angmgmaddlid - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > angmgmaddlid		Structured version Visualization version GIF version

Theorem angmgmaddlid 29272

Description: The left identity element for the addition of angles. (Contributed by Thierry Arnoux, 31-Aug-2026.)

**Hypotheses**
Ref	Expression
angmgmadd.p	⊢ 𝑃 = (Base‘𝐺)
angmgmadd.a	⊢ 𝐴 = {𝑑 ∈ (𝑃 ↑_m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}
angmgmadd.i	⊢ 𝐼 = (Itv‘𝐺)
angmgmadd.d	⊢ − = (dist‘𝐺)
angmgmadd.c	⊢ ∼ = (cgrA‘𝐺)
angmgmadd.l	⊢ 𝐿 = (LineG‘𝐺)
angmgmadd.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
angmgmadd.o	⊢ + = (𝑒 ∈ 𝐴, 𝑓 ∈ 𝐴 ↦ if((𝑒‘0) ∈ ((𝑒‘1)𝐿(𝑒‘2)), ⟨“(𝑓‘0)(𝑓‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩ ∼ 𝑒 ∧ ((𝑓‘1) − 𝑠) = ((𝑒‘1) − (𝑒‘0))))”⟩, ⟨“(𝑒‘0)(𝑒‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ 𝑓 ∧ ((𝑒‘1) − 𝑠) = ((𝑓‘1) − (𝑓‘0)) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))) ≠ ∅))”⟩))
angmgmaddlid.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
angmgmaddlid.y	⊢ (𝜑 → 𝑌 ∈ (𝑃 ∖ {𝑋}))
angmgmaddlid.e	⊢ (𝜑 → 𝐸 ∈ 𝐴)

**Assertion**
Ref	Expression
angmgmaddlid	⊢ (𝜑 → (⟨“𝑋𝑌𝑋”⟩ + 𝐸) ∼ 𝐸)

Distinct variable groups: − ,𝑑,𝑒,𝑓,𝑠 ∼ ,𝑠 𝐸,𝑑,𝑒,𝑓,𝑠 𝐺,𝑑,𝑒,𝑓,𝑠 𝐼,𝑑,𝑒,𝑓,𝑠 𝐿,𝑑,𝑒,𝑓,𝑠 𝑃,𝑑,𝑒,𝑓,𝑠 𝑋,𝑑,𝑒,𝑓,𝑠 𝑌,𝑑,𝑒,𝑓,𝑠 𝜑,𝑑,𝑒,𝑓,𝑠 Allowed substitution hints: 𝐴(𝑒, 𝑓, 𝑠, 𝑑) + (𝑒, 𝑓, 𝑠, 𝑑) ∼ (𝑒, 𝑓, 𝑑)

Proof of Theorem angmgmaddlid Dummy variables 𝑡 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp-6r 800	. . . . . . . . . . 11 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑤 ∈ 𝑃) ∧ 𝐸 = ⟨“𝑢𝑣𝑤”⟩) ∧ 𝑢 ≠ 𝑣) ∧ 𝑣 ≠ 𝑤) ∧ 𝑡 ∈ 𝑃) ∧ 𝑡((hlG‘𝐺)‘𝑣)𝑤) ∧ (𝑣 − 𝑡) = (𝑌 − 𝑋)) → 𝐸 = ⟨“𝑢𝑣𝑤”⟩)
2	1	oveq2d 7432	. . . . . . . . . 10 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑤 ∈ 𝑃) ∧ 𝐸 = ⟨“𝑢𝑣𝑤”⟩) ∧ 𝑢 ≠ 𝑣) ∧ 𝑣 ≠ 𝑤) ∧ 𝑡 ∈ 𝑃) ∧ 𝑡((hlG‘𝐺)‘𝑣)𝑤) ∧ (𝑣 − 𝑡) = (𝑌 − 𝑋)) → (⟨“𝑋𝑌𝑋”⟩ + 𝐸) = (⟨“𝑋𝑌𝑋”⟩ + ⟨“𝑢𝑣𝑤”⟩))
3		angmgmadd.p	. . . . . . . . . . 11 ⊢ 𝑃 = (Base‘𝐺)
4		angmgmadd.a	. . . . . . . . . . 11 ⊢ 𝐴 = {𝑑 ∈ (𝑃 ↑_m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}
5		angmgmadd.i	. . . . . . . . . . 11 ⊢ 𝐼 = (Itv‘𝐺)
6		angmgmadd.d	. . . . . . . . . . 11 ⊢ − = (dist‘𝐺)
7		angmgmadd.c	. . . . . . . . . . 11 ⊢ ∼ = (cgrA‘𝐺)
8		angmgmadd.l	. . . . . . . . . . 11 ⊢ 𝐿 = (LineG‘𝐺)
9		angmgmadd.g	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
10	9	ad9antr 755	. . . . . . . . . . 11 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑤 ∈ 𝑃) ∧ 𝐸 = ⟨“𝑢𝑣𝑤”⟩) ∧ 𝑢 ≠ 𝑣) ∧ 𝑣 ≠ 𝑤) ∧ 𝑡 ∈ 𝑃) ∧ 𝑡((hlG‘𝐺)‘𝑣)𝑤) ∧ (𝑣 − 𝑡) = (𝑌 − 𝑋)) → 𝐺 ∈ TarskiG)
11		simp-9r 806	. . . . . . . . . . 11 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑤 ∈ 𝑃) ∧ 𝐸 = ⟨“𝑢𝑣𝑤”⟩) ∧ 𝑢 ≠ 𝑣) ∧ 𝑣 ≠ 𝑤) ∧ 𝑡 ∈ 𝑃) ∧ 𝑡((hlG‘𝐺)‘𝑣)𝑤) ∧ (𝑣 − 𝑡) = (𝑌 − 𝑋)) → 𝑢 ∈ 𝑃)
12		simp-8r 804	. . . . . . . . . . 11 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑤 ∈ 𝑃) ∧ 𝐸 = ⟨“𝑢𝑣𝑤”⟩) ∧ 𝑢 ≠ 𝑣) ∧ 𝑣 ≠ 𝑤) ∧ 𝑡 ∈ 𝑃) ∧ 𝑡((hlG‘𝐺)‘𝑣)𝑤) ∧ (𝑣 − 𝑡) = (𝑌 − 𝑋)) → 𝑣 ∈ 𝑃)
13		simp-7r 802	. . . . . . . . . . 11 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑤 ∈ 𝑃) ∧ 𝐸 = ⟨“𝑢𝑣𝑤”⟩) ∧ 𝑢 ≠ 𝑣) ∧ 𝑣 ≠ 𝑤) ∧ 𝑡 ∈ 𝑃) ∧ 𝑡((hlG‘𝐺)‘𝑣)𝑤) ∧ (𝑣 − 𝑡) = (𝑌 − 𝑋)) → 𝑤 ∈ 𝑃)
14		angmgmaddlid.x	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
15	14	ad9antr 755	. . . . . . . . . . 11 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑤 ∈ 𝑃) ∧ 𝐸 = ⟨“𝑢𝑣𝑤”⟩) ∧ 𝑢 ≠ 𝑣) ∧ 𝑣 ≠ 𝑤) ∧ 𝑡 ∈ 𝑃) ∧ 𝑡((hlG‘𝐺)‘𝑣)𝑤) ∧ (𝑣 − 𝑡) = (𝑌 − 𝑋)) → 𝑋 ∈ 𝑃)
16		angmgmaddlid.y	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ∈ (𝑃 ∖ {𝑋}))
17	16	eldifad 3914	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
18	17	ad9antr 755	. . . . . . . . . . 11 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑤 ∈ 𝑃) ∧ 𝐸 = ⟨“𝑢𝑣𝑤”⟩) ∧ 𝑢 ≠ 𝑣) ∧ 𝑣 ≠ 𝑤) ∧ 𝑡 ∈ 𝑃) ∧ 𝑡((hlG‘𝐺)‘𝑣)𝑤) ∧ (𝑣 − 𝑡) = (𝑌 − 𝑋)) → 𝑌 ∈ 𝑃)
19		simp-5r 798	. . . . . . . . . . 11 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑤 ∈ 𝑃) ∧ 𝐸 = ⟨“𝑢𝑣𝑤”⟩) ∧ 𝑢 ≠ 𝑣) ∧ 𝑣 ≠ 𝑤) ∧ 𝑡 ∈ 𝑃) ∧ 𝑡((hlG‘𝐺)‘𝑣)𝑤) ∧ (𝑣 − 𝑡) = (𝑌 − 𝑋)) → 𝑢 ≠ 𝑣)
20		simp-4r 796	. . . . . . . . . . 11 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑤 ∈ 𝑃) ∧ 𝐸 = ⟨“𝑢𝑣𝑤”⟩) ∧ 𝑢 ≠ 𝑣) ∧ 𝑣 ≠ 𝑤) ∧ 𝑡 ∈ 𝑃) ∧ 𝑡((hlG‘𝐺)‘𝑣)𝑤) ∧ (𝑣 − 𝑡) = (𝑌 − 𝑋)) → 𝑣 ≠ 𝑤)
21	16	eldifsnbd 4752	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ≠ 𝑋)
22	21	necomd 3012	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
23	22	ad9antr 755	. . . . . . . . . . 11 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑤 ∈ 𝑃) ∧ 𝐸 = ⟨“𝑢𝑣𝑤”⟩) ∧ 𝑢 ≠ 𝑣) ∧ 𝑣 ≠ 𝑤) ∧ 𝑡 ∈ 𝑃) ∧ 𝑡((hlG‘𝐺)‘𝑣)𝑤) ∧ (𝑣 − 𝑡) = (𝑌 − 𝑋)) → 𝑋 ≠ 𝑌)
24	23	necomd 3012	. . . . . . . . . . 11 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑤 ∈ 𝑃) ∧ 𝐸 = ⟨“𝑢𝑣𝑤”⟩) ∧ 𝑢 ≠ 𝑣) ∧ 𝑣 ≠ 𝑤) ∧ 𝑡 ∈ 𝑃) ∧ 𝑡((hlG‘𝐺)‘𝑣)𝑤) ∧ (𝑣 − 𝑡) = (𝑌 − 𝑋)) → 𝑌 ≠ 𝑋)
25		angmgmadd.o	. . . . . . . . . . 11 ⊢ + = (𝑒 ∈ 𝐴, 𝑓 ∈ 𝐴 ↦ if((𝑒‘0) ∈ ((𝑒‘1)𝐿(𝑒‘2)), ⟨“(𝑓‘0)(𝑓‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩ ∼ 𝑒 ∧ ((𝑓‘1) − 𝑠) = ((𝑒‘1) − (𝑒‘0))))”⟩, ⟨“(𝑒‘0)(𝑒‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ 𝑓 ∧ ((𝑒‘1) − 𝑠) = ((𝑓‘1) − (𝑓‘0)) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))) ≠ ∅))”⟩))
26	3, 5, 8, 10, 18, 15, 24	tglinerflx2 28982	. . . . . . . . . . 11 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑤 ∈ 𝑃) ∧ 𝐸 = ⟨“𝑢𝑣𝑤”⟩) ∧ 𝑢 ≠ 𝑣) ∧ 𝑣 ≠ 𝑤) ∧ 𝑡 ∈ 𝑃) ∧ 𝑡((hlG‘𝐺)‘𝑣)𝑤) ∧ (𝑣 − 𝑡) = (𝑌 − 𝑋)) → 𝑋 ∈ (𝑌𝐿𝑋))
27		simpllr 788	. . . . . . . . . . 11 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑤 ∈ 𝑃) ∧ 𝐸 = ⟨“𝑢𝑣𝑤”⟩) ∧ 𝑢 ≠ 𝑣) ∧ 𝑣 ≠ 𝑤) ∧ 𝑡 ∈ 𝑃) ∧ 𝑡((hlG‘𝐺)‘𝑣)𝑤) ∧ (𝑣 − 𝑡) = (𝑌 − 𝑋)) → 𝑡 ∈ 𝑃)
28		eqid 2762	. . . . . . . . . . . 12 ⊢ (hlG‘𝐺) = (hlG‘𝐺)
29		simplr 781	. . . . . . . . . . . . 13 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑤 ∈ 𝑃) ∧ 𝐸 = ⟨“𝑢𝑣𝑤”⟩) ∧ 𝑢 ≠ 𝑣) ∧ 𝑣 ≠ 𝑤) ∧ 𝑡 ∈ 𝑃) ∧ 𝑡((hlG‘𝐺)‘𝑣)𝑤) ∧ (𝑣 − 𝑡) = (𝑌 − 𝑋)) → 𝑡((hlG‘𝐺)‘𝑣)𝑤)
30	3, 5, 28, 27, 13, 12, 10, 29	hlcomd 28950	. . . . . . . . . . . 12 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑤 ∈ 𝑃) ∧ 𝐸 = ⟨“𝑢𝑣𝑤”⟩) ∧ 𝑢 ≠ 𝑣) ∧ 𝑣 ≠ 𝑤) ∧ 𝑡 ∈ 𝑃) ∧ 𝑡((hlG‘𝐺)‘𝑣)𝑤) ∧ (𝑣 − 𝑡) = (𝑌 − 𝑋)) → 𝑤((hlG‘𝐺)‘𝑣)𝑡)
31	3, 5, 28, 15, 11, 18, 10, 23	hlid 28955	. . . . . . . . . . . 12 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑤 ∈ 𝑃) ∧ 𝐸 = ⟨“𝑢𝑣𝑤”⟩) ∧ 𝑢 ≠ 𝑣) ∧ 𝑣 ≠ 𝑤) ∧ 𝑡 ∈ 𝑃) ∧ 𝑡((hlG‘𝐺)‘𝑣)𝑤) ∧ (𝑣 − 𝑡) = (𝑌 − 𝑋)) → 𝑋((hlG‘𝐺)‘𝑌)𝑋)
32	3, 7, 28, 10, 30, 31, 12, 18	zerocgra 29211	. . . . . . . . . . 11 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑤 ∈ 𝑃) ∧ 𝐸 = ⟨“𝑢𝑣𝑤”⟩) ∧ 𝑢 ≠ 𝑣) ∧ 𝑣 ≠ 𝑤) ∧ 𝑡 ∈ 𝑃) ∧ 𝑡((hlG‘𝐺)‘𝑣)𝑤) ∧ (𝑣 − 𝑡) = (𝑌 − 𝑋)) → ⟨“𝑤𝑣𝑡”⟩ ∼ ⟨“𝑋𝑌𝑋”⟩)
33		simpr 490	. . . . . . . . . . 11 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑤 ∈ 𝑃) ∧ 𝐸 = ⟨“𝑢𝑣𝑤”⟩) ∧ 𝑢 ≠ 𝑣) ∧ 𝑣 ≠ 𝑤) ∧ 𝑡 ∈ 𝑃) ∧ 𝑡((hlG‘𝐺)‘𝑣)𝑤) ∧ (𝑣 − 𝑡) = (𝑌 − 𝑋)) → (𝑣 − 𝑡) = (𝑌 − 𝑋))
34	3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 15, 18, 15, 19, 20, 23, 24, 25, 26, 27, 32, 33	angmgmaddov2 29269	. . . . . . . . . 10 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑤 ∈ 𝑃) ∧ 𝐸 = ⟨“𝑢𝑣𝑤”⟩) ∧ 𝑢 ≠ 𝑣) ∧ 𝑣 ≠ 𝑤) ∧ 𝑡 ∈ 𝑃) ∧ 𝑡((hlG‘𝐺)‘𝑣)𝑤) ∧ (𝑣 − 𝑡) = (𝑌 − 𝑋)) → (⟨“𝑋𝑌𝑋”⟩ + ⟨“𝑢𝑣𝑤”⟩) = ⟨“𝑢𝑣𝑡”⟩)
35	2, 34	eqtrd 2797	. . . . . . . . 9 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑤 ∈ 𝑃) ∧ 𝐸 = ⟨“𝑢𝑣𝑤”⟩) ∧ 𝑢 ≠ 𝑣) ∧ 𝑣 ≠ 𝑤) ∧ 𝑡 ∈ 𝑃) ∧ 𝑡((hlG‘𝐺)‘𝑣)𝑤) ∧ (𝑣 − 𝑡) = (𝑌 − 𝑋)) → (⟨“𝑋𝑌𝑋”⟩ + 𝐸) = ⟨“𝑢𝑣𝑡”⟩)
36	7	eqcomi 2771	. . . . . . . . . . 11 ⊢ (cgrA‘𝐺) = ∼
37	36	a1i 11	. . . . . . . . . 10 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑤 ∈ 𝑃) ∧ 𝐸 = ⟨“𝑢𝑣𝑤”⟩) ∧ 𝑢 ≠ 𝑣) ∧ 𝑣 ≠ 𝑤) ∧ 𝑡 ∈ 𝑃) ∧ 𝑡((hlG‘𝐺)‘𝑣)𝑤) ∧ (𝑣 − 𝑡) = (𝑌 − 𝑋)) → (cgrA‘𝐺) = ∼ )
38	33	eqcomd 2768	. . . . . . . . . . . . 13 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑤 ∈ 𝑃) ∧ 𝐸 = ⟨“𝑢𝑣𝑤”⟩) ∧ 𝑢 ≠ 𝑣) ∧ 𝑣 ≠ 𝑤) ∧ 𝑡 ∈ 𝑃) ∧ 𝑡((hlG‘𝐺)‘𝑣)𝑤) ∧ (𝑣 − 𝑡) = (𝑌 − 𝑋)) → (𝑌 − 𝑋) = (𝑣 − 𝑡))
39	3, 6, 5, 10, 18, 15, 12, 27, 38, 24	tgcgrneq 28825	. . . . . . . . . . . 12 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑤 ∈ 𝑃) ∧ 𝐸 = ⟨“𝑢𝑣𝑤”⟩) ∧ 𝑢 ≠ 𝑣) ∧ 𝑣 ≠ 𝑤) ∧ 𝑡 ∈ 𝑃) ∧ 𝑡((hlG‘𝐺)‘𝑣)𝑤) ∧ (𝑣 − 𝑡) = (𝑌 − 𝑋)) → 𝑣 ≠ 𝑡)
40	3, 5, 10, 28, 11, 12, 27, 19, 39	cgraid 29206	. . . . . . . . . . 11 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑤 ∈ 𝑃) ∧ 𝐸 = ⟨“𝑢𝑣𝑤”⟩) ∧ 𝑢 ≠ 𝑣) ∧ 𝑣 ≠ 𝑤) ∧ 𝑡 ∈ 𝑃) ∧ 𝑡((hlG‘𝐺)‘𝑣)𝑤) ∧ (𝑣 − 𝑡) = (𝑌 − 𝑋)) → ⟨“𝑢𝑣𝑡”⟩(cgrA‘𝐺)⟨“𝑢𝑣𝑡”⟩)
41	3, 5, 28, 10, 11, 12, 27, 11, 12, 27, 40, 13, 30	cgrahl2 29204	. . . . . . . . . 10 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑤 ∈ 𝑃) ∧ 𝐸 = ⟨“𝑢𝑣𝑤”⟩) ∧ 𝑢 ≠ 𝑣) ∧ 𝑣 ≠ 𝑤) ∧ 𝑡 ∈ 𝑃) ∧ 𝑡((hlG‘𝐺)‘𝑣)𝑤) ∧ (𝑣 − 𝑡) = (𝑌 − 𝑋)) → ⟨“𝑢𝑣𝑡”⟩(cgrA‘𝐺)⟨“𝑢𝑣𝑤”⟩)
42	37, 41	breqdi 5122	. . . . . . . . 9 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑤 ∈ 𝑃) ∧ 𝐸 = ⟨“𝑢𝑣𝑤”⟩) ∧ 𝑢 ≠ 𝑣) ∧ 𝑣 ≠ 𝑤) ∧ 𝑡 ∈ 𝑃) ∧ 𝑡((hlG‘𝐺)‘𝑣)𝑤) ∧ (𝑣 − 𝑡) = (𝑌 − 𝑋)) → ⟨“𝑢𝑣𝑡”⟩ ∼ ⟨“𝑢𝑣𝑤”⟩)
43	35, 42	eqbrtrd 5131	. . . . . . . 8 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑤 ∈ 𝑃) ∧ 𝐸 = ⟨“𝑢𝑣𝑤”⟩) ∧ 𝑢 ≠ 𝑣) ∧ 𝑣 ≠ 𝑤) ∧ 𝑡 ∈ 𝑃) ∧ 𝑡((hlG‘𝐺)‘𝑣)𝑤) ∧ (𝑣 − 𝑡) = (𝑌 − 𝑋)) → (⟨“𝑋𝑌𝑋”⟩ + 𝐸) ∼ ⟨“𝑢𝑣𝑤”⟩)
44	43	anasss 472	. . . . . . 7 ⊢ (((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑤 ∈ 𝑃) ∧ 𝐸 = ⟨“𝑢𝑣𝑤”⟩) ∧ 𝑢 ≠ 𝑣) ∧ 𝑣 ≠ 𝑤) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡((hlG‘𝐺)‘𝑣)𝑤 ∧ (𝑣 − 𝑡) = (𝑌 − 𝑋))) → (⟨“𝑋𝑌𝑋”⟩ + 𝐸) ∼ ⟨“𝑢𝑣𝑤”⟩)
45		simp-5r 798	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑤 ∈ 𝑃) ∧ 𝐸 = ⟨“𝑢𝑣𝑤”⟩) ∧ 𝑢 ≠ 𝑣) ∧ 𝑣 ≠ 𝑤) → 𝑣 ∈ 𝑃)
46	17	ad6antr 749	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑤 ∈ 𝑃) ∧ 𝐸 = ⟨“𝑢𝑣𝑤”⟩) ∧ 𝑢 ≠ 𝑣) ∧ 𝑣 ≠ 𝑤) → 𝑌 ∈ 𝑃)
47	14	ad6antr 749	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑤 ∈ 𝑃) ∧ 𝐸 = ⟨“𝑢𝑣𝑤”⟩) ∧ 𝑢 ≠ 𝑣) ∧ 𝑣 ≠ 𝑤) → 𝑋 ∈ 𝑃)
48	9	ad6antr 749	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑤 ∈ 𝑃) ∧ 𝐸 = ⟨“𝑢𝑣𝑤”⟩) ∧ 𝑢 ≠ 𝑣) ∧ 𝑣 ≠ 𝑤) → 𝐺 ∈ TarskiG)
49		simp-4r 796	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑤 ∈ 𝑃) ∧ 𝐸 = ⟨“𝑢𝑣𝑤”⟩) ∧ 𝑢 ≠ 𝑣) ∧ 𝑣 ≠ 𝑤) → 𝑤 ∈ 𝑃)
50		simpr 490	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑤 ∈ 𝑃) ∧ 𝐸 = ⟨“𝑢𝑣𝑤”⟩) ∧ 𝑢 ≠ 𝑣) ∧ 𝑣 ≠ 𝑤) → 𝑣 ≠ 𝑤)
51	50	necomd 3012	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑤 ∈ 𝑃) ∧ 𝐸 = ⟨“𝑢𝑣𝑤”⟩) ∧ 𝑢 ≠ 𝑣) ∧ 𝑣 ≠ 𝑤) → 𝑤 ≠ 𝑣)
52	21	ad6antr 749	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑤 ∈ 𝑃) ∧ 𝐸 = ⟨“𝑢𝑣𝑤”⟩) ∧ 𝑢 ≠ 𝑣) ∧ 𝑣 ≠ 𝑤) → 𝑌 ≠ 𝑋)
53	3, 5, 28, 45, 46, 47, 48, 49, 6, 51, 52	hlcgrex 28962	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑤 ∈ 𝑃) ∧ 𝐸 = ⟨“𝑢𝑣𝑤”⟩) ∧ 𝑢 ≠ 𝑣) ∧ 𝑣 ≠ 𝑤) → ∃𝑡 ∈ 𝑃 (𝑡((hlG‘𝐺)‘𝑣)𝑤 ∧ (𝑣 − 𝑡) = (𝑌 − 𝑋)))
54	44, 53	r19.29a 3172	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑤 ∈ 𝑃) ∧ 𝐸 = ⟨“𝑢𝑣𝑤”⟩) ∧ 𝑢 ≠ 𝑣) ∧ 𝑣 ≠ 𝑤) → (⟨“𝑋𝑌𝑋”⟩ + 𝐸) ∼ ⟨“𝑢𝑣𝑤”⟩)
55		simpllr 788	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑤 ∈ 𝑃) ∧ 𝐸 = ⟨“𝑢𝑣𝑤”⟩) ∧ 𝑢 ≠ 𝑣) ∧ 𝑣 ≠ 𝑤) → 𝐸 = ⟨“𝑢𝑣𝑤”⟩)
56	54, 55	breqtrrd 5137	. . . . 5 ⊢ (((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑤 ∈ 𝑃) ∧ 𝐸 = ⟨“𝑢𝑣𝑤”⟩) ∧ 𝑢 ≠ 𝑣) ∧ 𝑣 ≠ 𝑤) → (⟨“𝑋𝑌𝑋”⟩ + 𝐸) ∼ 𝐸)
57	56	anasss 472	. . . 4 ⊢ ((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑤 ∈ 𝑃) ∧ 𝐸 = ⟨“𝑢𝑣𝑤”⟩) ∧ (𝑢 ≠ 𝑣 ∧ 𝑣 ≠ 𝑤)) → (⟨“𝑋𝑌𝑋”⟩ + 𝐸) ∼ 𝐸)
58	57	anasss 472	. . 3 ⊢ (((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ 𝑤 ∈ 𝑃) ∧ (𝐸 = ⟨“𝑢𝑣𝑤”⟩ ∧ (𝑢 ≠ 𝑣 ∧ 𝑣 ≠ 𝑤))) → (⟨“𝑋𝑌𝑋”⟩ + 𝐸) ∼ 𝐸)
59	58	r19.29an 3168	. 2 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ ∃𝑤 ∈ 𝑃 (𝐸 = ⟨“𝑢𝑣𝑤”⟩ ∧ (𝑢 ≠ 𝑣 ∧ 𝑣 ≠ 𝑤))) → (⟨“𝑋𝑌𝑋”⟩ + 𝐸) ∼ 𝐸)
60	3	fvexi 6896	. . 3 ⊢ 𝑃 ∈ V
61		angmgmaddlid.e	. . 3 ⊢ (𝜑 → 𝐸 ∈ 𝐴)
62	60, 4, 61	elcgrabasi 29255	. 2 ⊢ (𝜑 → ∃𝑢 ∈ 𝑃 ∃𝑣 ∈ 𝑃 ∃𝑤 ∈ 𝑃 (𝐸 = ⟨“𝑢𝑣𝑤”⟩ ∧ (𝑢 ≠ 𝑣 ∧ 𝑣 ≠ 𝑤)))
63	59, 62	r19.29vva 3224	1 ⊢ (𝜑 → (⟨“𝑋𝑌𝑋”⟩ + 𝐸) ∼ 𝐸)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2957 ∃wrex 3088 {crab 3414 ∖ cdif 3899 ∩ cin 3901 ∅c0 4282 ifcif 4485 {csn 4587 class class class wbr 5107 ‘cfv 6537 ℩crio 7372 (class class class)co 7416 ∈ cmpo 7418 ↑_m cmap 8829 0cc0 11127 1c1 11128 2c2 12322 3c3 12323 ..^cfzo 13711 ⟨“cs3 14915 Basecbs 17305 distcds 17355 TarskiGcstrkg 28769 Itvcitv 28775 LineGclng 28776 hlGchlg 28943 cgrAccgra 29194

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7866 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8458 df-oadd 8462 df-er 8699 df-map 8831 df-pm 8832 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-dju 9909 df-card 9947 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-nn 12261 df-2 12330 df-3 12331 df-n0 12532 df-xnn0 12605 df-z 12619 df-uz 12891 df-fz 13564 df-fzo 13712 df-hash 14397 df-word 14581 df-concat 14638 df-s1 14665 df-s2 14921 df-s3 14922 df-trkgc 28790 df-trkgb 28791 df-trkgcb 28792 df-trkg 28795 df-cgrg 28854 df-leg 28926 df-hlg 28944 df-cgra 29195

This theorem is used by: angmgmlem 29275

W3C validator