Theorem lnrot2 26092

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 2795	. . . . . 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 25957	. . . . 5 ⊢ (𝜑 → (𝑋 ∈ (𝑌𝐼𝑍) ↔ 𝑋 ∈ (𝑍𝐼𝑌)))
10		biidd 263	. . . . 5 ⊢ (𝜑 → (𝑌 ∈ (𝑋𝐼𝑍) ↔ 𝑌 ∈ (𝑋𝐼𝑍)))
11	2, 3, 4, 5, 6, 8, 7	tgbtwncomb 25957	. . . . 5 ⊢ (𝜑 → (𝑍 ∈ (𝑌𝐼𝑋) ↔ 𝑍 ∈ (𝑋𝐼𝑌)))
12	9, 10, 11	3orbi123d 1427	. . . 4 ⊢ (𝜑 → ((𝑋 ∈ (𝑌𝐼𝑍) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑍 ∈ (𝑌𝐼𝑋)) ↔ (𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑍 ∈ (𝑋𝐼𝑌))))
13		3orrot 1085	. . . 4 ⊢ ((𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)) ↔ (𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑍 ∈ (𝑋𝐼𝑌)))
14	12, 13	syl6bbr 290	. . 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 26021	. . 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 26021	. . 3 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
20	14, 17, 19	3bitr4d 312	. 2 ⊢ (𝜑 → (𝑋 ∈ (𝑌𝐿𝑍) ↔ 𝑍 ∈ (𝑋𝐿𝑌)))
21	1, 20	mpbid 233	1 ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌))

Syntax hints:   → wi 4   ∨ w3o 1079   = wceq 1522   ∈ wcel 2081   ≠ wne 2984  ‘cfv 6225  (class class class)co 7016  Basecbs 16312  distcds 16403  TarskiGcstrkg 25898  Itvcitv 25904  LineGclng 25905

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-iota 6189  df-fun 6227  df-fv 6233  df-ov 7019  df-oprab 7020  df-mpo 7021  df-trkgc 25916  df-trkgb 25917  df-trkgcb 25918  df-trkg 25921

This theorem is referenced by:  coltr  26115  mideulem2  26202