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Theorem tglnpt2 26444
 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 731 . . . . . 6 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋 = 𝑥) → 𝐺 ∈ TarskiG)
6 simp-4r 783 . . . . . 6 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋 = 𝑥) → 𝑥𝑃)
7 simpllr 775 . . . . . 6 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋 = 𝑥) → 𝑧𝑃)
8 simplrr 777 . . . . . 6 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋 = 𝑥) → 𝑥𝑧)
91, 2, 3, 5, 6, 7, 8tglinerflx2 26437 . . . . 5 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋 = 𝑥) → 𝑧 ∈ (𝑥𝐿𝑧))
10 simplrl 776 . . . . 5 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋 = 𝑥) → 𝐴 = (𝑥𝐿𝑧))
119, 10eleqtrrd 2919 . . . 4 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋 = 𝑥) → 𝑧𝐴)
12 simpr 488 . . . . 5 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋 = 𝑥) → 𝑋 = 𝑥)
1312, 8eqnetrd 3081 . . . 4 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋 = 𝑥) → 𝑋𝑧)
14 neeq2 3077 . . . . 5 (𝑦 = 𝑧 → (𝑋𝑦𝑋𝑧))
1514rspcev 3609 . . . 4 ((𝑧𝐴𝑋𝑧) → ∃𝑦𝐴 𝑋𝑦)
1611, 13, 15syl2anc 587 . . 3 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋 = 𝑥) → ∃𝑦𝐴 𝑋𝑦)
174ad4antr 731 . . . . . 6 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋𝑥) → 𝐺 ∈ TarskiG)
18 simp-4r 783 . . . . . 6 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋𝑥) → 𝑥𝑃)
19 simpllr 775 . . . . . 6 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋𝑥) → 𝑧𝑃)
20 simplrr 777 . . . . . 6 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋𝑥) → 𝑥𝑧)
211, 2, 3, 17, 18, 19, 20tglinerflx1 26436 . . . . 5 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋𝑥) → 𝑥 ∈ (𝑥𝐿𝑧))
22 simplrl 776 . . . . 5 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋𝑥) → 𝐴 = (𝑥𝐿𝑧))
2321, 22eleqtrrd 2919 . . . 4 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋𝑥) → 𝑥𝐴)
24 simpr 488 . . . 4 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋𝑥) → 𝑋𝑥)
25 neeq2 3077 . . . . 5 (𝑦 = 𝑥 → (𝑋𝑦𝑋𝑥))
2625rspcev 3609 . . . 4 ((𝑥𝐴𝑋𝑥) → ∃𝑦𝐴 𝑋𝑦)
2723, 24, 26syl2anc 587 . . 3 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋𝑥) → ∃𝑦𝐴 𝑋𝑦)
2816, 27pm2.61dane 3101 . 2 ((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) → ∃𝑦𝐴 𝑋𝑦)
29 tglnpt2.a . . 3 (𝜑𝐴 ∈ ran 𝐿)
301, 2, 3, 4, 29tgisline 26430 . 2 (𝜑 → ∃𝑥𝑃𝑧𝑃 (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧))
3128, 30r19.29vva 3327 1 (𝜑 → ∃𝑦𝐴 𝑋𝑦)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2115   ≠ wne 3014  ∃wrex 3134  ran crn 5544  ‘cfv 6345  (class class class)co 7151  Basecbs 16485  TarskiGcstrkg 26233  Itvcitv 26239  LineGclng 26240 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5054  df-opab 5116  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-iota 6304  df-fun 6347  df-fv 6353  df-ov 7154  df-oprab 7155  df-mpo 7156  df-trkgc 26251  df-trkgb 26252  df-trkgcb 26253  df-trkg 26256 This theorem is referenced by:  perpneq  26517  perpdrag  26531  oppperpex  26556  lnperpex  26606
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