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

Theorem tglnpt2 28664
Description: Find a second point on a line. (Contributed by Thierry Arnoux, 18-Oct-2019.)
Hypotheses
Ref Expression
tglnpt2.p 𝑃 = (Base‘𝐺)
tglnpt2.i 𝐼 = (Itv‘𝐺)
tglnpt2.l 𝐿 = (LineG‘𝐺)
tglnpt2.g (𝜑𝐺 ∈ TarskiG)
tglnpt2.a (𝜑𝐴 ∈ ran 𝐿)
tglnpt2.x (𝜑𝑋𝐴)
Assertion
Ref Expression
tglnpt2 (𝜑 → ∃𝑦𝐴 𝑋𝑦)
Distinct variable groups:   𝑦,𝐴   𝑦,𝑋
Allowed substitution hints:   𝜑(𝑦)   𝑃(𝑦)   𝐺(𝑦)   𝐼(𝑦)   𝐿(𝑦)

Proof of Theorem tglnpt2
Dummy variables 𝑥 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tglnpt2.p . . . . . 6 𝑃 = (Base‘𝐺)
2 tglnpt2.i . . . . . 6 𝐼 = (Itv‘𝐺)
3 tglnpt2.l . . . . . 6 𝐿 = (LineG‘𝐺)
4 tglnpt2.g . . . . . . 7 (𝜑𝐺 ∈ TarskiG)
54ad4antr 732 . . . . . 6 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋 = 𝑥) → 𝐺 ∈ TarskiG)
6 simp-4r 784 . . . . . 6 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋 = 𝑥) → 𝑥𝑃)
7 simpllr 776 . . . . . 6 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋 = 𝑥) → 𝑧𝑃)
8 simplrr 778 . . . . . 6 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋 = 𝑥) → 𝑥𝑧)
91, 2, 3, 5, 6, 7, 8tglinerflx2 28657 . . . . 5 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋 = 𝑥) → 𝑧 ∈ (𝑥𝐿𝑧))
10 simplrl 777 . . . . 5 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋 = 𝑥) → 𝐴 = (𝑥𝐿𝑧))
119, 10eleqtrrd 2842 . . . 4 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋 = 𝑥) → 𝑧𝐴)
12 simpr 484 . . . . 5 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋 = 𝑥) → 𝑋 = 𝑥)
1312, 8eqnetrd 3006 . . . 4 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋 = 𝑥) → 𝑋𝑧)
14 neeq2 3002 . . . . 5 (𝑦 = 𝑧 → (𝑋𝑦𝑋𝑧))
1514rspcev 3622 . . . 4 ((𝑧𝐴𝑋𝑧) → ∃𝑦𝐴 𝑋𝑦)
1611, 13, 15syl2anc 584 . . 3 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋 = 𝑥) → ∃𝑦𝐴 𝑋𝑦)
174ad4antr 732 . . . . . 6 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋𝑥) → 𝐺 ∈ TarskiG)
18 simp-4r 784 . . . . . 6 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋𝑥) → 𝑥𝑃)
19 simpllr 776 . . . . . 6 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋𝑥) → 𝑧𝑃)
20 simplrr 778 . . . . . 6 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋𝑥) → 𝑥𝑧)
211, 2, 3, 17, 18, 19, 20tglinerflx1 28656 . . . . 5 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋𝑥) → 𝑥 ∈ (𝑥𝐿𝑧))
22 simplrl 777 . . . . 5 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋𝑥) → 𝐴 = (𝑥𝐿𝑧))
2321, 22eleqtrrd 2842 . . . 4 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋𝑥) → 𝑥𝐴)
24 simpr 484 . . . 4 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋𝑥) → 𝑋𝑥)
25 neeq2 3002 . . . . 5 (𝑦 = 𝑥 → (𝑋𝑦𝑋𝑥))
2625rspcev 3622 . . . 4 ((𝑥𝐴𝑋𝑥) → ∃𝑦𝐴 𝑋𝑦)
2723, 24, 26syl2anc 584 . . 3 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋𝑥) → ∃𝑦𝐴 𝑋𝑦)
2816, 27pm2.61dane 3027 . 2 ((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) → ∃𝑦𝐴 𝑋𝑦)
29 tglnpt2.a . . 3 (𝜑𝐴 ∈ ran 𝐿)
301, 2, 3, 4, 29tgisline 28650 . 2 (𝜑 → ∃𝑥𝑃𝑧𝑃 (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧))
3128, 30r19.29vva 3214 1 (𝜑 → ∃𝑦𝐴 𝑋𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wne 2938  wrex 3068  ran crn 5690  cfv 6563  (class class class)co 7431  Basecbs 17245  TarskiGcstrkg 28450  Itvcitv 28456  LineGclng 28457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-trkgc 28471  df-trkgb 28472  df-trkgcb 28473  df-trkg 28476
This theorem is referenced by:  perpneq  28737  perpdrag  28751  oppperpex  28776  lnperpex  28826
  Copyright terms: Public domain W3C validator