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Theorem tglngval 28481
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 28476 . . . 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 7443 . 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 2986 . . . 4 (𝜑𝑌𝑋)
12 eldifsn 4795 . . . 4 (𝑌 ∈ (𝑃 ∖ {𝑋}) ↔ (𝑌𝑃𝑌𝑋))
139, 11, 12sylanbrc 581 . . 3 (𝜑𝑌 ∈ (𝑃 ∖ {𝑋}))
142fvexi 6917 . . . . 5 𝑃 ∈ V
1514rabex 5341 . . . 4 {𝑧𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))} ∈ V
1615a1i 11 . . 3 (𝜑 → {𝑧𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))} ∈ V)
17 oveq12 7435 . . . . . . 7 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑥𝐼𝑦) = (𝑋𝐼𝑌))
1817eleq2d 2812 . . . . . 6 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑧 ∈ (𝑥𝐼𝑦) ↔ 𝑧 ∈ (𝑋𝐼𝑌)))
19 simpl 481 . . . . . . 7 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑥 = 𝑋)
20 simpr 483 . . . . . . . 8 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑦 = 𝑌)
2120oveq2d 7442 . . . . . . 7 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑧𝐼𝑦) = (𝑧𝐼𝑌))
2219, 21eleq12d 2820 . . . . . 6 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑥 ∈ (𝑧𝐼𝑦) ↔ 𝑋 ∈ (𝑧𝐼𝑌)))
2319oveq1d 7441 . . . . . . 7 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑥𝐼𝑧) = (𝑋𝐼𝑧))
2420, 23eleq12d 2820 . . . . . 6 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑦 ∈ (𝑥𝐼𝑧) ↔ 𝑌 ∈ (𝑋𝐼𝑧)))
2518, 22, 243orbi123d 1432 . . . . 5 ((𝑥 = 𝑋𝑦 = 𝑌) → ((𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)) ↔ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))))
2625rabbidv 3427 . . . 4 ((𝑥 = 𝑋𝑦 = 𝑌) → {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} = {𝑧𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))})
27 sneq 4643 . . . . 5 (𝑥 = 𝑋 → {𝑥} = {𝑋})
2827difeq2d 4121 . . . 4 (𝑥 = 𝑋 → (𝑃 ∖ {𝑥}) = (𝑃 ∖ {𝑋}))
29 eqid 2726 . . . 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 7581 . . 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 1368 . 2 (𝜑 → (𝑋(𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})𝑌) = {𝑧𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))})
327, 31eqtrd 2766 1 (𝜑 → (𝑋𝐿𝑌) = {𝑧𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3o 1083   = wceq 1534  wcel 2099  wne 2930  {crab 3419  Vcvv 3462  cdif 3944  {csn 4633  cfv 6556  (class class class)co 7426  cmpo 7428  Basecbs 17215  TarskiGcstrkg 28357  Itvcitv 28363  LineGclng 28364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5306  ax-nul 5313  ax-pr 5435
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4916  df-br 5156  df-opab 5218  df-id 5582  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6508  df-fun 6558  df-fv 6564  df-ov 7429  df-oprab 7430  df-mpo 7431  df-trkg 28383
This theorem is referenced by:  tglnssp  28482  tgellng  28483  tgisline  28557
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