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Theorem tgelrnln 28462
Description: The property of being a proper line, generated by two distinct points. (Contributed by Thierry Arnoux, 25-May-2019.)
Hypotheses
Ref Expression
tglineelsb2.p 𝐡 = (Baseβ€˜πΊ)
tglineelsb2.i 𝐼 = (Itvβ€˜πΊ)
tglineelsb2.l 𝐿 = (LineGβ€˜πΊ)
tglineelsb2.g (πœ‘ β†’ 𝐺 ∈ TarskiG)
tgelrnln.x (πœ‘ β†’ 𝑋 ∈ 𝐡)
tgelrnln.y (πœ‘ β†’ π‘Œ ∈ 𝐡)
tgelrnln.d (πœ‘ β†’ 𝑋 β‰  π‘Œ)
Assertion
Ref Expression
tgelrnln (πœ‘ β†’ (π‘‹πΏπ‘Œ) ∈ ran 𝐿)

Proof of Theorem tgelrnln
StepHypRef Expression
1 df-ov 7429 . 2 (π‘‹πΏπ‘Œ) = (πΏβ€˜βŸ¨π‘‹, π‘ŒβŸ©)
2 tglineelsb2.g . . . 4 (πœ‘ β†’ 𝐺 ∈ TarskiG)
3 tglineelsb2.p . . . . 5 𝐡 = (Baseβ€˜πΊ)
4 tglineelsb2.l . . . . 5 𝐿 = (LineGβ€˜πΊ)
5 tglineelsb2.i . . . . 5 𝐼 = (Itvβ€˜πΊ)
63, 4, 5tglnfn 28379 . . . 4 (𝐺 ∈ TarskiG β†’ 𝐿 Fn ((𝐡 Γ— 𝐡) βˆ– I ))
72, 6syl 17 . . 3 (πœ‘ β†’ 𝐿 Fn ((𝐡 Γ— 𝐡) βˆ– I ))
8 tgelrnln.x . . . . 5 (πœ‘ β†’ 𝑋 ∈ 𝐡)
9 tgelrnln.y . . . . 5 (πœ‘ β†’ π‘Œ ∈ 𝐡)
108, 9opelxpd 5721 . . . 4 (πœ‘ β†’ βŸ¨π‘‹, π‘ŒβŸ© ∈ (𝐡 Γ— 𝐡))
11 tgelrnln.d . . . . 5 (πœ‘ β†’ 𝑋 β‰  π‘Œ)
12 df-br 5153 . . . . . . . 8 (𝑋 I π‘Œ ↔ βŸ¨π‘‹, π‘ŒβŸ© ∈ I )
13 ideqg 5858 . . . . . . . 8 (π‘Œ ∈ 𝐡 β†’ (𝑋 I π‘Œ ↔ 𝑋 = π‘Œ))
1412, 13bitr3id 284 . . . . . . 7 (π‘Œ ∈ 𝐡 β†’ (βŸ¨π‘‹, π‘ŒβŸ© ∈ I ↔ 𝑋 = π‘Œ))
1514necon3bbid 2975 . . . . . 6 (π‘Œ ∈ 𝐡 β†’ (Β¬ βŸ¨π‘‹, π‘ŒβŸ© ∈ I ↔ 𝑋 β‰  π‘Œ))
1615biimpar 476 . . . . 5 ((π‘Œ ∈ 𝐡 ∧ 𝑋 β‰  π‘Œ) β†’ Β¬ βŸ¨π‘‹, π‘ŒβŸ© ∈ I )
179, 11, 16syl2anc 582 . . . 4 (πœ‘ β†’ Β¬ βŸ¨π‘‹, π‘ŒβŸ© ∈ I )
1810, 17eldifd 3960 . . 3 (πœ‘ β†’ βŸ¨π‘‹, π‘ŒβŸ© ∈ ((𝐡 Γ— 𝐡) βˆ– I ))
19 fnfvelrn 7095 . . 3 ((𝐿 Fn ((𝐡 Γ— 𝐡) βˆ– I ) ∧ βŸ¨π‘‹, π‘ŒβŸ© ∈ ((𝐡 Γ— 𝐡) βˆ– I )) β†’ (πΏβ€˜βŸ¨π‘‹, π‘ŒβŸ©) ∈ ran 𝐿)
207, 18, 19syl2anc 582 . 2 (πœ‘ β†’ (πΏβ€˜βŸ¨π‘‹, π‘ŒβŸ©) ∈ ran 𝐿)
211, 20eqeltrid 2833 1 (πœ‘ β†’ (π‘‹πΏπ‘Œ) ∈ ran 𝐿)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   = wceq 1533   ∈ wcel 2098   β‰  wne 2937   βˆ– cdif 3946  βŸ¨cop 4638   class class class wbr 5152   I cid 5579   Γ— cxp 5680  ran crn 5683   Fn wfn 6548  β€˜cfv 6553  (class class class)co 7426  Basecbs 17189  TarskiGcstrkg 28259  Itvcitv 28265  LineGclng 28266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 8001  df-2nd 8002  df-trkg 28285
This theorem is referenced by:  tghilberti1  28469  tglineinteq  28477  colline  28481  tglowdim2ln  28483  footexALT  28550  footexlem2  28552  foot  28554  perprag  28558  colperpexlem3  28564  mideulem2  28566  midex  28569  outpasch  28587  lnopp2hpgb  28595  colopp  28601  lmieu  28616  lmimid  28626  hypcgrlem1  28631  hypcgrlem2  28632  lnperpex  28635  trgcopy  28636  trgcopyeulem  28637  acopy  28665  acopyeu  28666  tgasa1  28690
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