Theorem lnrot1 28907

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 = (Base‘𝐺)
btwnlng1.i	⊢ 𝐼 = (Itv‘𝐺)
btwnlng1.l	⊢ 𝐿 = (LineG‘𝐺)
btwnlng1.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
btwnlng1.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
btwnlng1.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
btwnlng1.z	⊢ (𝜑 → 𝑍 ∈ 𝑃)
btwnlng1.d	⊢ (𝜑 → 𝑋 ≠ 𝑌)
lnrot1.1	⊢ (𝜑 → 𝑌 ∈ (𝑍𝐿𝑋))
lnrot1.2	⊢ (𝜑 → 𝑍 ≠ 𝑋)

Assertion

Ref	Expression
lnrot1	⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌))

Proof of Theorem lnrot1

Step	Hyp	Ref	Expression
1		lnrot1.1	. 2 ⊢ (𝜑 → 𝑌 ∈ (𝑍𝐿𝑋))
2		btwnlng1.p	. . . . . 6 ⊢ 𝑃 = (Base‘𝐺)
3		eqid 2762	. . . . . 6 ⊢ (dist‘𝐺) = (dist‘𝐺)
4		btwnlng1.i	. . . . . 6 ⊢ 𝐼 = (Itv‘𝐺)
5		btwnlng1.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
6		btwnlng1.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
7		btwnlng1.z	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
8		btwnlng1.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
9	2, 3, 4, 5, 6, 7, 8	tgbtwncomb 28769	. . . . 5 ⊢ (𝜑 → (𝑍 ∈ (𝑌𝐼𝑋) ↔ 𝑍 ∈ (𝑋𝐼𝑌)))
10		biidd 265	. . . . 5 ⊢ (𝜑 → (𝑋 ∈ (𝑍𝐼𝑌) ↔ 𝑋 ∈ (𝑍𝐼𝑌)))
11	2, 3, 4, 5, 7, 6, 8	tgbtwncomb 28769	. . . . 5 ⊢ (𝜑 → (𝑌 ∈ (𝑍𝐼𝑋) ↔ 𝑌 ∈ (𝑋𝐼𝑍)))
12	9, 10, 11	3orbi123d 1462	. . . 4 ⊢ (𝜑 → ((𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑍𝐼𝑋)) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
13		3orrot 1107	. . . . 5 ⊢ ((𝑌 ∈ (𝑍𝐼𝑋) ∨ 𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑋 ∈ (𝑍𝐼𝑌)) ↔ (𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑍𝐼𝑋)))
14	13	a1i 11	. . . 4 ⊢ (𝜑 → ((𝑌 ∈ (𝑍𝐼𝑋) ∨ 𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑋 ∈ (𝑍𝐼𝑌)) ↔ (𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑍𝐼𝑋))))
15		btwnlng1.l	. . . . 5 ⊢ 𝐿 = (LineG‘𝐺)
16		btwnlng1.d	. . . . 5 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
17	2, 15, 4, 5, 8, 6, 16, 7	tgellng 28833	. . . 4 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
18	12, 14, 17	3bitr4rd 315	. . 3 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑌 ∈ (𝑍𝐼𝑋) ∨ 𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑋 ∈ (𝑍𝐼𝑌))))
19		lnrot1.2	. . . 4 ⊢ (𝜑 → 𝑍 ≠ 𝑋)
20	2, 15, 4, 5, 7, 8, 19, 6	tgellng 28833	. . 3 ⊢ (𝜑 → (𝑌 ∈ (𝑍𝐿𝑋) ↔ (𝑌 ∈ (𝑍𝐼𝑋) ∨ 𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑋 ∈ (𝑍𝐼𝑌))))
21	18, 20	bitr4d 285	. 2 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ 𝑌 ∈ (𝑍𝐿𝑋)))
22	1, 21	mpbird 260	1 ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∨ w3o 1101   = wceq 1569   ∈ wcel 2142   ≠ wne 2957  ‘cfv 6536  (class class class)co 7412  Basecbs 17275  distcds 17325  TarskiGcstrkg 28707  Itvcitv 28713  LineGclng 28714

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-sbc 3744  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7415  df-oprab 7416  df-mpo 7417  df-trkgc 28728  df-trkgb 28729  df-trkgcb 28730  df-trkg 28733

This theorem is used by:  tglineelsb2  28916  tglineneq  28929  coltr3  28933  hlperpnel  29017  opphllem4  29042  lmieu  29104  ragcgra  29157  prlngmid2  29222  prlngsymquadopp  29226