ncolcom - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ncolcom		Structured version Visualization version GIF version

Theorem ncolcom 28495

Description: Swapping 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
ncolcom	⊢ (𝜑 → ¬ (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋))

**Proof of Theorem ncolcom**
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 ∧ (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋)) → 𝐺 ∈ TarskiG)
7		tglngval.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
8	7	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋)) → 𝑌 ∈ 𝑃)
9		tglngval.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
10	9	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋)) → 𝑋 ∈ 𝑃)
11		tgcolg.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
12	11	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋)) → 𝑍 ∈ 𝑃)
13		simpr 484	. . 3 ⊢ ((𝜑 ∧ (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋)) → (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋))
14	2, 3, 4, 6, 8, 10, 12, 13	colcom 28492	. 2 ⊢ ((𝜑 ∧ (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋)) → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
15	1, 14	mtand 815	1 ⊢ (𝜑 → ¬ (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2109 ‘cfv 6514 (class class class)co 7390 Basecbs 17186 TarskiGcstrkg 28361 Itvcitv 28367 LineGclng 28368

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pr 5390

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-sbc 3757 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-br 5111 df-opab 5173 df-id 5536 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-iota 6467 df-fun 6516 df-fv 6522 df-ov 7393 df-oprab 7394 df-mpo 7395 df-trkgc 28382 df-trkgb 28383 df-trkgcb 28384 df-trkg 28387

This theorem is referenced by: ncolne2 28560 symquadlem 28623 midexlem 28626 outpasch 28689 acopyeu 28768 cgrg3col4 28787 tgasa1 28792

W3C validator