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Theorem isline2 35582
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 2771 . . 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 35547 . 2 (𝐾 ∈ Lat → (𝑋𝑁 ↔ ∃𝑝𝐴𝑞𝐴 (𝑝𝑞𝑋 = {𝑟𝐴𝑟(le‘𝐾)(𝑝 𝑞)})))
6 simpl 468 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑝𝐴𝑞𝐴)) → 𝐾 ∈ Lat)
7 eqid 2771 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
87, 3atbase 35098 . . . . . . . 8 (𝑝𝐴𝑝 ∈ (Base‘𝐾))
98ad2antrl 707 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑝𝐴𝑞𝐴)) → 𝑝 ∈ (Base‘𝐾))
107, 3atbase 35098 . . . . . . . 8 (𝑞𝐴𝑞 ∈ (Base‘𝐾))
1110ad2antll 708 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑝𝐴𝑞𝐴)) → 𝑞 ∈ (Base‘𝐾))
127, 2latjcl 17259 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) → (𝑝 𝑞) ∈ (Base‘𝐾))
136, 9, 11, 12syl3anc 1476 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑝𝐴𝑞𝐴)) → (𝑝 𝑞) ∈ (Base‘𝐾))
14 isline2.m . . . . . . 7 𝑀 = (pmap‘𝐾)
157, 1, 3, 14pmapval 35565 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑝 𝑞) ∈ (Base‘𝐾)) → (𝑀‘(𝑝 𝑞)) = {𝑟𝐴𝑟(le‘𝐾)(𝑝 𝑞)})
1613, 15syldan 579 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑝𝐴𝑞𝐴)) → (𝑀‘(𝑝 𝑞)) = {𝑟𝐴𝑟(le‘𝐾)(𝑝 𝑞)})
1716eqeq2d 2781 . . . 4 ((𝐾 ∈ Lat ∧ (𝑝𝐴𝑞𝐴)) → (𝑋 = (𝑀‘(𝑝 𝑞)) ↔ 𝑋 = {𝑟𝐴𝑟(le‘𝐾)(𝑝 𝑞)}))
1817anbi2d 614 . . 3 ((𝐾 ∈ Lat ∧ (𝑝𝐴𝑞𝐴)) → ((𝑝𝑞𝑋 = (𝑀‘(𝑝 𝑞))) ↔ (𝑝𝑞𝑋 = {𝑟𝐴𝑟(le‘𝐾)(𝑝 𝑞)})))
19182rexbidva 3204 . 2 (𝐾 ∈ Lat → (∃𝑝𝐴𝑞𝐴 (𝑝𝑞𝑋 = (𝑀‘(𝑝 𝑞))) ↔ ∃𝑝𝐴𝑞𝐴 (𝑝𝑞𝑋 = {𝑟𝐴𝑟(le‘𝐾)(𝑝 𝑞)})))
205, 19bitr4d 271 1 (𝐾 ∈ Lat → (𝑋𝑁 ↔ ∃𝑝𝐴𝑞𝐴 (𝑝𝑞𝑋 = (𝑀‘(𝑝 𝑞)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wne 2943  wrex 3062  {crab 3065   class class class wbr 4786  cfv 6031  (class class class)co 6793  Basecbs 16064  lecple 16156  joincjn 17152  Latclat 17253  Atomscatm 35072  Linesclines 35302  pmapcpmap 35305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-lub 17182  df-glb 17183  df-join 17184  df-meet 17185  df-lat 17254  df-ats 35076  df-lines 35309  df-pmap 35312
This theorem is referenced by:  isline3  35584  lncvrelatN  35589  linepsubclN  35759
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