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Theorem tglnpt2 25436
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 767 . . . . . 6 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋 = 𝑥) → 𝐺 ∈ TarskiG)
6 simp-4r 806 . . . . . 6 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋 = 𝑥) → 𝑥𝑃)
7 simpllr 798 . . . . . 6 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋 = 𝑥) → 𝑧𝑃)
8 simplrr 800 . . . . . 6 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋 = 𝑥) → 𝑥𝑧)
91, 2, 3, 5, 6, 7, 8tglinerflx2 25429 . . . . 5 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋 = 𝑥) → 𝑧 ∈ (𝑥𝐿𝑧))
10 simplrl 799 . . . . 5 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋 = 𝑥) → 𝐴 = (𝑥𝐿𝑧))
119, 10eleqtrrd 2701 . . . 4 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋 = 𝑥) → 𝑧𝐴)
12 simpr 477 . . . . 5 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋 = 𝑥) → 𝑋 = 𝑥)
1312, 8eqnetrd 2857 . . . 4 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋 = 𝑥) → 𝑋𝑧)
14 neeq2 2853 . . . . 5 (𝑦 = 𝑧 → (𝑋𝑦𝑋𝑧))
1514rspcev 3295 . . . 4 ((𝑧𝐴𝑋𝑧) → ∃𝑦𝐴 𝑋𝑦)
1611, 13, 15syl2anc 692 . . 3 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋 = 𝑥) → ∃𝑦𝐴 𝑋𝑦)
174ad4antr 767 . . . . . 6 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋𝑥) → 𝐺 ∈ TarskiG)
18 simp-4r 806 . . . . . 6 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋𝑥) → 𝑥𝑃)
19 simpllr 798 . . . . . 6 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋𝑥) → 𝑧𝑃)
20 simplrr 800 . . . . . 6 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋𝑥) → 𝑥𝑧)
211, 2, 3, 17, 18, 19, 20tglinerflx1 25428 . . . . 5 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋𝑥) → 𝑥 ∈ (𝑥𝐿𝑧))
22 simplrl 799 . . . . 5 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋𝑥) → 𝐴 = (𝑥𝐿𝑧))
2321, 22eleqtrrd 2701 . . . 4 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋𝑥) → 𝑥𝐴)
24 simpr 477 . . . 4 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋𝑥) → 𝑋𝑥)
25 neeq2 2853 . . . . 5 (𝑦 = 𝑥 → (𝑋𝑦𝑋𝑥))
2625rspcev 3295 . . . 4 ((𝑥𝐴𝑋𝑥) → ∃𝑦𝐴 𝑋𝑦)
2723, 24, 26syl2anc 692 . . 3 (((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) ∧ 𝑋𝑥) → ∃𝑦𝐴 𝑋𝑦)
2816, 27pm2.61dane 2877 . 2 ((((𝜑𝑥𝑃) ∧ 𝑧𝑃) ∧ (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧)) → ∃𝑦𝐴 𝑋𝑦)
29 tglnpt2.a . . 3 (𝜑𝐴 ∈ ran 𝐿)
301, 2, 3, 4, 29tgisline 25422 . 2 (𝜑 → ∃𝑥𝑃𝑧𝑃 (𝐴 = (𝑥𝐿𝑧) ∧ 𝑥𝑧))
3128, 30r19.29vva 3073 1 (𝜑 → ∃𝑦𝐴 𝑋𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wne 2790  wrex 2908  ran crn 5075  cfv 5847  (class class class)co 6604  Basecbs 15781  TarskiGcstrkg 25229  Itvcitv 25235  LineGclng 25236
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-trkgc 25247  df-trkgb 25248  df-trkgcb 25249  df-trkg 25252
This theorem is referenced by:  perpneq  25509  perpdrag  25520  oppperpex  25545  lnperpex  25595
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