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Theorem ncolrot2 28439

Description: Rotating non-colinear points. (Contributed by Thierry Arnoux, 19-Oct-2019.)

**Hypotheses**
Ref	Expression
tglngval.p	⊢ 𝑃 = (Base‘𝐺)
tglngval.l	⊢ 𝐿 = (LineG‘𝐺)
tglngval.i	⊢ 𝐼 = (Itv‘𝐺)
tglngval.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tglngval.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
tglngval.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
tgcolg.z	⊢ (𝜑 → 𝑍 ∈ 𝑃)
ncolrot	⊢ (𝜑 → ¬ (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))

**Assertion**
Ref	Expression
ncolrot2	⊢ (𝜑 → ¬ (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋))

**Proof of Theorem ncolrot2**
Step	Hyp	Ref	Expression
1		ncolrot	. 2 ⊢ (𝜑 → ¬ (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
2		tglngval.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
3		tglngval.l	. . 3 ⊢ 𝐿 = (LineG‘𝐺)
4		tglngval.i	. . 3 ⊢ 𝐼 = (Itv‘𝐺)
5		tglngval.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
6	5	adantr 479	. . 3 ⊢ ((𝜑 ∧ (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋)) → 𝐺 ∈ TarskiG)
7		tgcolg.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
8	7	adantr 479	. . 3 ⊢ ((𝜑 ∧ (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋)) → 𝑍 ∈ 𝑃)
9		tglngval.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
10	9	adantr 479	. . 3 ⊢ ((𝜑 ∧ (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋)) → 𝑋 ∈ 𝑃)
11		tglngval.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
12	11	adantr 479	. . 3 ⊢ ((𝜑 ∧ (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋)) → 𝑌 ∈ 𝑃)
13		simpr 483	. . 3 ⊢ ((𝜑 ∧ (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋)) → (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋))
14	2, 3, 4, 6, 8, 10, 12, 13	colrot1 28435	. 2 ⊢ ((𝜑 ∧ (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋)) → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
15	1, 14	mtand 814	1 ⊢ (𝜑 → ¬ (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∨ wo 845 = wceq 1533 ∈ wcel 2098 ‘cfv 6549 (class class class)co 7419 Basecbs 17183 TarskiGcstrkg 28303 Itvcitv 28309 LineGclng 28310

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6501 df-fun 6551 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-trkgc 28324 df-trkgb 28325 df-trkgcb 28326 df-trkg 28329

This theorem is referenced by: midexlem 28568 perpneq 28590 opphllem 28611 outpasch 28631 hlpasch 28632 trgcopy 28680 acopyeu 28710

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