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Theorem lneq2at 34583
Description: A line equals the join of any two of its distinct points (atoms). (Contributed by NM, 29-Apr-2012.)
Hypotheses
Ref Expression
lneq2at.b 𝐵 = (Base‘𝐾)
lneq2at.l = (le‘𝐾)
lneq2at.j = (join‘𝐾)
lneq2at.a 𝐴 = (Atoms‘𝐾)
lneq2at.n 𝑁 = (Lines‘𝐾)
lneq2at.m 𝑀 = (pmap‘𝐾)
Assertion
Ref Expression
lneq2at (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → 𝑋 = (𝑃 𝑄))

Proof of Theorem lneq2at
Dummy variables 𝑠 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp11 1089 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → 𝐾 ∈ HL)
2 simp12 1090 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → 𝑋𝐵)
31, 2jca 554 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → (𝐾 ∈ HL ∧ 𝑋𝐵))
4 simp13 1091 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → (𝑀𝑋) ∈ 𝑁)
5 lneq2at.b . . . . 5 𝐵 = (Base‘𝐾)
6 lneq2at.j . . . . 5 = (join‘𝐾)
7 lneq2at.a . . . . 5 𝐴 = (Atoms‘𝐾)
8 lneq2at.n . . . . 5 𝑁 = (Lines‘𝐾)
9 lneq2at.m . . . . 5 𝑀 = (pmap‘𝐾)
105, 6, 7, 8, 9isline3 34581 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵) → ((𝑀𝑋) ∈ 𝑁 ↔ ∃𝑟𝐴𝑠𝐴 (𝑟𝑠𝑋 = (𝑟 𝑠))))
1110biimpd 219 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵) → ((𝑀𝑋) ∈ 𝑁 → ∃𝑟𝐴𝑠𝐴 (𝑟𝑠𝑋 = (𝑟 𝑠))))
123, 4, 11sylc 65 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → ∃𝑟𝐴𝑠𝐴 (𝑟𝑠𝑋 = (𝑟 𝑠)))
13 simp3r 1088 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → 𝑋 = (𝑟 𝑠))
14 simp111 1188 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → 𝐾 ∈ HL)
15 simp121 1191 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → 𝑃𝐴)
16 simp122 1192 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → 𝑄𝐴)
1715, 16jca 554 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → (𝑃𝐴𝑄𝐴))
18 simp2 1060 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → (𝑟𝐴𝑠𝐴))
1914, 17, 183jca 1240 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → (𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑟𝐴𝑠𝐴)))
20 simp123 1193 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → 𝑃𝑄)
2119, 20jca 554 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑃𝑄))
22 hllat 34169 . . . . . . . . . . 11 (𝐾 ∈ HL → 𝐾 ∈ Lat)
231, 22syl 17 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → 𝐾 ∈ Lat)
24 simp21 1092 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → 𝑃𝐴)
255, 7atbase 34095 . . . . . . . . . . . 12 (𝑃𝐴𝑃𝐵)
2624, 25syl 17 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → 𝑃𝐵)
27 simp22 1093 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → 𝑄𝐴)
285, 7atbase 34095 . . . . . . . . . . . 12 (𝑄𝐴𝑄𝐵)
2927, 28syl 17 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → 𝑄𝐵)
3026, 29, 23jca 1240 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → (𝑃𝐵𝑄𝐵𝑋𝐵))
3123, 30jca 554 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → (𝐾 ∈ Lat ∧ (𝑃𝐵𝑄𝐵𝑋𝐵)))
32 simp3 1061 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → (𝑃 𝑋𝑄 𝑋))
33 lneq2at.l . . . . . . . . . . 11 = (le‘𝐾)
345, 33, 6latjle12 17002 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑃𝐵𝑄𝐵𝑋𝐵)) → ((𝑃 𝑋𝑄 𝑋) ↔ (𝑃 𝑄) 𝑋))
3534biimpd 219 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑃𝐵𝑄𝐵𝑋𝐵)) → ((𝑃 𝑋𝑄 𝑋) → (𝑃 𝑄) 𝑋))
3631, 32, 35sylc 65 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → (𝑃 𝑄) 𝑋)
37363ad2ant1 1080 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → (𝑃 𝑄) 𝑋)
3837, 13breqtrd 4649 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → (𝑃 𝑄) (𝑟 𝑠))
39 simpl1 1062 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑃𝑄) → 𝐾 ∈ HL)
40 simpl2l 1112 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑃𝑄) → 𝑃𝐴)
41 simpl2r 1113 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑃𝑄) → 𝑄𝐴)
42 simpr 477 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑃𝑄) → 𝑃𝑄)
43 simpl3 1064 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑃𝑄) → (𝑟𝐴𝑠𝐴))
4433, 6, 7ps-1 34282 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑟𝐴𝑠𝐴)) → ((𝑃 𝑄) (𝑟 𝑠) ↔ (𝑃 𝑄) = (𝑟 𝑠)))
4539, 40, 41, 42, 43, 44syl131anc 1336 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑃𝑄) → ((𝑃 𝑄) (𝑟 𝑠) ↔ (𝑃 𝑄) = (𝑟 𝑠)))
4645biimpd 219 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑃𝑄) → ((𝑃 𝑄) (𝑟 𝑠) → (𝑃 𝑄) = (𝑟 𝑠)))
4721, 38, 46sylc 65 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → (𝑃 𝑄) = (𝑟 𝑠))
4813, 47eqtr4d 2658 . . . 4 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → 𝑋 = (𝑃 𝑄))
49483exp 1261 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → ((𝑟𝐴𝑠𝐴) → ((𝑟𝑠𝑋 = (𝑟 𝑠)) → 𝑋 = (𝑃 𝑄))))
5049rexlimdvv 3032 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → (∃𝑟𝐴𝑠𝐴 (𝑟𝑠𝑋 = (𝑟 𝑠)) → 𝑋 = (𝑃 𝑄)))
5112, 50mpd 15 1 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → 𝑋 = (𝑃 𝑄))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wrex 2909   class class class wbr 4623  cfv 5857  (class class class)co 6615  Basecbs 15800  lecple 15888  joincjn 16884  Latclat 16985  Atomscatm 34069  HLchlt 34156  Linesclines 34299  pmapcpmap 34302
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 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
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 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-preset 16868  df-poset 16886  df-plt 16898  df-lub 16914  df-glb 16915  df-join 16916  df-meet 16917  df-p0 16979  df-lat 16986  df-clat 17048  df-oposet 33982  df-ol 33984  df-oml 33985  df-covers 34072  df-ats 34073  df-atl 34104  df-cvlat 34128  df-hlat 34157  df-lines 34306  df-pmap 34309
This theorem is referenced by:  lnjatN  34585  lncmp  34588  cdlema1N  34596
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