btwnhl1 - Metamath Proof Explorer

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Theorem btwnhl1 28702

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 2991	. 2 ⊢ (𝜑 → 𝐵 ≠ 𝐴)
4		btwnhl1.1	. . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐵))
5	4	orcd 880	. 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 28692	. 2 ⊢ (𝜑 → (𝐶(𝐾‘𝐴)𝐵 ↔ (𝐶 ≠ 𝐴 ∧ 𝐵 ≠ 𝐴 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐵 ∈ (𝐴𝐼𝐶)))))
14	1, 3, 5, 13	mpbir3and 1350	1 ⊢ (𝜑 → 𝐶(𝐾‘𝐴)𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∨ wo 854 = wceq 1548 ∈ wcel 2121 ≠ wne 2936 class class class wbr 5075 ‘cfv 6489 (class class class)co 7360 Basecbs 17174 TarskiGcstrkg 28517 Itvcitv 28523 hlGchlg 28690

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-ral 3056 df-rex 3066 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4842 df-iun 4926 df-br 5076 df-opab 5138 df-mpt 5157 df-id 5516 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-ov 7363 df-hlg 28691

This theorem is referenced by: outpasch 28845 hlpasch 28846 lnopp2hpgb 28853 dfcgra2 28920 cgranbtwn 34865

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