆)
7 dalem.ps . . . . . . . . . . 11 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
87dalemclccjdd 35477 . . . . . . . . . 10 (𝜓𝐶 (𝑐 𝑑))
983ad2ant3 1130 . . . . . . . . 9 ((𝜑𝑌 = 𝑍𝜓) → 𝐶 (𝑐 𝑑))
109adantr 472 . . . . . . . 8 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → 𝐶 (𝑐 𝑑))
11 simpr 479 . . . . . . . . . 10 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → 𝑐 = 𝐺)
12 dalem23.g . . . . . . . . . . . . 13 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
131dalemkelat 35413 . . . . . . . . . . . . . . 15 (𝜑𝐾 ∈ Lat)
14133ad2ant1 1128 . . . . . . . . . . . . . 14 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ Lat)
151dalemkehl 35412 . . . . . . . . . . . . . . . 16 (𝜑𝐾 ∈ HL)
16153ad2ant1 1128 . . . . . . . . . . . . . . 15 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
177dalemccea 35472 . . . . . . . . . . . . . . . 16 (𝜓𝑐𝐴)
18173ad2ant3 1130 . . . . . . . . . . . . . . 15 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝐴)
191dalempea 35415 . . . . . . . . . . . . . . . 16 (𝜑𝑃𝐴)
20193ad2ant1 1128 . . . . . . . . . . . . . . 15 ((𝜑𝑌 = 𝑍𝜓) → 𝑃𝐴)
21 eqid 2760 . . . . . . . . . . . . . . . 16 (Base‘𝐾) = (Base‘𝐾)
2221, 3, 4hlatjcl 35156 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ 𝑐𝐴𝑃𝐴) → (𝑐 𝑃) ∈ (Base‘𝐾))
2316, 18, 20, 22syl3anc 1477 . . . . . . . . . . . . . 14 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑃) ∈ (Base‘𝐾))
247dalemddea 35473 . . . . . . . . . . . . . . . 16 (𝜓𝑑𝐴)
25243ad2ant3 1130 . . . . . . . . . . . . . . 15 ((𝜑𝑌 = 𝑍𝜓) → 𝑑𝐴)
261dalemsea 35418 . . . . . . . . . . . . . . . 16 (𝜑𝑆𝐴)
27263ad2ant1 1128 . . . . . . . . . . . . . . 15 ((𝜑𝑌 = 𝑍𝜓) → 𝑆𝐴)
2821, 3, 4hlatjcl 35156 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ 𝑑𝐴𝑆𝐴) → (𝑑 𝑆) ∈ (Base‘𝐾))
2916, 25, 27, 28syl3anc 1477 . . . . . . . . . . . . . 14 ((𝜑𝑌 = 𝑍𝜓) → (𝑑 𝑆) ∈ (Base‘𝐾))
30 dalem23.m . . . . . . . . . . . . . . 15 = (meet‘𝐾)
3121, 2, 30latmle2 17278 . . . . . . . . . . . . . 14 ((𝐾 ∈ Lat ∧ (𝑐 𝑃) ∈ (Base‘𝐾) ∧ (𝑑 𝑆) ∈ (Base‘𝐾)) → ((𝑐 𝑃) (𝑑 𝑆)) (𝑑 𝑆))
3214, 23, 29, 31syl3anc 1477 . . . . . . . . . . . . 13 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 𝑃) (𝑑 𝑆)) (𝑑 𝑆))
3312, 32syl5eqbr 4839 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑍𝜓) → 𝐺 (𝑑 𝑆))
343, 4hlatjcom 35157 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑑𝐴𝑆𝐴) → (𝑑 𝑆) = (𝑆 𝑑))
3516, 25, 27, 34syl3anc 1477 . . . . . . . . . . . 12 ((𝜑𝑌 = 𝑍𝜓) → (𝑑 𝑆) = (𝑆 𝑑))
3633, 35breqtrd 4830 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → 𝐺 (𝑆 𝑑))
3736adantr 472 . . . . . . . . . 10 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → 𝐺 (𝑆 𝑑))
3811, 37eqbrtrd 4826 . . . . . . . . 9 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → 𝑐 (𝑆 𝑑))
392, 3, 4hlatlej2 35165 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑑𝐴) → 𝑑 (𝑆 𝑑))
4016, 27, 25, 39syl3anc 1477 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → 𝑑 (𝑆 𝑑))
4140adantr 472 . . . . . . . . 9 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → 𝑑 (𝑆 𝑑))
427, 4dalemcceb 35478 . . . . . . . . . . . 12 (𝜓𝑐 ∈ (Base‘𝐾))
43423ad2ant3 1130 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → 𝑐 ∈ (Base‘𝐾))
4421, 4atbase 35079 . . . . . . . . . . . . 13 (𝑑𝐴𝑑 ∈ (Base‘𝐾))
4524, 44syl 17 . . . . . . . . . . . 12 (𝜓𝑑 ∈ (Base‘𝐾))
46453ad2ant3 1130 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → 𝑑 ∈ (Base‘𝐾))
4721, 3, 4hlatjcl 35156 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑑𝐴) → (𝑆 𝑑) ∈ (Base‘𝐾))
4816, 27, 25, 47syl3anc 1477 . . . . . . . . . . 11 ((𝜑𝑌 = 𝑍𝜓) → (𝑆 𝑑) ∈ (Base‘𝐾))
4921, 2, 3latjle12 17263 . . . . . . . . . . 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 1479 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → ((𝑐 (𝑆 𝑑) ∧ 𝑑 (𝑆 𝑑)) ↔ (𝑐 𝑑) (𝑆 𝑑)))
5150adantr 472 . . . . . . . . 9 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → ((𝑐 (𝑆 𝑑) ∧ 𝑑 (𝑆 𝑑)) ↔ (𝑐 𝑑) (𝑆 𝑑)))
5238, 41, 51mpbi2and 994 . . . . . . . 8 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → (𝑐 𝑑) (𝑆 𝑑))
531, 4dalemceb 35427 . . . . . . . . . . 11 (𝜑𝐶 ∈ (Base‘𝐾))
54533ad2ant1 1128 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → 𝐶 ∈ (Base‘𝐾))
5521, 3, 4hlatjcl 35156 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑐𝐴𝑑𝐴) → (𝑐 𝑑) ∈ (Base‘𝐾))
5616, 18, 25, 55syl3anc 1477 . . . . . . . . . 10 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 𝑑) ∈ (Base‘𝐾))
5721, 2lattr 17257 . . . . . . . . . 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 1479 . . . . . . . . 9 ((𝜑𝑌 = 𝑍𝜓) → ((𝐶 (𝑐 𝑑) ∧ (𝑐 𝑑) (𝑆 𝑑)) → 𝐶 (𝑆 𝑑)))
5958adantr 472 . . . . . . . 8 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → ((𝐶 (𝑐 𝑑) ∧ (𝑐 𝑑) (𝑆 𝑑)) → 𝐶 (𝑆 𝑑)))
6010, 52, 59mp2and 717 . . . . . . 7 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → 𝐶 (𝑆 𝑑))
61 dalem23.o . . . . . . . . . . 11 𝑂 = (LPlanes‘𝐾)
621, 61dalemyeb 35438 . . . . . . . . . 10 (𝜑𝑌 ∈ (Base‘𝐾))
63623ad2ant1 1128 . . . . . . . . 9 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 ∈ (Base‘𝐾))
6421, 2, 30latmlem1 17282 . . . . . . . . 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 1479 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → (𝐶 (𝑆 𝑑) → (𝐶 𝑌) ((𝑆 𝑑) 𝑌)))
6665adantr 472 . . . . . . 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 35469 . . . . . . . . 9 ((𝜑𝑌 = 𝑍) → 𝐶 𝑌)
71703adant3 1127 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝐶 𝑌)
7221, 2, 30latleeqm1 17280 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝐶 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (𝐶 𝑌 ↔ (𝐶 𝑌) = 𝐶))
7314, 54, 63, 72syl3anc 1477 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → (𝐶 𝑌 ↔ (𝐶 𝑌) = 𝐶))
7471, 73mpbid 222 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → (𝐶 𝑌) = 𝐶)
7574adantr 472 . . . . . 6 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → (𝐶 𝑌) = 𝐶)
761, 2, 3, 4, 69dalemsly 35444 . . . . . . . . 9 ((𝜑𝑌 = 𝑍) → 𝑆 𝑌)
77763adant3 1127 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝑆 𝑌)
787dalem-ddly 35475 . . . . . . . . 9 (𝜓 → ¬ 𝑑 𝑌)
79783ad2ant3 1130 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → ¬ 𝑑 𝑌)
8021, 2, 3, 30, 42atjm 35234 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑆𝐴𝑑𝐴𝑌 ∈ (Base‘𝐾)) ∧ (𝑆 𝑌 ∧ ¬ 𝑑 𝑌)) → ((𝑆 𝑑) 𝑌) = 𝑆)
8116, 27, 25, 63, 77, 79, 80syl132anc 1495 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → ((𝑆 𝑑) 𝑌) = 𝑆)
8281adantr 472 . . . . . 6 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → ((𝑆 𝑑) 𝑌) = 𝑆)
8367, 75, 823brtr3d 4835 . . . . 5 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → 𝐶 𝑆)
84 hlatl 35150 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
8515, 84syl 17 . . . . . . . 8 (𝜑𝐾 ∈ AtLat)
861, 2, 3, 4, 61, 68dalemcea 35449 . . . . . . . 8 (𝜑𝐶𝐴)
872, 4atcmp 35101 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝐶𝐴𝑆𝐴) → (𝐶 𝑆𝐶 = 𝑆))
8885, 86, 26, 87syl3anc 1477 . . . . . . 7 (𝜑 → (𝐶 𝑆𝐶 = 𝑆))
89883ad2ant1 1128 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → (𝐶 𝑆𝐶 = 𝑆))
9089adantr 472 . . . . 5 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → (𝐶 𝑆𝐶 = 𝑆))
9183, 90mpbid 222 . . . 4 (((𝜑𝑌 = 𝑍𝜓) ∧ 𝑐 = 𝐺) → 𝐶 = 𝑆)
9291ex 449 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (𝑐 = 𝐺𝐶 = 𝑆))
9392necon3d 2953 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝐶𝑆𝑐𝐺))
946, 93mpd 15 1 ((𝜑𝑌 = 𝑍𝜓) → 𝑐𝐺)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932   class class class wbr 4804  cfv 6049  (class class class)co 6813  Basecbs 16059  lecple 16150  joincjn 17145  meetcmee 17146  Latclat 17246  Atomscatm 35053  AtLatcal 35054  HLchlt 35140  LPlanesclpl 35281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-preset 17129  df-poset 17147  df-plt 17159  df-lub 17175  df-glb 17176  df-join 17177  df-meet 17178  df-p0 17240  df-lat 17247  df-clat 17309  df-oposet 34966  df-ol 34968  df-oml 34969  df-covers 35056  df-ats 35057  df-atl 35088  df-cvlat 35112  df-hlat 35141  df-llines 35287  df-lplanes 35288
This theorem is referenced by:  dalem28  35489  dalem31N  35492
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