Theorem trgcopyeu 28894

Description: Triangle construction: a copy of a given triangle can always be constructed in such a way that one side is lying on a half-line, and the third vertex is on a given half-plane: uniqueness part. Second part of Theorem 10.16 of [Schwabhauser] p. 92. (Contributed by Thierry Arnoux, 8-Aug-2020.)

Hypotheses

Ref	Expression
trgcopy.p	⊢ 𝑃 = (Base‘𝐺)
trgcopy.m	⊢ − = (dist‘𝐺)
trgcopy.i	⊢ 𝐼 = (Itv‘𝐺)
trgcopy.l	⊢ 𝐿 = (LineG‘𝐺)
trgcopy.k	⊢ 𝐾 = (hlG‘𝐺)
trgcopy.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
trgcopy.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
trgcopy.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
trgcopy.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
trgcopy.d	⊢ (𝜑 → 𝐷 ∈ 𝑃)
trgcopy.e	⊢ (𝜑 → 𝐸 ∈ 𝑃)
trgcopy.f	⊢ (𝜑 → 𝐹 ∈ 𝑃)
trgcopy.1	⊢ (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))
trgcopy.2	⊢ (𝜑 → ¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))
trgcopy.3	⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸))

Assertion

Ref	Expression
trgcopyeu	⊢ (𝜑 → ∃!𝑓 ∈ 𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹))

Distinct variable groups:   − ,𝑓   𝐴,𝑓   𝐵,𝑓   𝐶,𝑓   𝐷,𝑓   𝑓,𝐸   𝑓,𝐹   𝑓,𝐺   𝑓,𝐼   𝑓,𝐿   𝑃,𝑓   𝜑,𝑓   𝑓,𝐾

Proof of Theorem trgcopyeu

Dummy variables 𝑎 𝑏 𝑘 𝑡 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		trgcopy.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
2		trgcopy.m	. . 3 ⊢ − = (dist‘𝐺)
3		trgcopy.i	. . 3 ⊢ 𝐼 = (Itv‘𝐺)
4		trgcopy.l	. . 3 ⊢ 𝐿 = (LineG‘𝐺)
5		trgcopy.k	. . 3 ⊢ 𝐾 = (hlG‘𝐺)
6		trgcopy.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7		trgcopy.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
8		trgcopy.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
9		trgcopy.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
10		trgcopy.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
11		trgcopy.e	. . 3 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
12		trgcopy.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑃)
13		trgcopy.1	. . 3 ⊢ (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))
14		trgcopy.2	. . 3 ⊢ (𝜑 → ¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))
15		trgcopy.3	. . 3 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸))
16	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15	trgcopy 28892	. 2 ⊢ (𝜑 → ∃𝑓 ∈ 𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹))
17	6	ad5antr 735	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ 𝑘 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩) ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) → 𝐺 ∈ TarskiG)
18	7	ad5antr 735	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ 𝑘 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩) ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) → 𝐴 ∈ 𝑃)
19	8	ad5antr 735	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ 𝑘 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩) ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) → 𝐵 ∈ 𝑃)
20	9	ad5antr 735	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ 𝑘 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩) ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) → 𝐶 ∈ 𝑃)
21	10	ad5antr 735	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ 𝑘 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩) ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) → 𝐷 ∈ 𝑃)
22	11	ad5antr 735	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ 𝑘 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩) ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) → 𝐸 ∈ 𝑃)
23	12	ad5antr 735	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ 𝑘 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩) ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) → 𝐹 ∈ 𝑃)
24	13	ad5antr 735	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ 𝑘 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩) ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))
25	14	ad5antr 735	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ 𝑘 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩) ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) → ¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))
26	15	ad5antr 735	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ 𝑘 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩) ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) → (𝐴 − 𝐵) = (𝐷 − 𝐸))
27		simpl 482	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → 𝑥 = 𝑎)
28	27	eleq1d 2822	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝑥 ∈ (𝑃 ∖ (𝐷𝐿𝐸)) ↔ 𝑎 ∈ (𝑃 ∖ (𝐷𝐿𝐸))))
29		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → 𝑦 = 𝑏)
30	29	eleq1d 2822	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝑦 ∈ (𝑃 ∖ (𝐷𝐿𝐸)) ↔ 𝑏 ∈ (𝑃 ∖ (𝐷𝐿𝐸))))
31	28, 30	anbi12d 633	. . . . . . . . 9 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ((𝑥 ∈ (𝑃 ∖ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ (𝑃 ∖ (𝐷𝐿𝐸))) ↔ (𝑎 ∈ (𝑃 ∖ (𝐷𝐿𝐸)) ∧ 𝑏 ∈ (𝑃 ∖ (𝐷𝐿𝐸)))))
32		simpr 484	. . . . . . . . . . 11 ⊢ (((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) ∧ 𝑧 = 𝑡) → 𝑧 = 𝑡)
33		simpll 767	. . . . . . . . . . . 12 ⊢ (((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) ∧ 𝑧 = 𝑡) → 𝑥 = 𝑎)
34		simplr 769	. . . . . . . . . . . 12 ⊢ (((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) ∧ 𝑧 = 𝑡) → 𝑦 = 𝑏)
35	33, 34	oveq12d 7382	. . . . . . . . . . 11 ⊢ (((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) ∧ 𝑧 = 𝑡) → (𝑥𝐼𝑦) = (𝑎𝐼𝑏))
36	32, 35	eleq12d 2831	. . . . . . . . . 10 ⊢ (((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) ∧ 𝑧 = 𝑡) → (𝑧 ∈ (𝑥𝐼𝑦) ↔ 𝑡 ∈ (𝑎𝐼𝑏)))
37	36	cbvrexdva 3219	. . . . . . . . 9 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (∃𝑧 ∈ (𝐷𝐿𝐸)𝑧 ∈ (𝑥𝐼𝑦) ↔ ∃𝑡 ∈ (𝐷𝐿𝐸)𝑡 ∈ (𝑎𝐼𝑏)))
38	31, 37	anbi12d 633	. . . . . . . 8 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (((𝑥 ∈ (𝑃 ∖ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ (𝑃 ∖ (𝐷𝐿𝐸))) ∧ ∃𝑧 ∈ (𝐷𝐿𝐸)𝑧 ∈ (𝑥𝐼𝑦)) ↔ ((𝑎 ∈ (𝑃 ∖ (𝐷𝐿𝐸)) ∧ 𝑏 ∈ (𝑃 ∖ (𝐷𝐿𝐸))) ∧ ∃𝑡 ∈ (𝐷𝐿𝐸)𝑡 ∈ (𝑎𝐼𝑏))))
39	38	cbvopabv 5159	. . . . . . 7 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑃 ∖ (𝐷𝐿𝐸)) ∧ 𝑦 ∈ (𝑃 ∖ (𝐷𝐿𝐸))) ∧ ∃𝑧 ∈ (𝐷𝐿𝐸)𝑧 ∈ (𝑥𝐼𝑦))} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝐷𝐿𝐸)) ∧ 𝑏 ∈ (𝑃 ∖ (𝐷𝐿𝐸))) ∧ ∃𝑡 ∈ (𝐷𝐿𝐸)𝑡 ∈ (𝑎𝐼𝑏))}
40		simp-5r 786	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ 𝑘 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩) ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) → 𝑓 ∈ 𝑃)
41		simp-4r 784	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ 𝑘 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩) ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) → 𝑘 ∈ 𝑃)
42		simpllr 776	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ 𝑘 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩) ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) → (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹))
43	42	simpld 494	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ 𝑘 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩) ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩)
44		simplr 769	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ 𝑘 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩) ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩)
45	42	simprd 495	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ 𝑘 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩) ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) → 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)
46		simpr 484	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ 𝑘 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩) ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) → 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)
47	1, 2, 3, 4, 5, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 39, 40, 41, 43, 44, 45, 46	trgcopyeulem 28893	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ 𝑘 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩) ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) → 𝑓 = 𝑘)
48	47	anasss 466	. . . . 5 ⊢ (((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ 𝑘 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩ ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) → 𝑓 = 𝑘)
49	48	expl 457	. . . 4 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ 𝑘 ∈ 𝑃) → (((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩ ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) → 𝑓 = 𝑘))
50	49	anasss 466	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑃 ∧ 𝑘 ∈ 𝑃)) → (((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩ ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) → 𝑓 = 𝑘))
51	50	ralrimivva 3181	. 2 ⊢ (𝜑 → ∀𝑓 ∈ 𝑃 ∀𝑘 ∈ 𝑃 (((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩ ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) → 𝑓 = 𝑘))
52		eqidd 2738	. . . . . 6 ⊢ (𝑓 = 𝑘 → 𝐷 = 𝐷)
53		eqidd 2738	. . . . . 6 ⊢ (𝑓 = 𝑘 → 𝐸 = 𝐸)
54		id 22	. . . . . 6 ⊢ (𝑓 = 𝑘 → 𝑓 = 𝑘)
55	52, 53, 54	s3eqd 14823	. . . . 5 ⊢ (𝑓 = 𝑘 → ⟨“𝐷𝐸𝑓”⟩ = ⟨“𝐷𝐸𝑘”⟩)
56	55	breq2d 5098	. . . 4 ⊢ (𝑓 = 𝑘 → (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ↔ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩))
57		breq1 5089	. . . 4 ⊢ (𝑓 = 𝑘 → (𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹 ↔ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹))
58	56, 57	anbi12d 633	. . 3 ⊢ (𝑓 = 𝑘 → ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) ↔ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩ ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)))
59	58	reu4 3678	. 2 ⊢ (∃!𝑓 ∈ 𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) ↔ (∃𝑓 ∈ 𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) ∧ ∀𝑓 ∈ 𝑃 ∀𝑘 ∈ 𝑃 (((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩ ∧ 𝑘((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)) → 𝑓 = 𝑘)))
60	16, 51, 59	sylanbrc 584	1 ⊢ (𝜑 → ∃!𝑓 ∈ 𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ∧ 𝑓((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  ∃!wreu 3341   ∖ cdif 3887   class class class wbr 5086  {copab 5148  ‘cfv 6496  (class class class)co 7364  ⟨“cs3 14801  Basecbs 17176  distcds 17226  TarskiGcstrkg 28515  Itvcitv 28521  LineGclng 28522  cgrGccgrg 28598  hlGchlg 28688  hpGchpg 28845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5306  ax-pr 5374  ax-un 7686  ax-cnex 11091  ax-resscn 11092  ax-1cn 11093  ax-icn 11094  ax-addcl 11095  ax-addrcl 11096  ax-mulcl 11097  ax-mulrcl 11098  ax-mulcom 11099  ax-addass 11100  ax-mulass 11101  ax-distr 11102  ax-i2m1 11103  ax-1ne0 11104  ax-1rid 11105  ax-rnegex 11106  ax-rrecex 11107  ax-cnre 11108  ax-pre-lttri 11109  ax-pre-lttrn 11110  ax-pre-ltadd 11111  ax-pre-mulgt0 11112

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5523  df-eprel 5528  df-po 5536  df-so 5537  df-fr 5581  df-we 5583  df-xp 5634  df-rel 5635  df-cnv 5636  df-co 5637  df-dm 5638  df-rn 5639  df-res 5640  df-ima 5641  df-pred 6263  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7321  df-ov 7367  df-oprab 7368  df-mpo 7369  df-om 7815  df-1st 7939  df-2nd 7940  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-oadd 8406  df-er 8640  df-map 8772  df-pm 8773  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-dju 9822  df-card 9860  df-pnf 11178  df-mnf 11179  df-xr 11180  df-ltxr 11181  df-le 11182  df-sub 11376  df-neg 11377  df-nn 12172  df-2 12241  df-3 12242  df-n0 12435  df-xnn0 12508  df-z 12522  df-uz 12786  df-fz 13459  df-fzo 13606  df-hash 14290  df-word 14473  df-concat 14530  df-s1 14556  df-s2 14807  df-s3 14808  df-trkgc 28536  df-trkgb 28537  df-trkgcb 28538  df-trkgld 28540  df-trkg 28541  df-cgrg 28599  df-ismt 28621  df-leg 28671  df-hlg 28689  df-mir 28741  df-rag 28782  df-perpg 28784  df-hpg 28846  df-mid 28862  df-lmi 28863

This theorem is referenced by: (None)