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Theorem tglngval 26039
Description: The line going through points 𝑋 and 𝑌. (Contributed by Thierry Arnoux, 28-Mar-2019.)
Hypotheses
Ref Expression
tglngval.p 𝑃 = (Base‘𝐺)
tglngval.l 𝐿 = (LineG‘𝐺)
tglngval.i 𝐼 = (Itv‘𝐺)
tglngval.g (𝜑𝐺 ∈ TarskiG)
tglngval.x (𝜑𝑋𝑃)
tglngval.y (𝜑𝑌𝑃)
tglngval.z (𝜑𝑋𝑌)
Assertion
Ref Expression
tglngval (𝜑 → (𝑋𝐿𝑌) = {𝑧𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))})
Distinct variable groups:   𝑧,𝐺   𝑧,𝐼   𝑧,𝑃   𝑧,𝑋   𝑧,𝑌   𝜑,𝑧
Allowed substitution hint:   𝐿(𝑧)

Proof of Theorem tglngval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tglngval.g . . . 4 (𝜑𝐺 ∈ TarskiG)
2 tglngval.p . . . . 5 𝑃 = (Base‘𝐺)
3 tglngval.l . . . . 5 𝐿 = (LineG‘𝐺)
4 tglngval.i . . . . 5 𝐼 = (Itv‘𝐺)
52, 3, 4tglng 26034 . . . 4 (𝐺 ∈ TarskiG → 𝐿 = (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
61, 5syl 17 . . 3 (𝜑𝐿 = (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
76oveqd 6993 . 2 (𝜑 → (𝑋𝐿𝑌) = (𝑋(𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})𝑌))
8 tglngval.x . . 3 (𝜑𝑋𝑃)
9 tglngval.y . . . 4 (𝜑𝑌𝑃)
10 tglngval.z . . . . 5 (𝜑𝑋𝑌)
1110necomd 3023 . . . 4 (𝜑𝑌𝑋)
12 eldifsn 4593 . . . 4 (𝑌 ∈ (𝑃 ∖ {𝑋}) ↔ (𝑌𝑃𝑌𝑋))
139, 11, 12sylanbrc 575 . . 3 (𝜑𝑌 ∈ (𝑃 ∖ {𝑋}))
142fvexi 6513 . . . . 5 𝑃 ∈ V
1514rabex 5091 . . . 4 {𝑧𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))} ∈ V
1615a1i 11 . . 3 (𝜑 → {𝑧𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))} ∈ V)
17 oveq12 6985 . . . . . . 7 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑥𝐼𝑦) = (𝑋𝐼𝑌))
1817eleq2d 2852 . . . . . 6 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑧 ∈ (𝑥𝐼𝑦) ↔ 𝑧 ∈ (𝑋𝐼𝑌)))
19 simpl 475 . . . . . . 7 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑥 = 𝑋)
20 simpr 477 . . . . . . . 8 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑦 = 𝑌)
2120oveq2d 6992 . . . . . . 7 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑧𝐼𝑦) = (𝑧𝐼𝑌))
2219, 21eleq12d 2861 . . . . . 6 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑥 ∈ (𝑧𝐼𝑦) ↔ 𝑋 ∈ (𝑧𝐼𝑌)))
2319oveq1d 6991 . . . . . . 7 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑥𝐼𝑧) = (𝑋𝐼𝑧))
2420, 23eleq12d 2861 . . . . . 6 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑦 ∈ (𝑥𝐼𝑧) ↔ 𝑌 ∈ (𝑋𝐼𝑧)))
2518, 22, 243orbi123d 1414 . . . . 5 ((𝑥 = 𝑋𝑦 = 𝑌) → ((𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)) ↔ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))))
2625rabbidv 3404 . . . 4 ((𝑥 = 𝑋𝑦 = 𝑌) → {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} = {𝑧𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))})
27 sneq 4451 . . . . 5 (𝑥 = 𝑋 → {𝑥} = {𝑋})
2827difeq2d 3990 . . . 4 (𝑥 = 𝑋 → (𝑃 ∖ {𝑥}) = (𝑃 ∖ {𝑋}))
29 eqid 2779 . . . 4 (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) = (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})
3026, 28, 29ovmpox 7119 . . 3 ((𝑋𝑃𝑌 ∈ (𝑃 ∖ {𝑋}) ∧ {𝑧𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))} ∈ V) → (𝑋(𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})𝑌) = {𝑧𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))})
318, 13, 16, 30syl3anc 1351 . 2 (𝜑 → (𝑋(𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})𝑌) = {𝑧𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))})
327, 31eqtrd 2815 1 (𝜑 → (𝑋𝐿𝑌) = {𝑧𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  w3o 1067   = wceq 1507  wcel 2050  wne 2968  {crab 3093  Vcvv 3416  cdif 3827  {csn 4441  cfv 6188  (class class class)co 6976  cmpo 6978  Basecbs 16339  TarskiGcstrkg 25918  Itvcitv 25924  LineGclng 25925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-sep 5060  ax-nul 5067  ax-pr 5186
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3418  df-sbc 3683  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-iota 6152  df-fun 6190  df-fv 6196  df-ov 6979  df-oprab 6980  df-mpo 6981  df-trkg 25941
This theorem is referenced by:  tglnssp  26040  tgellng  26041  tgisline  26115
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