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Theorem isline4N 34540
Description: Definition of line in terms of original lattice elements. (Contributed by NM, 16-Jun-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
isline4.b 𝐵 = (Base‘𝐾)
isline4.c 𝐶 = ( ⋖ ‘𝐾)
isline4.a 𝐴 = (Atoms‘𝐾)
isline4.n 𝑁 = (Lines‘𝐾)
isline4.m 𝑀 = (pmap‘𝐾)
Assertion
Ref Expression
isline4N ((𝐾 ∈ HL ∧ 𝑋𝐵) → ((𝑀𝑋) ∈ 𝑁 ↔ ∃𝑝𝐴 𝑝𝐶𝑋))
Distinct variable groups:   𝐴,𝑝   𝐵,𝑝   𝐾,𝑝   𝑀,𝑝   𝑋,𝑝
Allowed substitution hints:   𝐶(𝑝)   𝑁(𝑝)

Proof of Theorem isline4N
Dummy variable 𝑞 is distinct from all other variables.
StepHypRef Expression
1 isline4.b . . 3 𝐵 = (Base‘𝐾)
2 eqid 2621 . . 3 (join‘𝐾) = (join‘𝐾)
3 isline4.a . . 3 𝐴 = (Atoms‘𝐾)
4 isline4.n . . 3 𝑁 = (Lines‘𝐾)
5 isline4.m . . 3 𝑀 = (pmap‘𝐾)
61, 2, 3, 4, 5isline3 34539 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵) → ((𝑀𝑋) ∈ 𝑁 ↔ ∃𝑝𝐴𝑞𝐴 (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞))))
7 simpll 789 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ 𝑝𝐴) → 𝐾 ∈ HL)
81, 3atbase 34053 . . . . . 6 (𝑝𝐴𝑝𝐵)
98adantl 482 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ 𝑝𝐴) → 𝑝𝐵)
10 simplr 791 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ 𝑝𝐴) → 𝑋𝐵)
11 eqid 2621 . . . . . 6 (le‘𝐾) = (le‘𝐾)
12 isline4.c . . . . . 6 𝐶 = ( ⋖ ‘𝐾)
131, 11, 2, 12, 3cvrval3 34176 . . . . 5 ((𝐾 ∈ HL ∧ 𝑝𝐵𝑋𝐵) → (𝑝𝐶𝑋 ↔ ∃𝑞𝐴𝑞(le‘𝐾)𝑝 ∧ (𝑝(join‘𝐾)𝑞) = 𝑋)))
147, 9, 10, 13syl3anc 1323 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ 𝑝𝐴) → (𝑝𝐶𝑋 ↔ ∃𝑞𝐴𝑞(le‘𝐾)𝑝 ∧ (𝑝(join‘𝐾)𝑞) = 𝑋)))
15 hlatl 34124 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
1615ad3antrrr 765 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ 𝑝𝐴) ∧ 𝑞𝐴) → 𝐾 ∈ AtLat)
17 simpr 477 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ 𝑝𝐴) ∧ 𝑞𝐴) → 𝑞𝐴)
18 simplr 791 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ 𝑝𝐴) ∧ 𝑞𝐴) → 𝑝𝐴)
1911, 3atncmp 34076 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑞𝐴𝑝𝐴) → (¬ 𝑞(le‘𝐾)𝑝𝑞𝑝))
2016, 17, 18, 19syl3anc 1323 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ 𝑝𝐴) ∧ 𝑞𝐴) → (¬ 𝑞(le‘𝐾)𝑝𝑞𝑝))
21 necom 2843 . . . . . . 7 (𝑞𝑝𝑝𝑞)
2220, 21syl6bb 276 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ 𝑝𝐴) ∧ 𝑞𝐴) → (¬ 𝑞(le‘𝐾)𝑝𝑝𝑞))
23 eqcom 2628 . . . . . . 7 ((𝑝(join‘𝐾)𝑞) = 𝑋𝑋 = (𝑝(join‘𝐾)𝑞))
2423a1i 11 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ 𝑝𝐴) ∧ 𝑞𝐴) → ((𝑝(join‘𝐾)𝑞) = 𝑋𝑋 = (𝑝(join‘𝐾)𝑞)))
2522, 24anbi12d 746 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ 𝑝𝐴) ∧ 𝑞𝐴) → ((¬ 𝑞(le‘𝐾)𝑝 ∧ (𝑝(join‘𝐾)𝑞) = 𝑋) ↔ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞))))
2625rexbidva 3042 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ 𝑝𝐴) → (∃𝑞𝐴𝑞(le‘𝐾)𝑝 ∧ (𝑝(join‘𝐾)𝑞) = 𝑋) ↔ ∃𝑞𝐴 (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞))))
2714, 26bitrd 268 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ 𝑝𝐴) → (𝑝𝐶𝑋 ↔ ∃𝑞𝐴 (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞))))
2827rexbidva 3042 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (∃𝑝𝐴 𝑝𝐶𝑋 ↔ ∃𝑝𝐴𝑞𝐴 (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞))))
296, 28bitr4d 271 1 ((𝐾 ∈ HL ∧ 𝑋𝐵) → ((𝑀𝑋) ∈ 𝑁 ↔ ∃𝑝𝐴 𝑝𝐶𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wne 2790  wrex 2908   class class class wbr 4613  cfv 5847  (class class class)co 6604  Basecbs 15781  lecple 15869  joincjn 16865  ccvr 34026  Atomscatm 34027  AtLatcal 34028  HLchlt 34114  Linesclines 34257  pmapcpmap 34260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-preset 16849  df-poset 16867  df-plt 16879  df-lub 16895  df-glb 16896  df-join 16897  df-meet 16898  df-p0 16960  df-lat 16967  df-clat 17029  df-oposet 33940  df-ol 33942  df-oml 33943  df-covers 34030  df-ats 34031  df-atl 34062  df-cvlat 34086  df-hlat 34115  df-lines 34264  df-pmap 34267
This theorem is referenced by: (None)
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