Theorem lnrot1 27871

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 2732	. . . . . 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 27737	. . . . 5 ⊢ (𝜑 → (𝑍 ∈ (𝑌𝐼𝑋) ↔ 𝑍 ∈ (𝑋𝐼𝑌)))
10		biidd 261	. . . . 5 ⊢ (𝜑 → (𝑋 ∈ (𝑍𝐼𝑌) ↔ 𝑋 ∈ (𝑍𝐼𝑌)))
11	2, 3, 4, 5, 7, 6, 8	tgbtwncomb 27737	. . . . 5 ⊢ (𝜑 → (𝑌 ∈ (𝑍𝐼𝑋) ↔ 𝑌 ∈ (𝑋𝐼𝑍)))
12	9, 10, 11	3orbi123d 1435	. . . 4 ⊢ (𝜑 → ((𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑍𝐼𝑋)) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
13		3orrot 1092	. . . . 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 27801	. . . 4 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
18	12, 14, 17	3bitr4rd 311	. . 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 27801	. . 3 ⊢ (𝜑 → (𝑌 ∈ (𝑍𝐿𝑋) ↔ (𝑌 ∈ (𝑍𝐼𝑋) ∨ 𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑋 ∈ (𝑍𝐼𝑌))))
21	18, 20	bitr4d 281	. 2 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ 𝑌 ∈ (𝑍𝐿𝑋)))
22	1, 21	mpbird 256	1 ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌))

Syntax hints:   → wi 4   ↔ wb 205   ∨ w3o 1086   = wceq 1541   ∈ wcel 2106   ≠ wne 2940  ‘cfv 6543  (class class class)co 7408  Basecbs 17143  distcds 17205  TarskiGcstrkg 27675  Itvcitv 27681  LineGclng 27682

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 5299  ax-nul 5306  ax-pr 5427

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 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-trkgc 27696  df-trkgb 27697  df-trkgcb 27698  df-trkg 27701

This theorem is referenced by:  tglineelsb2  27880  tglineneq  27892  coltr3  27896  hlperpnel  27973  opphllem4  27998  lmieu  28032