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Theorem angmgmaddlid 29326

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 7424	. . . . . . . . . 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 3910	. . . . . . . . . . . 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 4748	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ≠ 𝑋)
22	21	necomd 3010	. . . . . . . . . . . 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 3010	. . . . . . . . . . 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 29036	. . . . . . . . . . 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 2760	. . . . . . . . . . . 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 29004	. . . . . . . . . . . 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 29009	. . . . . . . . . . . 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 29265	. . . . . . . . . . 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 29323	. . . . . . . . . 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 2795	. . . . . . . . 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 2769	. . . . . . . . . . 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 2766	. . . . . . . . . . . . 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 28879	. . . . . . . . . . . 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 29260	. . . . . . . . . . 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 29258	. . . . . . . . . 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 5117	. . . . . . . . 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 5126	. . . . . . . 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 3010	. . . . . . . 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 29016	. . . . . . 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 3170	. . . . . 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 5132	. . . . 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 3166	. 2 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑣 ∈ 𝑃) ∧ ∃𝑤 ∈ 𝑃 (𝐸 = ⟨“𝑢𝑣𝑤”⟩ ∧ (𝑢 ≠ 𝑣 ∧ 𝑣 ≠ 𝑤))) → (⟨“𝑋𝑌𝑋”⟩ + 𝐸) ∼ 𝐸)
60	3	fvexi 6887	. . 3 ⊢ 𝑃 ∈ V
61		angmgmaddlid.e	. . 3 ⊢ (𝜑 → 𝐸 ∈ 𝐴)
62	60, 4, 61	elcgrabasi 29309	. 2 ⊢ (𝜑 → ∃𝑢 ∈ 𝑃 ∃𝑣 ∈ 𝑃 ∃𝑤 ∈ 𝑃 (𝐸 = ⟨“𝑢𝑣𝑤”⟩ ∧ (𝑢 ≠ 𝑣 ∧ 𝑣 ≠ 𝑤)))
63	59, 62	r19.29vva 3222	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 2955 ∃wrex 3086 {crab 3412 ∖ cdif 3895 ∩ cin 3897 ∅c0 4278 ifcif 4481 {csn 4583 class class class wbr 5102 ‘cfv 6527 ℩crio 7364 (class class class)co 7408 ∈ cmpo 7410 ↑_m cmap 8825 0cc0 11172 1c1 11173 2c2 12367 3c3 12368 ..^cfzo 13757 ⟨“cs3 14961 Basecbs 17349 distcds 17399 TarskiGcstrkg 28823 Itvcitv 28829 LineGclng 28830 hlGchlg 28997 cgrAccgra 29248

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 2213 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7734 ax-cnex 11228 ax-resscn 11229 ax-1cn 11230 ax-icn 11231 ax-addcl 11232 ax-addrcl 11233 ax-mulcl 11234 ax-mulrcl 11235 ax-mulcom 11236 ax-addass 11237 ax-mulass 11238 ax-distr 11239 ax-i2m1 11240 ax-1ne0 11241 ax-1rid 11242 ax-rnegex 11243 ax-rrecex 11244 ax-cnre 11245 ax-pre-lttri 11246 ax-pre-lttrn 11247 ax-pre-ltadd 11248 ax-pre-mulgt0 11249

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 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6293 df-ord 6354 df-on 6355 df-lim 6356 df-suc 6357 df-iota 6483 df-fun 6529 df-fn 6530 df-f 6531 df-f1 6532 df-fo 6533 df-f1o 6534 df-fv 6535 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7861 df-1st 7984 df-2nd 7985 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8454 df-oadd 8458 df-er 8695 df-map 8827 df-pm 8828 df-en 8952 df-dom 8953 df-sdom 8954 df-fin 8955 df-dju 9954 df-card 9992 df-pnf 11317 df-mnf 11318 df-xr 11319 df-ltxr 11320 df-le 11321 df-sub 11515 df-neg 11516 df-nn 12306 df-2 12375 df-3 12376 df-n0 12577 df-xnn0 12650 df-z 12664 df-uz 12936 df-fz 13610 df-fzo 13758 df-hash 14443 df-word 14627 df-concat 14684 df-s1 14711 df-s2 14967 df-s3 14968 df-trkgc 28844 df-trkgb 28845 df-trkgcb 28846 df-trkg 28849 df-cgrg 28908 df-leg 28980 df-hlg 28998 df-cgra 29249

This theorem is used by: angmgmlem 29329

W3C validator