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Theorem tglnpt2 28649
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 28642 . . . . 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 2844 . . . 4 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋 = 𝑥) → 𝑧𝐴)
12 simpr 484 . . . . 5 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋 = 𝑥) → 𝑋 = 𝑥)
1312, 8eqnetrd 3008 . . . 4 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋 = 𝑥) → 𝑋𝑧)
14 neeq2 3004 . . . . 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 28641 . . . . 5 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋𝑥) → 𝑥 ∈ (𝑥𝐿𝑧))
22 simplrl 777 . . . . 5 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋𝑥) → 𝐴 = (𝑥𝐿𝑧))
2321, 22eleqtrrd 2844 . . . 4 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋𝑥) → 𝑥𝐴)
24 simpr 484 . . . 4 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋𝑥) → 𝑋𝑥)
25 neeq2 3004 . . . . 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 3029 . 2 ((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) → ∃𝑦𝐴 𝑋𝑦)
29 tglnpt2.a . . 3 (𝜑𝐴 ∈ ran 𝐿)
301, 2, 3, 4, 29tgisline 28635 . 2 (𝜑 → ∃𝑥𝑃𝑧𝑃 (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧))
3128, 30r19.29vva 3216 1 (𝜑 → ∃𝑦𝐴 𝑋𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wne 2940  wrex 3070  ran crn 5686  cfv 6561  (class class class)co 7431  Basecbs 17247  TarskiGcstrkg 28435  Itvcitv 28441  LineGclng 28442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-trkgc 28456  df-trkgb 28457  df-trkgcb 28458  df-trkg 28461
This theorem is referenced by:  perpneq  28722  perpdrag  28736  oppperpex  28761  lnperpex  28811
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