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 29422

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 7459	. . . . . . . . . 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 4054	. . . . . . . . . . . 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 4749	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ≠ 𝑋)
22	21	necomd 3014	. . . . . . . . . . . 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 3014	. . . . . . . . . . 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 29132	. . . . . . . . . . 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 2763	. . . . . . . . . . . 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 29100	. . . . . . . . . . . 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 29105	. . . . . . . . . . . 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 29361	. . . . . . . . . . 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 29419	. . . . . . . . . 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 2798	. . . . . . . . 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 2772	. . . . . . . . . . 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 2769	. . . . . . . . . . . . 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 28975	. . . . . . . . . . . 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 29356	. . . . . . . . . . 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 29354	. . . . . . . . . 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 5121	. . . . . . . . 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 5130	. . . . . . . 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 3014	. . . . . . . 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 29112	. . . . . . 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 3174	. . . . . 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 5136	. . . . 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 3170	. 2 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ ∃𝑤 ∈ 𝑃 (𝐸 = ⟨“𝑢𝑣𝑤”⟩ ∧ (𝑢 ≠ 𝑣 ∧ 𝑣 ≠ 𝑤))) → (⟨“𝑋𝑌𝑋”⟩ + 𝐸) ∼ 𝐸)
60	3	fvexi 6922	. . 3 ⊢ 𝑃 ∈ V
61		angmgmaddlid.e	. . 3 ⊢ (𝜑 → 𝐸 ∈ 𝐴)
62	60, 4, 61	elcgrabasi 29405	. 2 ⊢ (𝜑 → ∃𝑢 ∈ 𝑃 ∃𝑣 ∈ 𝑃 ∃𝑤 ∈ 𝑃 (𝐸 = ⟨“𝑢𝑣𝑤”⟩ ∧ (𝑢 ≠ 𝑣 ∧ 𝑣 ≠ 𝑤)))
63	59, 62	r19.29vva 3226	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 2147 ≠ wne 2959 ∃wrex 3090 {crab 3416 ∖ cdif 4048 ∩ cin 4092 ∅c0 4281 ifcif 4485 {csn 4588 class class class wbr 5106 ‘cfv 6558 ℩crio 7399 (class class class)co 7443 ∈ cmpo 7445 ↑_m cmap 8875 0cc0 11227 1c1 11228 2c2 12424 3c3 12425 ..^cfzo 13815 ⟨“cs3 15021 Basecbs 17416 distcds 17466 TarskiGcstrkg 28919 Itvcitv 28925 LineGclng 28926 hlGchlg 29093 cgrAccgra 29344

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-cnex 11283 ax-resscn 11284 ax-1cn 11285 ax-icn 11286 ax-addcl 11287 ax-addrcl 11288 ax-mulcl 11289 ax-mulrcl 11290 ax-mulcom 11291 ax-addass 11292 ax-mulass 11293 ax-distr 11294 ax-i2m1 11295 ax-1ne0 11296 ax-1rid 11297 ax-rnegex 11298 ax-rrecex 11299 ax-cnre 11300 ax-pre-lttri 11301 ax-pre-lttrn 11302 ax-pre-ltadd 11303 ax-pre-mulgt0 11304

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6322 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-om 7902 df-1st 8025 df-2nd 8026 df-frecs 8327 df-wrecs 8358 df-recs 8407 df-rdg 8446 df-1o 8504 df-oadd 8508 df-er 8745 df-map 8877 df-pm 8878 df-en 9002 df-dom 9003 df-sdom 9004 df-fin 9005 df-dju 10009 df-card 10047 df-pnf 11372 df-mnf 11373 df-xr 11374 df-ltxr 11375 df-le 11376 df-sub 11570 df-neg 11571 df-nn 12363 df-2 12432 df-3 12433 df-n0 12634 df-xnn0 12707 df-z 12721 df-uz 12993 df-fz 13667 df-fzo 13816 df-hash 14502 df-word 14686 df-concat 14743 df-s1 14770 df-s2 15027 df-s3 15028 df-trkgc 28940 df-trkgb 28941 df-trkgcb 28942 df-trkg 28945 df-cgrg 29004 df-leg 29076 df-hlg 29094 df-cgra 29345

This theorem is used by: angmgmlem 29425

W3C validator