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Theorem tglnpt2 28159
Description: Find a second point on a line. (Contributed by Thierry Arnoux, 18-Oct-2019.)
Hypotheses
Ref Expression
tglnpt2.p 𝑃 = (Baseβ€˜πΊ)
tglnpt2.i 𝐼 = (Itvβ€˜πΊ)
tglnpt2.l 𝐿 = (LineGβ€˜πΊ)
tglnpt2.g (πœ‘ β†’ 𝐺 ∈ TarskiG)
tglnpt2.a (πœ‘ β†’ 𝐴 ∈ ran 𝐿)
tglnpt2.x (πœ‘ β†’ 𝑋 ∈ 𝐴)
Assertion
Ref Expression
tglnpt2 (πœ‘ β†’ βˆƒπ‘¦ ∈ 𝐴 𝑋 β‰  𝑦)
Distinct variable groups:   𝑦,𝐴   𝑦,𝑋
Allowed substitution hints:   πœ‘(𝑦)   𝑃(𝑦)   𝐺(𝑦)   𝐼(𝑦)   𝐿(𝑦)

Proof of Theorem tglnpt2
Dummy variables π‘₯ 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tglnpt2.p . . . . . 6 𝑃 = (Baseβ€˜πΊ)
2 tglnpt2.i . . . . . 6 𝐼 = (Itvβ€˜πΊ)
3 tglnpt2.l . . . . . 6 𝐿 = (LineGβ€˜πΊ)
4 tglnpt2.g . . . . . . 7 (πœ‘ β†’ 𝐺 ∈ TarskiG)
54ad4antr 728 . . . . . 6 (((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ (𝐴 = (π‘₯𝐿𝑧) ∧ π‘₯ β‰  𝑧)) ∧ 𝑋 = π‘₯) β†’ 𝐺 ∈ TarskiG)
6 simp-4r 780 . . . . . 6 (((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ (𝐴 = (π‘₯𝐿𝑧) ∧ π‘₯ β‰  𝑧)) ∧ 𝑋 = π‘₯) β†’ π‘₯ ∈ 𝑃)
7 simpllr 772 . . . . . 6 (((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ (𝐴 = (π‘₯𝐿𝑧) ∧ π‘₯ β‰  𝑧)) ∧ 𝑋 = π‘₯) β†’ 𝑧 ∈ 𝑃)
8 simplrr 774 . . . . . 6 (((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ (𝐴 = (π‘₯𝐿𝑧) ∧ π‘₯ β‰  𝑧)) ∧ 𝑋 = π‘₯) β†’ π‘₯ β‰  𝑧)
91, 2, 3, 5, 6, 7, 8tglinerflx2 28152 . . . . 5 (((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ (𝐴 = (π‘₯𝐿𝑧) ∧ π‘₯ β‰  𝑧)) ∧ 𝑋 = π‘₯) β†’ 𝑧 ∈ (π‘₯𝐿𝑧))
10 simplrl 773 . . . . 5 (((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ (𝐴 = (π‘₯𝐿𝑧) ∧ π‘₯ β‰  𝑧)) ∧ 𝑋 = π‘₯) β†’ 𝐴 = (π‘₯𝐿𝑧))
119, 10eleqtrrd 2834 . . . 4 (((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ (𝐴 = (π‘₯𝐿𝑧) ∧ π‘₯ β‰  𝑧)) ∧ 𝑋 = π‘₯) β†’ 𝑧 ∈ 𝐴)
12 simpr 483 . . . . 5 (((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ (𝐴 = (π‘₯𝐿𝑧) ∧ π‘₯ β‰  𝑧)) ∧ 𝑋 = π‘₯) β†’ 𝑋 = π‘₯)
1312, 8eqnetrd 3006 . . . 4 (((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ (𝐴 = (π‘₯𝐿𝑧) ∧ π‘₯ β‰  𝑧)) ∧ 𝑋 = π‘₯) β†’ 𝑋 β‰  𝑧)
14 neeq2 3002 . . . . 5 (𝑦 = 𝑧 β†’ (𝑋 β‰  𝑦 ↔ 𝑋 β‰  𝑧))
1514rspcev 3611 . . . 4 ((𝑧 ∈ 𝐴 ∧ 𝑋 β‰  𝑧) β†’ βˆƒπ‘¦ ∈ 𝐴 𝑋 β‰  𝑦)
1611, 13, 15syl2anc 582 . . 3 (((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ (𝐴 = (π‘₯𝐿𝑧) ∧ π‘₯ β‰  𝑧)) ∧ 𝑋 = π‘₯) β†’ βˆƒπ‘¦ ∈ 𝐴 𝑋 β‰  𝑦)
174ad4antr 728 . . . . . 6 (((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ (𝐴 = (π‘₯𝐿𝑧) ∧ π‘₯ β‰  𝑧)) ∧ 𝑋 β‰  π‘₯) β†’ 𝐺 ∈ TarskiG)
18 simp-4r 780 . . . . . 6 (((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ (𝐴 = (π‘₯𝐿𝑧) ∧ π‘₯ β‰  𝑧)) ∧ 𝑋 β‰  π‘₯) β†’ π‘₯ ∈ 𝑃)
19 simpllr 772 . . . . . 6 (((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ (𝐴 = (π‘₯𝐿𝑧) ∧ π‘₯ β‰  𝑧)) ∧ 𝑋 β‰  π‘₯) β†’ 𝑧 ∈ 𝑃)
20 simplrr 774 . . . . . 6 (((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ (𝐴 = (π‘₯𝐿𝑧) ∧ π‘₯ β‰  𝑧)) ∧ 𝑋 β‰  π‘₯) β†’ π‘₯ β‰  𝑧)
211, 2, 3, 17, 18, 19, 20tglinerflx1 28151 . . . . 5 (((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ (𝐴 = (π‘₯𝐿𝑧) ∧ π‘₯ β‰  𝑧)) ∧ 𝑋 β‰  π‘₯) β†’ π‘₯ ∈ (π‘₯𝐿𝑧))
22 simplrl 773 . . . . 5 (((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ (𝐴 = (π‘₯𝐿𝑧) ∧ π‘₯ β‰  𝑧)) ∧ 𝑋 β‰  π‘₯) β†’ 𝐴 = (π‘₯𝐿𝑧))
2321, 22eleqtrrd 2834 . . . 4 (((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ (𝐴 = (π‘₯𝐿𝑧) ∧ π‘₯ β‰  𝑧)) ∧ 𝑋 β‰  π‘₯) β†’ π‘₯ ∈ 𝐴)
24 simpr 483 . . . 4 (((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ (𝐴 = (π‘₯𝐿𝑧) ∧ π‘₯ β‰  𝑧)) ∧ 𝑋 β‰  π‘₯) β†’ 𝑋 β‰  π‘₯)
25 neeq2 3002 . . . . 5 (𝑦 = π‘₯ β†’ (𝑋 β‰  𝑦 ↔ 𝑋 β‰  π‘₯))
2625rspcev 3611 . . . 4 ((π‘₯ ∈ 𝐴 ∧ 𝑋 β‰  π‘₯) β†’ βˆƒπ‘¦ ∈ 𝐴 𝑋 β‰  𝑦)
2723, 24, 26syl2anc 582 . . 3 (((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ (𝐴 = (π‘₯𝐿𝑧) ∧ π‘₯ β‰  𝑧)) ∧ 𝑋 β‰  π‘₯) β†’ βˆƒπ‘¦ ∈ 𝐴 𝑋 β‰  𝑦)
2816, 27pm2.61dane 3027 . 2 ((((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) ∧ (𝐴 = (π‘₯𝐿𝑧) ∧ π‘₯ β‰  𝑧)) β†’ βˆƒπ‘¦ ∈ 𝐴 𝑋 β‰  𝑦)
29 tglnpt2.a . . 3 (πœ‘ β†’ 𝐴 ∈ ran 𝐿)
301, 2, 3, 4, 29tgisline 28145 . 2 (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝑃 βˆƒπ‘§ ∈ 𝑃 (𝐴 = (π‘₯𝐿𝑧) ∧ π‘₯ β‰  𝑧))
3128, 30r19.29vva 3211 1 (πœ‘ β†’ βˆƒπ‘¦ ∈ 𝐴 𝑋 β‰  𝑦)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104   β‰  wne 2938  βˆƒwrex 3068  ran crn 5676  β€˜cfv 6542  (class class class)co 7411  Basecbs 17148  TarskiGcstrkg 27945  Itvcitv 27951  LineGclng 27952
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-trkgc 27966  df-trkgb 27967  df-trkgcb 27968  df-trkg 27971
This theorem is referenced by:  perpneq  28232  perpdrag  28246  oppperpex  28271  lnperpex  28321
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