btwnhl1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > btwnhl1		Structured version Visualization version GIF version

Theorem btwnhl1 28592

Description: Deduce half-line from betweenness. (Contributed by Thierry Arnoux, 4-Mar-2020.)

**Hypotheses**
Ref	Expression
ishlg.p	⊢ 𝑃 = (Base‘𝐺)
ishlg.i	⊢ 𝐼 = (Itv‘𝐺)
ishlg.k	⊢ 𝐾 = (hlG‘𝐺)
ishlg.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
ishlg.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
ishlg.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
hlln.1	⊢ (𝜑 → 𝐺 ∈ TarskiG)
hltr.d	⊢ (𝜑 → 𝐷 ∈ 𝑃)
btwnhl1.1	⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐵))
btwnhl1.2	⊢ (𝜑 → 𝐴 ≠ 𝐵)
btwnhl1.3	⊢ (𝜑 → 𝐶 ≠ 𝐴)

**Assertion**
Ref	Expression
btwnhl1	⊢ (𝜑 → 𝐶(𝐾‘𝐴)𝐵)

**Proof of Theorem btwnhl1**
Step	Hyp	Ref	Expression
1		btwnhl1.3	. 2 ⊢ (𝜑 → 𝐶 ≠ 𝐴)
2		btwnhl1.2	. . 3 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
3	2	necomd 2980	. 2 ⊢ (𝜑 → 𝐵 ≠ 𝐴)
4		btwnhl1.1	. . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐵))
5	4	orcd 873	. 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐵 ∈ (𝐴𝐼𝐶)))
6		ishlg.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
7		ishlg.i	. . 3 ⊢ 𝐼 = (Itv‘𝐺)
8		ishlg.k	. . 3 ⊢ 𝐾 = (hlG‘𝐺)
9		ishlg.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
10		ishlg.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
11		ishlg.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
12		hlln.1	. . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
13	6, 7, 8, 9, 10, 11, 12	ishlg 28582	. 2 ⊢ (𝜑 → (𝐶(𝐾‘𝐴)𝐵 ↔ (𝐶 ≠ 𝐴 ∧ 𝐵 ≠ 𝐴 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐵 ∈ (𝐴𝐼𝐶)))))
14	1, 3, 5, 13	mpbir3and 1343	1 ⊢ (𝜑 → 𝐶(𝐾‘𝐴)𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∨ wo 847 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 class class class wbr 5102 ‘cfv 6499 (class class class)co 7369 Basecbs 17155 TarskiGcstrkg 28407 Itvcitv 28413 hlGchlg 28580

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 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-ov 7372 df-hlg 28581

This theorem is referenced by: outpasch 28735 hlpasch 28736 lnopp2hpgb 28743 dfcgra2 28810

W3C validator