MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tgelrnln Structured version   Visualization version   GIF version

Theorem tgelrnln 28389
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 7408 . 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 28306 . . . 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 5708 . . . 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 5142 . . . . . . . 8 (𝑋 I π‘Œ ↔ βŸ¨π‘‹, π‘ŒβŸ© ∈ I )
13 ideqg 5845 . . . . . . . 8 (π‘Œ ∈ 𝐡 β†’ (𝑋 I π‘Œ ↔ 𝑋 = π‘Œ))
1412, 13bitr3id 285 . . . . . . 7 (π‘Œ ∈ 𝐡 β†’ (βŸ¨π‘‹, π‘ŒβŸ© ∈ I ↔ 𝑋 = π‘Œ))
1514necon3bbid 2972 . . . . . 6 (π‘Œ ∈ 𝐡 β†’ (Β¬ βŸ¨π‘‹, π‘ŒβŸ© ∈ I ↔ 𝑋 β‰  π‘Œ))
1615biimpar 477 . . . . 5 ((π‘Œ ∈ 𝐡 ∧ 𝑋 β‰  π‘Œ) β†’ Β¬ βŸ¨π‘‹, π‘ŒβŸ© ∈ I )
179, 11, 16syl2anc 583 . . . 4 (πœ‘ β†’ Β¬ βŸ¨π‘‹, π‘ŒβŸ© ∈ I )
1810, 17eldifd 3954 . . 3 (πœ‘ β†’ βŸ¨π‘‹, π‘ŒβŸ© ∈ ((𝐡 Γ— 𝐡) βˆ– I ))
19 fnfvelrn 7076 . . 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 583 . 2 (πœ‘ β†’ (πΏβ€˜βŸ¨π‘‹, π‘ŒβŸ©) ∈ ran 𝐿)
211, 20eqeltrid 2831 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 2934   βˆ– cdif 3940  βŸ¨cop 4629   class class class wbr 5141   I cid 5566   Γ— cxp 5667  ran crn 5670   Fn wfn 6532  β€˜cfv 6537  (class class class)co 7405  Basecbs 17153  TarskiGcstrkg 28186  Itvcitv 28192  LineGclng 28193
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-trkg 28212
This theorem is referenced by:  tghilberti1  28396  tglineinteq  28404  colline  28408  tglowdim2ln  28410  footexALT  28477  footexlem2  28479  foot  28481  perprag  28485  colperpexlem3  28491  mideulem2  28493  midex  28496  outpasch  28514  lnopp2hpgb  28522  colopp  28528  lmieu  28543  lmimid  28553  hypcgrlem1  28558  hypcgrlem2  28559  lnperpex  28562  trgcopy  28563  trgcopyeulem  28564  acopy  28592  acopyeu  28593  tgasa1  28617
  Copyright terms: Public domain W3C validator