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Theorem tglngval 25191
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 25186 . . . 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 6543 . 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 2836 . . . 4 (𝜑𝑌𝑋)
12 eldifsn 4259 . . . 4 (𝑌 ∈ (𝑃 ∖ {𝑋}) ↔ (𝑌𝑃𝑌𝑋))
139, 11, 12sylanbrc 694 . . 3 (𝜑𝑌 ∈ (𝑃 ∖ {𝑋}))
14 fvex 6097 . . . . . 6 (Base‘𝐺) ∈ V
152, 14eqeltri 2683 . . . . 5 𝑃 ∈ V
1615rabex 4734 . . . 4 {𝑧𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))} ∈ V
1716a1i 11 . . 3 (𝜑 → {𝑧𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))} ∈ V)
18 oveq12 6535 . . . . . . 7 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑥𝐼𝑦) = (𝑋𝐼𝑌))
1918eleq2d 2672 . . . . . 6 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑧 ∈ (𝑥𝐼𝑦) ↔ 𝑧 ∈ (𝑋𝐼𝑌)))
20 simpl 471 . . . . . . 7 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑥 = 𝑋)
21 simpr 475 . . . . . . . 8 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑦 = 𝑌)
2221oveq2d 6542 . . . . . . 7 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑧𝐼𝑦) = (𝑧𝐼𝑌))
2320, 22eleq12d 2681 . . . . . 6 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑥 ∈ (𝑧𝐼𝑦) ↔ 𝑋 ∈ (𝑧𝐼𝑌)))
2420oveq1d 6541 . . . . . . 7 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑥𝐼𝑧) = (𝑋𝐼𝑧))
2521, 24eleq12d 2681 . . . . . 6 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑦 ∈ (𝑥𝐼𝑧) ↔ 𝑌 ∈ (𝑋𝐼𝑧)))
2619, 23, 253orbi123d 1389 . . . . 5 ((𝑥 = 𝑋𝑦 = 𝑌) → ((𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)) ↔ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))))
2726rabbidv 3163 . . . 4 ((𝑥 = 𝑋𝑦 = 𝑌) → {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} = {𝑧𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))})
28 sneq 4134 . . . . 5 (𝑥 = 𝑋 → {𝑥} = {𝑋})
2928difeq2d 3689 . . . 4 (𝑥 = 𝑋 → (𝑃 ∖ {𝑥}) = (𝑃 ∖ {𝑋}))
30 eqid 2609 . . . 4 (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) = (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})
3127, 29, 30ovmpt2x 6664 . . 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))})
328, 13, 17, 31syl3anc 1317 . 2 (𝜑 → (𝑋(𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})𝑌) = {𝑧𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))})
337, 32eqtrd 2643 1 (𝜑 → (𝑋𝐿𝑌) = {𝑧𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3o 1029   = wceq 1474  wcel 1976  wne 2779  {crab 2899  Vcvv 3172  cdif 3536  {csn 4124  cfv 5789  (class class class)co 6526  cmpt2 6528  Basecbs 15643  TarskiGcstrkg 25073  Itvcitv 25079  LineGclng 25080
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-iota 5753  df-fun 5791  df-fv 5797  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-trkg 25096
This theorem is referenced by:  tglnssp  25192  tgellng  25193  tgisline  25267
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