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Theorem isline2 35844
 Description: Definition of line in terms of projective map. (Contributed by NM, 25-Jan-2012.)
Hypotheses
Ref Expression
isline2.j = (join‘𝐾)
isline2.a 𝐴 = (Atoms‘𝐾)
isline2.n 𝑁 = (Lines‘𝐾)
isline2.m 𝑀 = (pmap‘𝐾)
Assertion
Ref Expression
isline2 (𝐾 ∈ Lat → (𝑋𝑁 ↔ ∃𝑝𝐴𝑞𝐴 (𝑝𝑞𝑋 = (𝑀‘(𝑝 𝑞)))))
Distinct variable groups:   𝑞,𝑝,𝐴   𝐾,𝑝,𝑞   𝑋,𝑝,𝑞
Allowed substitution hints:   (𝑞,𝑝)   𝑀(𝑞,𝑝)   𝑁(𝑞,𝑝)

Proof of Theorem isline2
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 eqid 2825 . . 3 (le‘𝐾) = (le‘𝐾)
2 isline2.j . . 3 = (join‘𝐾)
3 isline2.a . . 3 𝐴 = (Atoms‘𝐾)
4 isline2.n . . 3 𝑁 = (Lines‘𝐾)
51, 2, 3, 4isline 35809 . 2 (𝐾 ∈ Lat → (𝑋𝑁 ↔ ∃𝑝𝐴𝑞𝐴 (𝑝𝑞𝑋 = {𝑟𝐴𝑟(le‘𝐾)(𝑝 𝑞)})))
6 simpl 476 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑝𝐴𝑞𝐴)) → 𝐾 ∈ Lat)
7 eqid 2825 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
87, 3atbase 35359 . . . . . . . 8 (𝑝𝐴𝑝 ∈ (Base‘𝐾))
98ad2antrl 719 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑝𝐴𝑞𝐴)) → 𝑝 ∈ (Base‘𝐾))
107, 3atbase 35359 . . . . . . . 8 (𝑞𝐴𝑞 ∈ (Base‘𝐾))
1110ad2antll 720 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑝𝐴𝑞𝐴)) → 𝑞 ∈ (Base‘𝐾))
127, 2latjcl 17411 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) → (𝑝 𝑞) ∈ (Base‘𝐾))
136, 9, 11, 12syl3anc 1494 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑝𝐴𝑞𝐴)) → (𝑝 𝑞) ∈ (Base‘𝐾))
14 isline2.m . . . . . . 7 𝑀 = (pmap‘𝐾)
157, 1, 3, 14pmapval 35827 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑝 𝑞) ∈ (Base‘𝐾)) → (𝑀‘(𝑝 𝑞)) = {𝑟𝐴𝑟(le‘𝐾)(𝑝 𝑞)})
1613, 15syldan 585 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑝𝐴𝑞𝐴)) → (𝑀‘(𝑝 𝑞)) = {𝑟𝐴𝑟(le‘𝐾)(𝑝 𝑞)})
1716eqeq2d 2835 . . . 4 ((𝐾 ∈ Lat ∧ (𝑝𝐴𝑞𝐴)) → (𝑋 = (𝑀‘(𝑝 𝑞)) ↔ 𝑋 = {𝑟𝐴𝑟(le‘𝐾)(𝑝 𝑞)}))
1817anbi2d 622 . . 3 ((𝐾 ∈ Lat ∧ (𝑝𝐴𝑞𝐴)) → ((𝑝𝑞𝑋 = (𝑀‘(𝑝 𝑞))) ↔ (𝑝𝑞𝑋 = {𝑟𝐴𝑟(le‘𝐾)(𝑝 𝑞)})))
19182rexbidva 3266 . 2 (𝐾 ∈ Lat → (∃𝑝𝐴𝑞𝐴 (𝑝𝑞𝑋 = (𝑀‘(𝑝 𝑞))) ↔ ∃𝑝𝐴𝑞𝐴 (𝑝𝑞𝑋 = {𝑟𝐴𝑟(le‘𝐾)(𝑝 𝑞)})))
205, 19bitr4d 274 1 (𝐾 ∈ Lat → (𝑋𝑁 ↔ ∃𝑝𝐴𝑞𝐴 (𝑝𝑞𝑋 = (𝑀‘(𝑝 𝑞)))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1656   ∈ wcel 2164   ≠ wne 2999  ∃wrex 3118  {crab 3121   class class class wbr 4875  ‘cfv 6127  (class class class)co 6910  Basecbs 16229  lecple 16319  joincjn 17304  Latclat 17405  Atomscatm 35333  Linesclines 35564  pmapcpmap 35567 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-lub 17334  df-glb 17335  df-join 17336  df-meet 17337  df-lat 17406  df-ats 35337  df-lines 35571  df-pmap 35574 This theorem is referenced by:  isline3  35846  lncvrelatN  35851  linepsubclN  36021
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