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Theorem tglngval 27791
Description: The line going through points 𝑋 and π‘Œ. (Contributed by Thierry Arnoux, 28-Mar-2019.)
Hypotheses
Ref Expression
tglngval.p 𝑃 = (Baseβ€˜πΊ)
tglngval.l 𝐿 = (LineGβ€˜πΊ)
tglngval.i 𝐼 = (Itvβ€˜πΊ)
tglngval.g (πœ‘ β†’ 𝐺 ∈ TarskiG)
tglngval.x (πœ‘ β†’ 𝑋 ∈ 𝑃)
tglngval.y (πœ‘ β†’ π‘Œ ∈ 𝑃)
tglngval.z (πœ‘ β†’ 𝑋 β‰  π‘Œ)
Assertion
Ref Expression
tglngval (πœ‘ β†’ (π‘‹πΏπ‘Œ) = {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘‹πΌπ‘Œ) ∨ 𝑋 ∈ (π‘§πΌπ‘Œ) ∨ π‘Œ ∈ (𝑋𝐼𝑧))})
Distinct variable groups:   𝑧,𝐺   𝑧,𝐼   𝑧,𝑃   𝑧,𝑋   𝑧,π‘Œ   πœ‘,𝑧
Allowed substitution hint:   𝐿(𝑧)
Proof of Theorem tglngval
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tglngval.g . . . 4 (πœ‘ β†’ 𝐺 ∈ TarskiG)
2 tglngval.p . . . . 5 𝑃 = (Baseβ€˜πΊ)
3 tglngval.l . . . . 5 𝐿 = (LineGβ€˜πΊ)
4 tglngval.i . . . . 5 𝐼 = (Itvβ€˜πΊ)
52, 3, 4tglng 27786 . . . 4 (𝐺 ∈ TarskiG β†’ 𝐿 = (π‘₯ ∈ 𝑃, 𝑦 ∈ (𝑃 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))}))
61, 5syl 17 . . 3 (πœ‘ β†’ 𝐿 = (π‘₯ ∈ 𝑃, 𝑦 ∈ (𝑃 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))}))
76oveqd 7422 . 2 (πœ‘ β†’ (π‘‹πΏπ‘Œ) = (𝑋(π‘₯ ∈ 𝑃, 𝑦 ∈ (𝑃 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))})π‘Œ))
8 tglngval.x . . 3 (πœ‘ β†’ 𝑋 ∈ 𝑃)
9 tglngval.y . . . 4 (πœ‘ β†’ π‘Œ ∈ 𝑃)
10 tglngval.z . . . . 5 (πœ‘ β†’ 𝑋 β‰  π‘Œ)
1110necomd 2996 . . . 4 (πœ‘ β†’ π‘Œ β‰  𝑋)
12 eldifsn 4789 . . . 4 (π‘Œ ∈ (𝑃 βˆ– {𝑋}) ↔ (π‘Œ ∈ 𝑃 ∧ π‘Œ β‰  𝑋))
139, 11, 12sylanbrc 583 . . 3 (πœ‘ β†’ π‘Œ ∈ (𝑃 βˆ– {𝑋}))
142fvexi 6902 . . . . 5 𝑃 ∈ V
1514rabex 5331 . . . 4 {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘‹πΌπ‘Œ) ∨ 𝑋 ∈ (π‘§πΌπ‘Œ) ∨ π‘Œ ∈ (𝑋𝐼𝑧))} ∈ V
1615a1i 11 . . 3 (πœ‘ β†’ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘‹πΌπ‘Œ) ∨ 𝑋 ∈ (π‘§πΌπ‘Œ) ∨ π‘Œ ∈ (𝑋𝐼𝑧))} ∈ V)
17 oveq12 7414 . . . . . . 7 ((π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ) β†’ (π‘₯𝐼𝑦) = (π‘‹πΌπ‘Œ))
1817eleq2d 2819 . . . . . 6 ((π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ) β†’ (𝑧 ∈ (π‘₯𝐼𝑦) ↔ 𝑧 ∈ (π‘‹πΌπ‘Œ)))
19 simpl 483 . . . . . . 7 ((π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ) β†’ π‘₯ = 𝑋)
20 simpr 485 . . . . . . . 8 ((π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ) β†’ 𝑦 = π‘Œ)
2120oveq2d 7421 . . . . . . 7 ((π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ) β†’ (𝑧𝐼𝑦) = (π‘§πΌπ‘Œ))
2219, 21eleq12d 2827 . . . . . 6 ((π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ) β†’ (π‘₯ ∈ (𝑧𝐼𝑦) ↔ 𝑋 ∈ (π‘§πΌπ‘Œ)))
2319oveq1d 7420 . . . . . . 7 ((π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ) β†’ (π‘₯𝐼𝑧) = (𝑋𝐼𝑧))
2420, 23eleq12d 2827 . . . . . 6 ((π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ) β†’ (𝑦 ∈ (π‘₯𝐼𝑧) ↔ π‘Œ ∈ (𝑋𝐼𝑧)))
2518, 22, 243orbi123d 1435 . . . . 5 ((π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ) β†’ ((𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧)) ↔ (𝑧 ∈ (π‘‹πΌπ‘Œ) ∨ 𝑋 ∈ (π‘§πΌπ‘Œ) ∨ π‘Œ ∈ (𝑋𝐼𝑧))))
2625rabbidv 3440 . . . 4 ((π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ) β†’ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))} = {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘‹πΌπ‘Œ) ∨ 𝑋 ∈ (π‘§πΌπ‘Œ) ∨ π‘Œ ∈ (𝑋𝐼𝑧))})
27 sneq 4637 . . . . 5 (π‘₯ = 𝑋 β†’ {π‘₯} = {𝑋})
2827difeq2d 4121 . . . 4 (π‘₯ = 𝑋 β†’ (𝑃 βˆ– {π‘₯}) = (𝑃 βˆ– {𝑋}))
29 eqid 2732 . . . 4 (π‘₯ ∈ 𝑃, 𝑦 ∈ (𝑃 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))}) = (π‘₯ ∈ 𝑃, 𝑦 ∈ (𝑃 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))})
3026, 28, 29ovmpox 7557 . . 3 ((𝑋 ∈ 𝑃 ∧ π‘Œ ∈ (𝑃 βˆ– {𝑋}) ∧ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘‹πΌπ‘Œ) ∨ 𝑋 ∈ (π‘§πΌπ‘Œ) ∨ π‘Œ ∈ (𝑋𝐼𝑧))} ∈ V) β†’ (𝑋(π‘₯ ∈ 𝑃, 𝑦 ∈ (𝑃 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))})π‘Œ) = {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘‹πΌπ‘Œ) ∨ 𝑋 ∈ (π‘§πΌπ‘Œ) ∨ π‘Œ ∈ (𝑋𝐼𝑧))})
318, 13, 16, 30syl3anc 1371 . 2 (πœ‘ β†’ (𝑋(π‘₯ ∈ 𝑃, 𝑦 ∈ (𝑃 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))})π‘Œ) = {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘‹πΌπ‘Œ) ∨ 𝑋 ∈ (π‘§πΌπ‘Œ) ∨ π‘Œ ∈ (𝑋𝐼𝑧))})
327, 31eqtrd 2772 1 (πœ‘ β†’ (π‘‹πΏπ‘Œ) = {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘‹πΌπ‘Œ) ∨ 𝑋 ∈ (π‘§πΌπ‘Œ) ∨ π‘Œ ∈ (𝑋𝐼𝑧))})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∨ w3o 1086   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  {crab 3432  Vcvv 3474   βˆ– cdif 3944  {csn 4627  β€˜cfv 6540  (class class class)co 7405   ∈ cmpo 7407  Basecbs 17140  TarskiGcstrkg 27667  Itvcitv 27673  LineGclng 27674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  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-iota 6492  df-fun 6542  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-trkg 27693
This theorem is referenced by:  tglnssp  27792  tgellng  27793  tgisline  27867
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