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Theorem ncolrot2 28586

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 480	. . 3 ⊢ ((𝜑 ∧ (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋)) → 𝐺 ∈ TarskiG)
7		tgcolg.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
8	7	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋)) → 𝑍 ∈ 𝑃)
9		tglngval.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
10	9	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋)) → 𝑋 ∈ 𝑃)
11		tglngval.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
12	11	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋)) → 𝑌 ∈ 𝑃)
13		simpr 484	. . 3 ⊢ ((𝜑 ∧ (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋)) → (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋))
14	2, 3, 4, 6, 8, 10, 12, 13	colrot1 28582	. 2 ⊢ ((𝜑 ∧ (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋)) → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
15	1, 14	mtand 816	1 ⊢ (𝜑 → ¬ (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 TarskiGcstrkg 28450 Itvcitv 28456 LineGclng 28457

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-trkgc 28471 df-trkgb 28472 df-trkgcb 28473 df-trkg 28476

This theorem is referenced by: midexlem 28715 perpneq 28737 opphllem 28758 outpasch 28778 hlpasch 28779 trgcopy 28827 acopyeu 28857

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