Theorem trgcopyeu 28859

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 28857	. 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 2820	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝑥 ∈ (𝑃 ∖ (𝐷𝐿𝐸)) ↔ 𝑎 ∈ (𝑃 ∖ (𝐷𝐿𝐸))))
29		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → 𝑦 = 𝑏)
30	29	eleq1d 2820	. . . . . . . . . 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 7376	. . . . . . . . . . 11 ⊢ (((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) ∧ 𝑧 = 𝑡) → (𝑥𝐼𝑦) = (𝑎𝐼𝑏))
36	32, 35	eleq12d 2829	. . . . . . . . . 10 ⊢ (((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) ∧ 𝑧 = 𝑡) → (𝑧 ∈ (𝑥𝐼𝑦) ↔ 𝑡 ∈ (𝑎𝐼𝑏)))
37	36	cbvrexdva 3216	. . . . . . . . 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 5170	. . . . . . 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 28858	. . . . . 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 3178	. 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 2736	. . . . . 6 ⊢ (𝑓 = 𝑘 → 𝐷 = 𝐷)
53		eqidd 2736	. . . . . 6 ⊢ (𝑓 = 𝑘 → 𝐸 = 𝐸)
54		id 22	. . . . . 6 ⊢ (𝑓 = 𝑘 → 𝑓 = 𝑘)
55	52, 53, 54	s3eqd 14789	. . . . 5 ⊢ (𝑓 = 𝑘 → ⟨“𝐷𝐸𝑓”⟩ = ⟨“𝐷𝐸𝑘”⟩)
56	55	breq2d 5109	. . . 4 ⊢ (𝑓 = 𝑘 → (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ↔ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩))
57		breq1 5100	. . . 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 3688	. 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 3050  ∃wrex 3059  ∃!wreu 3347   ∖ cdif 3897   class class class wbr 5097  {copab 5159  ‘cfv 6491  (class class class)co 7358  ⟨“cs3 14767  Basecbs 17138  distcds 17188  TarskiGcstrkg 28480  Itvcitv 28486  LineGclng 28487  cgrGccgrg 28563  hlGchlg 28653  hpGchpg 28810

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 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

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 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3349  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4863  df-int 4902  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6258  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-oadd 8401  df-er 8635  df-map 8767  df-pm 8768  df-en 8886  df-dom 8887  df-sdom 8888  df-fin 8889  df-dju 9815  df-card 9853  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12148  df-2 12210  df-3 12211  df-n0 12404  df-xnn0 12477  df-z 12491  df-uz 12754  df-fz 13426  df-fzo 13573  df-hash 14256  df-word 14439  df-concat 14496  df-s1 14522  df-s2 14773  df-s3 14774  df-trkgc 28501  df-trkgb 28502  df-trkgcb 28503  df-trkgld 28505  df-trkg 28506  df-cgrg 28564  df-ismt 28586  df-leg 28636  df-hlg 28654  df-mir 28706  df-rag 28747  df-perpg 28749  df-hpg 28811  df-mid 28827  df-lmi 28828

This theorem is referenced by: (None)