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Theorem lnrot2 28441
Description: Rotating the points defining a line. Part of Theorem 4.11 of [Schwabhauser] p. 34. (Contributed by Thierry Arnoux, 3-Apr-2019.)
Hypotheses
Ref Expression
btwnlng1.p 𝑃 = (Baseβ€˜πΊ)
btwnlng1.i 𝐼 = (Itvβ€˜πΊ)
btwnlng1.l 𝐿 = (LineGβ€˜πΊ)
btwnlng1.g (πœ‘ β†’ 𝐺 ∈ TarskiG)
btwnlng1.x (πœ‘ β†’ 𝑋 ∈ 𝑃)
btwnlng1.y (πœ‘ β†’ π‘Œ ∈ 𝑃)
btwnlng1.z (πœ‘ β†’ 𝑍 ∈ 𝑃)
btwnlng1.d (πœ‘ β†’ 𝑋 β‰  π‘Œ)
lnrot2.1 (πœ‘ β†’ 𝑋 ∈ (π‘ŒπΏπ‘))
lnrot2.2 (πœ‘ β†’ π‘Œ β‰  𝑍)
Assertion
Ref Expression
lnrot2 (πœ‘ β†’ 𝑍 ∈ (π‘‹πΏπ‘Œ))

Proof of Theorem lnrot2
StepHypRef Expression
1 lnrot2.1 . 2 (πœ‘ β†’ 𝑋 ∈ (π‘ŒπΏπ‘))
2 btwnlng1.p . . . . . 6 𝑃 = (Baseβ€˜πΊ)
3 eqid 2728 . . . . . 6 (distβ€˜πΊ) = (distβ€˜πΊ)
4 btwnlng1.i . . . . . 6 𝐼 = (Itvβ€˜πΊ)
5 btwnlng1.g . . . . . 6 (πœ‘ β†’ 𝐺 ∈ TarskiG)
6 btwnlng1.y . . . . . 6 (πœ‘ β†’ π‘Œ ∈ 𝑃)
7 btwnlng1.x . . . . . 6 (πœ‘ β†’ 𝑋 ∈ 𝑃)
8 btwnlng1.z . . . . . 6 (πœ‘ β†’ 𝑍 ∈ 𝑃)
92, 3, 4, 5, 6, 7, 8tgbtwncomb 28306 . . . . 5 (πœ‘ β†’ (𝑋 ∈ (π‘ŒπΌπ‘) ↔ 𝑋 ∈ (π‘πΌπ‘Œ)))
10 biidd 262 . . . . 5 (πœ‘ β†’ (π‘Œ ∈ (𝑋𝐼𝑍) ↔ π‘Œ ∈ (𝑋𝐼𝑍)))
112, 3, 4, 5, 6, 8, 7tgbtwncomb 28306 . . . . 5 (πœ‘ β†’ (𝑍 ∈ (π‘ŒπΌπ‘‹) ↔ 𝑍 ∈ (π‘‹πΌπ‘Œ)))
129, 10, 113orbi123d 1432 . . . 4 (πœ‘ β†’ ((𝑋 ∈ (π‘ŒπΌπ‘) ∨ π‘Œ ∈ (𝑋𝐼𝑍) ∨ 𝑍 ∈ (π‘ŒπΌπ‘‹)) ↔ (𝑋 ∈ (π‘πΌπ‘Œ) ∨ π‘Œ ∈ (𝑋𝐼𝑍) ∨ 𝑍 ∈ (π‘‹πΌπ‘Œ))))
13 3orrot 1090 . . . 4 ((𝑍 ∈ (π‘‹πΌπ‘Œ) ∨ 𝑋 ∈ (π‘πΌπ‘Œ) ∨ π‘Œ ∈ (𝑋𝐼𝑍)) ↔ (𝑋 ∈ (π‘πΌπ‘Œ) ∨ π‘Œ ∈ (𝑋𝐼𝑍) ∨ 𝑍 ∈ (π‘‹πΌπ‘Œ)))
1412, 13bitr4di 289 . . 3 (πœ‘ β†’ ((𝑋 ∈ (π‘ŒπΌπ‘) ∨ π‘Œ ∈ (𝑋𝐼𝑍) ∨ 𝑍 ∈ (π‘ŒπΌπ‘‹)) ↔ (𝑍 ∈ (π‘‹πΌπ‘Œ) ∨ 𝑋 ∈ (π‘πΌπ‘Œ) ∨ π‘Œ ∈ (𝑋𝐼𝑍))))
15 btwnlng1.l . . . 4 𝐿 = (LineGβ€˜πΊ)
16 lnrot2.2 . . . 4 (πœ‘ β†’ π‘Œ β‰  𝑍)
172, 15, 4, 5, 6, 8, 16, 7tgellng 28370 . . 3 (πœ‘ β†’ (𝑋 ∈ (π‘ŒπΏπ‘) ↔ (𝑋 ∈ (π‘ŒπΌπ‘) ∨ π‘Œ ∈ (𝑋𝐼𝑍) ∨ 𝑍 ∈ (π‘ŒπΌπ‘‹))))
18 btwnlng1.d . . . 4 (πœ‘ β†’ 𝑋 β‰  π‘Œ)
192, 15, 4, 5, 7, 6, 18, 8tgellng 28370 . . 3 (πœ‘ β†’ (𝑍 ∈ (π‘‹πΏπ‘Œ) ↔ (𝑍 ∈ (π‘‹πΌπ‘Œ) ∨ 𝑋 ∈ (π‘πΌπ‘Œ) ∨ π‘Œ ∈ (𝑋𝐼𝑍))))
2014, 17, 193bitr4d 311 . 2 (πœ‘ β†’ (𝑋 ∈ (π‘ŒπΏπ‘) ↔ 𝑍 ∈ (π‘‹πΏπ‘Œ)))
211, 20mpbid 231 1 (πœ‘ β†’ 𝑍 ∈ (π‘‹πΏπ‘Œ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∨ w3o 1084   = wceq 1534   ∈ wcel 2099   β‰  wne 2937  β€˜cfv 6548  (class class class)co 7420  Basecbs 17180  distcds 17242  TarskiGcstrkg 28244  Itvcitv 28250  LineGclng 28251
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 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-trkgc 28265  df-trkgb 28266  df-trkgcb 28267  df-trkg 28270
This theorem is referenced by:  coltr  28464  mideulem2  28551
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