Theorem lnrot2 28527

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)
lnrot2.1	⊢ (𝜑 → 𝑋 ∈ (𝑌𝐿𝑍))
lnrot2.2	⊢ (𝜑 → 𝑌 ≠ 𝑍)

Assertion

Ref	Expression
lnrot2	⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌))

Proof of Theorem lnrot2

Step	Hyp	Ref	Expression
1		lnrot2.1	. 2 ⊢ (𝜑 → 𝑋 ∈ (𝑌𝐿𝑍))
2		btwnlng1.p	. . . . . 6 ⊢ 𝑃 = (Base‘𝐺)
3		eqid 2729	. . . . . 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.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
8		btwnlng1.z	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
9	2, 3, 4, 5, 6, 7, 8	tgbtwncomb 28392	. . . . 5 ⊢ (𝜑 → (𝑋 ∈ (𝑌𝐼𝑍) ↔ 𝑋 ∈ (𝑍𝐼𝑌)))
10		biidd 262	. . . . 5 ⊢ (𝜑 → (𝑌 ∈ (𝑋𝐼𝑍) ↔ 𝑌 ∈ (𝑋𝐼𝑍)))
11	2, 3, 4, 5, 6, 8, 7	tgbtwncomb 28392	. . . . 5 ⊢ (𝜑 → (𝑍 ∈ (𝑌𝐼𝑋) ↔ 𝑍 ∈ (𝑋𝐼𝑌)))
12	9, 10, 11	3orbi123d 1437	. . . 4 ⊢ (𝜑 → ((𝑋 ∈ (𝑌𝐼𝑍) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑍 ∈ (𝑌𝐼𝑋)) ↔ (𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑍 ∈ (𝑋𝐼𝑌))))
13		3orrot 1091	. . . 4 ⊢ ((𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)) ↔ (𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑍 ∈ (𝑋𝐼𝑌)))
14	12, 13	bitr4di 289	. . 3 ⊢ (𝜑 → ((𝑋 ∈ (𝑌𝐼𝑍) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑍 ∈ (𝑌𝐼𝑋)) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
15		btwnlng1.l	. . . 4 ⊢ 𝐿 = (LineG‘𝐺)
16		lnrot2.2	. . . 4 ⊢ (𝜑 → 𝑌 ≠ 𝑍)
17	2, 15, 4, 5, 6, 8, 16, 7	tgellng 28456	. . 3 ⊢ (𝜑 → (𝑋 ∈ (𝑌𝐿𝑍) ↔ (𝑋 ∈ (𝑌𝐼𝑍) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑍 ∈ (𝑌𝐼𝑋))))
18		btwnlng1.d	. . . 4 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
19	2, 15, 4, 5, 7, 6, 18, 8	tgellng 28456	. . 3 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
20	14, 17, 19	3bitr4d 311	. 2 ⊢ (𝜑 → (𝑋 ∈ (𝑌𝐿𝑍) ↔ 𝑍 ∈ (𝑋𝐿𝑌)))
21	1, 20	mpbid 232	1 ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌))

Syntax hints:   → wi 4   ∨ w3o 1085   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ‘cfv 6499  (class class class)co 7369  Basecbs 17155  distcds 17205  TarskiGcstrkg 28330  Itvcitv 28336  LineGclng 28337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6452  df-fun 6501  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-trkgc 28351  df-trkgb 28352  df-trkgcb 28353  df-trkg 28356

This theorem is referenced by:  coltr  28550  mideulem2  28637