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 25501

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 476	. . 3 ⊢ ((𝜑 ∧ (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋)) → (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋))
14	2, 3, 4, 6, 8, 10, 12, 13	colcom 25498	. 2 ⊢ ((𝜑 ∧ (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋)) → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
15	1, 14	mtand 692	1 ⊢ (𝜑 → ¬ (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∨ wo 382 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ‘cfv 5926 (class class class)co 6690 Basecbs 15904 TarskiGcstrkg 25374 Itvcitv 25380 LineGclng 25381

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-iota 5889 df-fun 5928 df-fv 5934 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-trkgc 25392 df-trkgb 25393 df-trkgcb 25394 df-trkg 25397

This theorem is referenced by: ncolne2 25566 symquadlem 25629 midexlem 25632 outpasch 25692 acopyeu 25770 cgrg3col4 25779 tgasa1 25784

W3C validator