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Theorem hlcgrex 27898

Description: Construct a point on a half-line, at a given distance of its origin. (Contributed by Thierry Arnoux, 1-Aug-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)
hlcgrex.m	⊢ − = (dist‘𝐺)
hlcgrex.1	⊢ (𝜑 → 𝐷 ≠ 𝐴)
hlcgrex.2	⊢ (𝜑 → 𝐵 ≠ 𝐶)

**Assertion**
Ref	Expression
hlcgrex	⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝑥(𝐾‘𝐴)𝐷 ∧ (𝐴 − 𝑥) = (𝐵 − 𝐶)))

Distinct variable groups: 𝑥, − 𝑥,𝐴 𝑥,𝐵 𝑥,𝐶 𝑥,𝐷 𝑥,𝐾 𝑥,𝐼 𝑥,𝑃 𝜑,𝑥 Allowed substitution hint: 𝐺(𝑥)

Proof of Theorem hlcgrex Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ishlg.p	. . . 4 ⊢ 𝑃 = (Base‘𝐺)
2		hlcgrex.m	. . . 4 ⊢ − = (dist‘𝐺)
3		ishlg.i	. . . 4 ⊢ 𝐼 = (Itv‘𝐺)
4		hlln.1	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	ad2antrr 725	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐺 ∈ TarskiG)
6		simplr 768	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝑦 ∈ 𝑃)
7		ishlg.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
8	7	ad2antrr 725	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ∈ 𝑃)
9		ishlg.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
10	9	ad2antrr 725	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐵 ∈ 𝑃)
11		ishlg.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
12	11	ad2antrr 725	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐶 ∈ 𝑃)
13	1, 2, 3, 5, 6, 8, 10, 12	axtgsegcon 27746	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → ∃𝑥 ∈ 𝑃 (𝐴 ∈ (𝑦𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐶)))
14	5	ad2antrr 725	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝑦𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐶))) → 𝐺 ∈ TarskiG)
15	10	ad2antrr 725	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝑦𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐶))) → 𝐵 ∈ 𝑃)
16	12	ad2antrr 725	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝑦𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐶))) → 𝐶 ∈ 𝑃)
17		simplr 768	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝑦𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐶))) → 𝑥 ∈ 𝑃)
18	8	ad2antrr 725	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝑦𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐶))) → 𝐴 ∈ 𝑃)
19		simprr 772	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝑦𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐶))) → (𝐴 − 𝑥) = (𝐵 − 𝐶))
20	1, 2, 3, 14, 18, 17, 15, 16, 19	tgcgrcoml 27761	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝑦𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐶))) → (𝑥 − 𝐴) = (𝐵 − 𝐶))
21	20	eqcomd 2739	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝑦𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐶))) → (𝐵 − 𝐶) = (𝑥 − 𝐴))
22		hlcgrex.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
23	22	ad4antr 731	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝑦𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐶))) → 𝐵 ≠ 𝐶)
24	1, 2, 3, 14, 15, 16, 17, 18, 21, 23	tgcgrneq 27765	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝑦𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐶))) → 𝑥 ≠ 𝐴)
25		hlcgrex.1	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ≠ 𝐴)
26	25	ad4antr 731	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝑦𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐶))) → 𝐷 ≠ 𝐴)
27	6	ad2antrr 725	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝑦𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐶))) → 𝑦 ∈ 𝑃)
28		hltr.d	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
29	28	ad4antr 731	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝑦𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐶))) → 𝐷 ∈ 𝑃)
30		simpllr 775	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝑦𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐶))) → (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦))
31	30	simprd 497	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝑦𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐶))) → 𝐴 ≠ 𝑦)
32	31	necomd 2997	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝑦𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐶))) → 𝑦 ≠ 𝐴)
33		simprl 770	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝑦𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐶))) → 𝐴 ∈ (𝑦𝐼𝑥))
34	30	simpld 496	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝑦𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐶))) → 𝐴 ∈ (𝐷𝐼𝑦))
35	1, 2, 3, 14, 29, 18, 27, 34	tgbtwncom 27770	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝑦𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐶))) → 𝐴 ∈ (𝑦𝐼𝐷))
36	1, 3, 14, 27, 18, 17, 29, 32, 33, 35	tgbtwnconn2 27858	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝑦𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐶))) → (𝑥 ∈ (𝐴𝐼𝐷) ∨ 𝐷 ∈ (𝐴𝐼𝑥)))
37		ishlg.k	. . . . . . . 8 ⊢ 𝐾 = (hlG‘𝐺)
38	1, 3, 37, 17, 29, 18, 14	ishlg 27884	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝑦𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐶))) → (𝑥(𝐾‘𝐴)𝐷 ↔ (𝑥 ≠ 𝐴 ∧ 𝐷 ≠ 𝐴 ∧ (𝑥 ∈ (𝐴𝐼𝐷) ∨ 𝐷 ∈ (𝐴𝐼𝑥)))))
39	24, 26, 36, 38	mpbir3and 1343	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝑦𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐶))) → 𝑥(𝐾‘𝐴)𝐷)
40	39, 19	jca 513	. . . . 5 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝑦𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐶))) → (𝑥(𝐾‘𝐴)𝐷 ∧ (𝐴 − 𝑥) = (𝐵 − 𝐶)))
41	40	ex 414	. . . 4 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) ∧ 𝑥 ∈ 𝑃) → ((𝐴 ∈ (𝑦𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐶)) → (𝑥(𝐾‘𝐴)𝐷 ∧ (𝐴 − 𝑥) = (𝐵 − 𝐶))))
42	41	reximdva 3169	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → (∃𝑥 ∈ 𝑃 (𝐴 ∈ (𝑦𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐶)) → ∃𝑥 ∈ 𝑃 (𝑥(𝐾‘𝐴)𝐷 ∧ (𝐴 − 𝑥) = (𝐵 − 𝐶))))
43	13, 42	mpd 15	. 2 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → ∃𝑥 ∈ 𝑃 (𝑥(𝐾‘𝐴)𝐷 ∧ (𝐴 − 𝑥) = (𝐵 − 𝐶)))
44	1	fvexi 6906	. . . . 5 ⊢ 𝑃 ∈ V
45	44	a1i 11	. . . 4 ⊢ (𝜑 → 𝑃 ∈ V)
46	45, 9, 11, 22	nehash2 14435	. . 3 ⊢ (𝜑 → 2 ≤ (♯‘𝑃))
47	1, 2, 3, 4, 28, 7, 46	tgbtwndiff 27788	. 2 ⊢ (𝜑 → ∃𝑦 ∈ 𝑃 (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦))
48	43, 47	r19.29a 3163	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝑥(𝐾‘𝐴)𝐷 ∧ (𝐴 − 𝑥) = (𝐵 − 𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∃wrex 3071 Vcvv 3475 class class class wbr 5149 ‘cfv 6544 (class class class)co 7409 Basecbs 17144 distcds 17206 TarskiGcstrkg 27709 Itvcitv 27715 hlGchlg 27882

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-oadd 8470 df-er 8703 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-dju 9896 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-xnn0 12545 df-z 12559 df-uz 12823 df-fz 13485 df-fzo 13628 df-hash 14291 df-word 14465 df-concat 14521 df-s1 14546 df-s2 14799 df-s3 14800 df-trkgc 27730 df-trkgb 27731 df-trkgcb 27732 df-trkg 27735 df-cgrg 27793 df-hlg 27883

This theorem is referenced by: hlcgreu 27900 trgcopy 28086 cgraswap 28102 cgracom 28104 cgratr 28105 acopy 28115 acopyeu 28116 tgasa1 28140

W3C validator