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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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