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Theorem lnrot1 26984

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 2738	. . . . . 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 26850	. . . . 5 ⊢ (𝜑 → (𝑍 ∈ (𝑌𝐼𝑋) ↔ 𝑍 ∈ (𝑋𝐼𝑌)))
10		biidd 261	. . . . 5 ⊢ (𝜑 → (𝑋 ∈ (𝑍𝐼𝑌) ↔ 𝑋 ∈ (𝑍𝐼𝑌)))
11	2, 3, 4, 5, 7, 6, 8	tgbtwncomb 26850	. . . . 5 ⊢ (𝜑 → (𝑌 ∈ (𝑍𝐼𝑋) ↔ 𝑌 ∈ (𝑋𝐼𝑍)))
12	9, 10, 11	3orbi123d 1434	. . . 4 ⊢ (𝜑 → ((𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑍𝐼𝑋)) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
13		3orrot 1091	. . . . 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 26914	. . . 4 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
18	12, 14, 17	3bitr4rd 312	. . 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 26914	. . 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))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∨ w3o 1085 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ‘cfv 6433 (class class class)co 7275 Basecbs 16912 distcds 16971 TarskiGcstrkg 26788 Itvcitv 26794 LineGclng 26795

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-iota 6391 df-fun 6435 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-trkgc 26809 df-trkgb 26810 df-trkgcb 26811 df-trkg 26814

This theorem is referenced by: tglineelsb2 26993 tglineneq 27005 coltr3 27009 hlperpnel 27086 opphllem4 27111 lmieu 27145

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