MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tglnpt2 Structured version   Visualization version   GIF version

Theorem tglnpt2 27892
Description: Find a second point on a line. (Contributed by Thierry Arnoux, 18-Oct-2019.)
Hypotheses
Ref Expression
tglnpt2.p 𝑃 = (Baseβ€˜πΊ)
tglnpt2.i 𝐼 = (Itvβ€˜πΊ)
tglnpt2.l 𝐿 = (LineGβ€˜πΊ)
tglnpt2.g (πœ‘ β†’ 𝐺 ∈ TarskiG)
tglnpt2.a (πœ‘ β†’ 𝐴 ∈ ran 𝐿)
tglnpt2.x (πœ‘ β†’ 𝑋 ∈ 𝐴)
Assertion
Ref Expression
tglnpt2 (πœ‘ β†’ βˆƒπ‘¦ ∈ 𝐴 𝑋 β‰  𝑦)
Distinct variable groups:   𝑦,𝐴   𝑦,𝑋
Allowed substitution hints:   πœ‘(𝑦)   𝑃(𝑦)   𝐺(𝑦)   𝐼(𝑦)   𝐿(𝑦)

Proof of Theorem tglnpt2
Dummy variables π‘₯ 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tglnpt2.p . . . . . 6 𝑃 = (Baseβ€˜πΊ)
2 tglnpt2.i . . . . . 6 𝐼 = (Itvβ€˜πΊ)
3 tglnpt2.l . . . . . 6 𝐿 = (LineGβ€˜πΊ)
4 tglnpt2.g . . . . . . 7 (πœ‘ β†’ 𝐺 ∈ TarskiG)
54ad4antr 731 . . . . . 6 (((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ (𝐴 = (π‘₯𝐿𝑧) ∧ π‘₯ β‰  𝑧)) ∧ 𝑋 = π‘₯) β†’ 𝐺 ∈ TarskiG)
6 simp-4r 783 . . . . . 6 (((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ (𝐴 = (π‘₯𝐿𝑧) ∧ π‘₯ β‰  𝑧)) ∧ 𝑋 = π‘₯) β†’ π‘₯ ∈ 𝑃)
7 simpllr 775 . . . . . 6 (((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ (𝐴 = (π‘₯𝐿𝑧) ∧ π‘₯ β‰  𝑧)) ∧ 𝑋 = π‘₯) β†’ 𝑧 ∈ 𝑃)
8 simplrr 777 . . . . . 6 (((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ (𝐴 = (π‘₯𝐿𝑧) ∧ π‘₯ β‰  𝑧)) ∧ 𝑋 = π‘₯) β†’ π‘₯ β‰  𝑧)
91, 2, 3, 5, 6, 7, 8tglinerflx2 27885 . . . . 5 (((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ (𝐴 = (π‘₯𝐿𝑧) ∧ π‘₯ β‰  𝑧)) ∧ 𝑋 = π‘₯) β†’ 𝑧 ∈ (π‘₯𝐿𝑧))
10 simplrl 776 . . . . 5 (((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ (𝐴 = (π‘₯𝐿𝑧) ∧ π‘₯ β‰  𝑧)) ∧ 𝑋 = π‘₯) β†’ 𝐴 = (π‘₯𝐿𝑧))
119, 10eleqtrrd 2837 . . . 4 (((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ (𝐴 = (π‘₯𝐿𝑧) ∧ π‘₯ β‰  𝑧)) ∧ 𝑋 = π‘₯) β†’ 𝑧 ∈ 𝐴)
12 simpr 486 . . . . 5 (((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ (𝐴 = (π‘₯𝐿𝑧) ∧ π‘₯ β‰  𝑧)) ∧ 𝑋 = π‘₯) β†’ 𝑋 = π‘₯)
1312, 8eqnetrd 3009 . . . 4 (((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ (𝐴 = (π‘₯𝐿𝑧) ∧ π‘₯ β‰  𝑧)) ∧ 𝑋 = π‘₯) β†’ 𝑋 β‰  𝑧)
14 neeq2 3005 . . . . 5 (𝑦 = 𝑧 β†’ (𝑋 β‰  𝑦 ↔ 𝑋 β‰  𝑧))
1514rspcev 3613 . . . 4 ((𝑧 ∈ 𝐴 ∧ 𝑋 β‰  𝑧) β†’ βˆƒπ‘¦ ∈ 𝐴 𝑋 β‰  𝑦)
1611, 13, 15syl2anc 585 . . 3 (((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ (𝐴 = (π‘₯𝐿𝑧) ∧ π‘₯ β‰  𝑧)) ∧ 𝑋 = π‘₯) β†’ βˆƒπ‘¦ ∈ 𝐴 𝑋 β‰  𝑦)
174ad4antr 731 . . . . . 6 (((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ (𝐴 = (π‘₯𝐿𝑧) ∧ π‘₯ β‰  𝑧)) ∧ 𝑋 β‰  π‘₯) β†’ 𝐺 ∈ TarskiG)
18 simp-4r 783 . . . . . 6 (((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ (𝐴 = (π‘₯𝐿𝑧) ∧ π‘₯ β‰  𝑧)) ∧ 𝑋 β‰  π‘₯) β†’ π‘₯ ∈ 𝑃)
19 simpllr 775 . . . . . 6 (((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ (𝐴 = (π‘₯𝐿𝑧) ∧ π‘₯ β‰  𝑧)) ∧ 𝑋 β‰  π‘₯) β†’ 𝑧 ∈ 𝑃)
20 simplrr 777 . . . . . 6 (((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ (𝐴 = (π‘₯𝐿𝑧) ∧ π‘₯ β‰  𝑧)) ∧ 𝑋 β‰  π‘₯) β†’ π‘₯ β‰  𝑧)
211, 2, 3, 17, 18, 19, 20tglinerflx1 27884 . . . . 5 (((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ (𝐴 = (π‘₯𝐿𝑧) ∧ π‘₯ β‰  𝑧)) ∧ 𝑋 β‰  π‘₯) β†’ π‘₯ ∈ (π‘₯𝐿𝑧))
22 simplrl 776 . . . . 5 (((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ (𝐴 = (π‘₯𝐿𝑧) ∧ π‘₯ β‰  𝑧)) ∧ 𝑋 β‰  π‘₯) β†’ 𝐴 = (π‘₯𝐿𝑧))
2321, 22eleqtrrd 2837 . . . 4 (((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ (𝐴 = (π‘₯𝐿𝑧) ∧ π‘₯ β‰  𝑧)) ∧ 𝑋 β‰  π‘₯) β†’ π‘₯ ∈ 𝐴)
24 simpr 486 . . . 4 (((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ (𝐴 = (π‘₯𝐿𝑧) ∧ π‘₯ β‰  𝑧)) ∧ 𝑋 β‰  π‘₯) β†’ 𝑋 β‰  π‘₯)
25 neeq2 3005 . . . . 5 (𝑦 = π‘₯ β†’ (𝑋 β‰  𝑦 ↔ 𝑋 β‰  π‘₯))
2625rspcev 3613 . . . 4 ((π‘₯ ∈ 𝐴 ∧ 𝑋 β‰  π‘₯) β†’ βˆƒπ‘¦ ∈ 𝐴 𝑋 β‰  𝑦)
2723, 24, 26syl2anc 585 . . 3 (((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ (𝐴 = (π‘₯𝐿𝑧) ∧ π‘₯ β‰  𝑧)) ∧ 𝑋 β‰  π‘₯) β†’ βˆƒπ‘¦ ∈ 𝐴 𝑋 β‰  𝑦)
2816, 27pm2.61dane 3030 . 2 ((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ (𝐴 = (π‘₯𝐿𝑧) ∧ π‘₯ β‰  𝑧)) β†’ βˆƒπ‘¦ ∈ 𝐴 𝑋 β‰  𝑦)
29 tglnpt2.a . . 3 (πœ‘ β†’ 𝐴 ∈ ran 𝐿)
301, 2, 3, 4, 29tgisline 27878 . 2 (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝑃 βˆƒπ‘§ ∈ 𝑃 (𝐴 = (π‘₯𝐿𝑧) ∧ π‘₯ β‰  𝑧))
3128, 30r19.29vva 3214 1 (πœ‘ β†’ βˆƒπ‘¦ ∈ 𝐴 𝑋 β‰  𝑦)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆƒwrex 3071  ran crn 5678  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  TarskiGcstrkg 27678  Itvcitv 27684  LineGclng 27685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-trkgc 27699  df-trkgb 27700  df-trkgcb 27701  df-trkg 27704
This theorem is referenced by:  perpneq  27965  perpdrag  27979  oppperpex  28004  lnperpex  28054
  Copyright terms: Public domain W3C validator