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Theorem ncolrot2 28649

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 481	. . 3 ⊢ ((𝜑 ∧ (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋)) → 𝐺 ∈ TarskiG)
7		tgcolg.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
8	7	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋)) → 𝑍 ∈ 𝑃)
9		tglngval.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
10	9	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋)) → 𝑋 ∈ 𝑃)
11		tglngval.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
12	11	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋)) → 𝑌 ∈ 𝑃)
13		simpr 485	. . 3 ⊢ ((𝜑 ∧ (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋)) → (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋))
14	2, 3, 4, 6, 8, 10, 12, 13	colrot1 28645	. 2 ⊢ ((𝜑 ∧ (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋)) → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
15	1, 14	mtand 821	1 ⊢ (𝜑 → ¬ (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∨ wo 853 = wceq 1547 ∈ wcel 2119 ‘cfv 6485 (class class class)co 7356 Basecbs 17170 TarskiGcstrkg 28513 Itvcitv 28519 LineGclng 28520

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-sep 5218 ax-nul 5228 ax-pr 5362

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-ral 3054 df-rex 3064 df-rab 3392 df-v 3433 df-sbc 3724 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-br 5073 df-opab 5135 df-id 5513 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-iota 6441 df-fun 6487 df-fv 6493 df-ov 7359 df-oprab 7360 df-mpo 7361 df-trkgc 28534 df-trkgb 28535 df-trkgcb 28536 df-trkg 28539

This theorem is referenced by: midexlem 28778 perpneq 28800 opphllem 28821 outpasch 28841 hlpasch 28842 trgcopy 28890 acopyeu 28920

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