Theorem trgcopyeu 28883

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 28881	. 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 7379	. . . . . . . . . . 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 3218	. . . . . . . . 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 5172	. . . . . . 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 28882	. . . . . 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 3180	. 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 14792	. . . . 5 ⊢ (𝑓 = 𝑘 → ⟨“𝐷𝐸𝑓”⟩ = ⟨“𝐷𝐸𝑘”⟩)
56	55	breq2d 5111	. . . 4 ⊢ (𝑓 = 𝑘 → (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑓”⟩ ↔ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑘”⟩))
57		breq1 5102	. . . 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 3690	. 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 3061  ∃!wreu 3349   ∖ cdif 3899   class class class wbr 5099  {copab 5161  ‘cfv 6493  (class class class)co 7361  ⟨“cs3 14770  Basecbs 17141  distcds 17191  TarskiGcstrkg 28504  Itvcitv 28510  LineGclng 28511  cgrGccgrg 28587  hlGchlg 28677  hpGchpg 28834

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 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7683  ax-cnex 11087  ax-resscn 11088  ax-1cn 11089  ax-icn 11090  ax-addcl 11091  ax-addrcl 11092  ax-mulcl 11093  ax-mulrcl 11094  ax-mulcom 11095  ax-addass 11096  ax-mulass 11097  ax-distr 11098  ax-i2m1 11099  ax-1ne0 11100  ax-1rid 11101  ax-rnegex 11102  ax-rrecex 11103  ax-cnre 11104  ax-pre-lttri 11105  ax-pre-lttrn 11106  ax-pre-ltadd 11107  ax-pre-mulgt0 11108

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 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-1st 7936  df-2nd 7937  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-1o 8400  df-oadd 8404  df-er 8638  df-map 8770  df-pm 8771  df-en 8889  df-dom 8890  df-sdom 8891  df-fin 8892  df-dju 9818  df-card 9856  df-pnf 11173  df-mnf 11174  df-xr 11175  df-ltxr 11176  df-le 11177  df-sub 11371  df-neg 11372  df-nn 12151  df-2 12213  df-3 12214  df-n0 12407  df-xnn0 12480  df-z 12494  df-uz 12757  df-fz 13429  df-fzo 13576  df-hash 14259  df-word 14442  df-concat 14499  df-s1 14525  df-s2 14776  df-s3 14777  df-trkgc 28525  df-trkgb 28526  df-trkgcb 28527  df-trkgld 28529  df-trkg 28530  df-cgrg 28588  df-ismt 28610  df-leg 28660  df-hlg 28678  df-mir 28730  df-rag 28771  df-perpg 28773  df-hpg 28835  df-mid 28851  df-lmi 28852

This theorem is referenced by: (None)