isleag - Metamath Proof Explorer

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

Theorem isleag 26641

Description: Geometrical "less than" property for angles. Definition 11.27 of [Schwabhauser] p. 102. (Contributed by Thierry Arnoux, 7-Oct-2020.)

**Hypotheses**
Ref	Expression
isleag.p	⊢ 𝑃 = (Base‘𝐺)
isleag.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
isleag.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
isleag.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
isleag.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
isleag.d	⊢ (𝜑 → 𝐷 ∈ 𝑃)
isleag.e	⊢ (𝜑 → 𝐸 ∈ 𝑃)
isleag.f	⊢ (𝜑 → 𝐹 ∈ 𝑃)

**Assertion**
Ref	Expression
isleag	⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩(≤_∠‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ ∃𝑥 ∈ 𝑃 (𝑥(inA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑥”⟩)))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 𝑥,𝐶 𝑥,𝐷 𝑥,𝐸 𝑥,𝐹 𝑥,𝐺 𝑥,𝑃 𝜑,𝑥

Proof of Theorem isleag Dummy variables 𝑎 𝑏 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isleag.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
2		isleag.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
3		isleag.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
4	1, 2, 3	s3cld 14225	. . . 4 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑃)
5		s3len 14247	. . . 4 ⊢ (♯‘⟨“𝐴𝐵𝐶”⟩) = 3
6		isleag.p	. . . . . 6 ⊢ 𝑃 = (Base‘𝐺)
7	6	fvexi 6659	. . . . 5 ⊢ 𝑃 ∈ V
8		3nn0 11903	. . . . 5 ⊢ 3 ∈ ℕ₀
9		wrdmap 13889	. . . . 5 ⊢ ((𝑃 ∈ V ∧ 3 ∈ ℕ₀) → ((⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑃 ∧ (♯‘⟨“𝐴𝐵𝐶”⟩) = 3) ↔ ⟨“𝐴𝐵𝐶”⟩ ∈ (𝑃 ↑_m (0..^3))))
10	7, 8, 9	mp2an 691	. . . 4 ⊢ ((⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑃 ∧ (♯‘⟨“𝐴𝐵𝐶”⟩) = 3) ↔ ⟨“𝐴𝐵𝐶”⟩ ∈ (𝑃 ↑_m (0..^3)))
11	4, 5, 10	sylanblc 592	. . 3 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (𝑃 ↑_m (0..^3)))
12		isleag.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
13		isleag.e	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
14		isleag.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝑃)
15	12, 13, 14	s3cld 14225	. . . 4 ⊢ (𝜑 → ⟨“𝐷𝐸𝐹”⟩ ∈ Word 𝑃)
16		s3len 14247	. . . 4 ⊢ (♯‘⟨“𝐷𝐸𝐹”⟩) = 3
17		wrdmap 13889	. . . . 5 ⊢ ((𝑃 ∈ V ∧ 3 ∈ ℕ₀) → ((⟨“𝐷𝐸𝐹”⟩ ∈ Word 𝑃 ∧ (♯‘⟨“𝐷𝐸𝐹”⟩) = 3) ↔ ⟨“𝐷𝐸𝐹”⟩ ∈ (𝑃 ↑_m (0..^3))))
18	7, 8, 17	mp2an 691	. . . 4 ⊢ ((⟨“𝐷𝐸𝐹”⟩ ∈ Word 𝑃 ∧ (♯‘⟨“𝐷𝐸𝐹”⟩) = 3) ↔ ⟨“𝐷𝐸𝐹”⟩ ∈ (𝑃 ↑_m (0..^3)))
19	15, 16, 18	sylanblc 592	. . 3 ⊢ (𝜑 → ⟨“𝐷𝐸𝐹”⟩ ∈ (𝑃 ↑_m (0..^3)))
20	11, 19	jca 515	. 2 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝑃 ↑_m (0..^3)) ∧ ⟨“𝐷𝐸𝐹”⟩ ∈ (𝑃 ↑_m (0..^3))))
21		isleag.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
22		elex 3459	. . . . 5 ⊢ (𝐺 ∈ TarskiG → 𝐺 ∈ V)
23		fveq2 6645	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
24	23, 6	eqtr4di 2851	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝑃)
25	24	oveq1d 7150	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → ((Base‘𝑔) ↑_m (0..^3)) = (𝑃 ↑_m (0..^3)))
26	25	eleq2d 2875	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑎 ∈ ((Base‘𝑔) ↑_m (0..^3)) ↔ 𝑎 ∈ (𝑃 ↑_m (0..^3))))
27	25	eleq2d 2875	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑏 ∈ ((Base‘𝑔) ↑_m (0..^3)) ↔ 𝑏 ∈ (𝑃 ↑_m (0..^3))))
28	26, 27	anbi12d 633	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ((𝑎 ∈ ((Base‘𝑔) ↑_m (0..^3)) ∧ 𝑏 ∈ ((Base‘𝑔) ↑_m (0..^3))) ↔ (𝑎 ∈ (𝑃 ↑_m (0..^3)) ∧ 𝑏 ∈ (𝑃 ↑_m (0..^3)))))
29		fveq2 6645	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (inA‘𝑔) = (inA‘𝐺))
30	29	breqd 5041	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑥(inA‘𝑔)⟨“(𝑏‘0)(𝑏‘1)(𝑏‘2)”⟩ ↔ 𝑥(inA‘𝐺)⟨“(𝑏‘0)(𝑏‘1)(𝑏‘2)”⟩))
31		fveq2 6645	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (cgrA‘𝑔) = (cgrA‘𝐺))
32	31	breqd 5041	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (⟨“(𝑎‘0)(𝑎‘1)(𝑎‘2)”⟩(cgrA‘𝑔)⟨“(𝑏‘0)(𝑏‘1)𝑥”⟩ ↔ ⟨“(𝑎‘0)(𝑎‘1)(𝑎‘2)”⟩(cgrA‘𝐺)⟨“(𝑏‘0)(𝑏‘1)𝑥”⟩))
33	30, 32	anbi12d 633	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ((𝑥(inA‘𝑔)⟨“(𝑏‘0)(𝑏‘1)(𝑏‘2)”⟩ ∧ ⟨“(𝑎‘0)(𝑎‘1)(𝑎‘2)”⟩(cgrA‘𝑔)⟨“(𝑏‘0)(𝑏‘1)𝑥”⟩) ↔ (𝑥(inA‘𝐺)⟨“(𝑏‘0)(𝑏‘1)(𝑏‘2)”⟩ ∧ ⟨“(𝑎‘0)(𝑎‘1)(𝑎‘2)”⟩(cgrA‘𝐺)⟨“(𝑏‘0)(𝑏‘1)𝑥”⟩)))
34	24, 33	rexeqbidv 3355	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (∃𝑥 ∈ (Base‘𝑔)(𝑥(inA‘𝑔)⟨“(𝑏‘0)(𝑏‘1)(𝑏‘2)”⟩ ∧ ⟨“(𝑎‘0)(𝑎‘1)(𝑎‘2)”⟩(cgrA‘𝑔)⟨“(𝑏‘0)(𝑏‘1)𝑥”⟩) ↔ ∃𝑥 ∈ 𝑃 (𝑥(inA‘𝐺)⟨“(𝑏‘0)(𝑏‘1)(𝑏‘2)”⟩ ∧ ⟨“(𝑎‘0)(𝑎‘1)(𝑎‘2)”⟩(cgrA‘𝐺)⟨“(𝑏‘0)(𝑏‘1)𝑥”⟩)))
35	28, 34	anbi12d 633	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (((𝑎 ∈ ((Base‘𝑔) ↑_m (0..^3)) ∧ 𝑏 ∈ ((Base‘𝑔) ↑_m (0..^3))) ∧ ∃𝑥 ∈ (Base‘𝑔)(𝑥(inA‘𝑔)⟨“(𝑏‘0)(𝑏‘1)(𝑏‘2)”⟩ ∧ ⟨“(𝑎‘0)(𝑎‘1)(𝑎‘2)”⟩(cgrA‘𝑔)⟨“(𝑏‘0)(𝑏‘1)𝑥”⟩)) ↔ ((𝑎 ∈ (𝑃 ↑_m (0..^3)) ∧ 𝑏 ∈ (𝑃 ↑_m (0..^3))) ∧ ∃𝑥 ∈ 𝑃 (𝑥(inA‘𝐺)⟨“(𝑏‘0)(𝑏‘1)(𝑏‘2)”⟩ ∧ ⟨“(𝑎‘0)(𝑎‘1)(𝑎‘2)”⟩(cgrA‘𝐺)⟨“(𝑏‘0)(𝑏‘1)𝑥”⟩))))
36	35	opabbidv 5096	. . . . . 6 ⊢ (𝑔 = 𝐺 → {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ((Base‘𝑔) ↑_m (0..^3)) ∧ 𝑏 ∈ ((Base‘𝑔) ↑_m (0..^3))) ∧ ∃𝑥 ∈ (Base‘𝑔)(𝑥(inA‘𝑔)⟨“(𝑏‘0)(𝑏‘1)(𝑏‘2)”⟩ ∧ ⟨“(𝑎‘0)(𝑎‘1)(𝑎‘2)”⟩(cgrA‘𝑔)⟨“(𝑏‘0)(𝑏‘1)𝑥”⟩))} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ↑_m (0..^3)) ∧ 𝑏 ∈ (𝑃 ↑_m (0..^3))) ∧ ∃𝑥 ∈ 𝑃 (𝑥(inA‘𝐺)⟨“(𝑏‘0)(𝑏‘1)(𝑏‘2)”⟩ ∧ ⟨“(𝑎‘0)(𝑎‘1)(𝑎‘2)”⟩(cgrA‘𝐺)⟨“(𝑏‘0)(𝑏‘1)𝑥”⟩))})
37		df-leag 26640	. . . . . 6 ⊢ ≤_∠ = (𝑔 ∈ V ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ((Base‘𝑔) ↑_m (0..^3)) ∧ 𝑏 ∈ ((Base‘𝑔) ↑_m (0..^3))) ∧ ∃𝑥 ∈ (Base‘𝑔)(𝑥(inA‘𝑔)⟨“(𝑏‘0)(𝑏‘1)(𝑏‘2)”⟩ ∧ ⟨“(𝑎‘0)(𝑎‘1)(𝑎‘2)”⟩(cgrA‘𝑔)⟨“(𝑏‘0)(𝑏‘1)𝑥”⟩))})
38		ovex 7168	. . . . . . . 8 ⊢ (𝑃 ↑_m (0..^3)) ∈ V
39	38, 38	xpex 7456	. . . . . . 7 ⊢ ((𝑃 ↑_m (0..^3)) × (𝑃 ↑_m (0..^3))) ∈ V
40		opabssxp 5607	. . . . . . 7 ⊢ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ↑_m (0..^3)) ∧ 𝑏 ∈ (𝑃 ↑_m (0..^3))) ∧ ∃𝑥 ∈ 𝑃 (𝑥(inA‘𝐺)⟨“(𝑏‘0)(𝑏‘1)(𝑏‘2)”⟩ ∧ ⟨“(𝑎‘0)(𝑎‘1)(𝑎‘2)”⟩(cgrA‘𝐺)⟨“(𝑏‘0)(𝑏‘1)𝑥”⟩))} ⊆ ((𝑃 ↑_m (0..^3)) × (𝑃 ↑_m (0..^3)))
41	39, 40	ssexi 5190	. . . . . 6 ⊢ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ↑_m (0..^3)) ∧ 𝑏 ∈ (𝑃 ↑_m (0..^3))) ∧ ∃𝑥 ∈ 𝑃 (𝑥(inA‘𝐺)⟨“(𝑏‘0)(𝑏‘1)(𝑏‘2)”⟩ ∧ ⟨“(𝑎‘0)(𝑎‘1)(𝑎‘2)”⟩(cgrA‘𝐺)⟨“(𝑏‘0)(𝑏‘1)𝑥”⟩))} ∈ V
42	36, 37, 41	fvmpt 6745	. . . . 5 ⊢ (𝐺 ∈ V → (≤_∠‘𝐺) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ↑_m (0..^3)) ∧ 𝑏 ∈ (𝑃 ↑_m (0..^3))) ∧ ∃𝑥 ∈ 𝑃 (𝑥(inA‘𝐺)⟨“(𝑏‘0)(𝑏‘1)(𝑏‘2)”⟩ ∧ ⟨“(𝑎‘0)(𝑎‘1)(𝑎‘2)”⟩(cgrA‘𝐺)⟨“(𝑏‘0)(𝑏‘1)𝑥”⟩))})
43	21, 22, 42	3syl 18	. . . 4 ⊢ (𝜑 → (≤_∠‘𝐺) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ↑_m (0..^3)) ∧ 𝑏 ∈ (𝑃 ↑_m (0..^3))) ∧ ∃𝑥 ∈ 𝑃 (𝑥(inA‘𝐺)⟨“(𝑏‘0)(𝑏‘1)(𝑏‘2)”⟩ ∧ ⟨“(𝑎‘0)(𝑎‘1)(𝑎‘2)”⟩(cgrA‘𝐺)⟨“(𝑏‘0)(𝑏‘1)𝑥”⟩))})
44	43	breqd 5041	. . 3 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩(≤_∠‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ ⟨“𝐴𝐵𝐶”⟩{⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ↑_m (0..^3)) ∧ 𝑏 ∈ (𝑃 ↑_m (0..^3))) ∧ ∃𝑥 ∈ 𝑃 (𝑥(inA‘𝐺)⟨“(𝑏‘0)(𝑏‘1)(𝑏‘2)”⟩ ∧ ⟨“(𝑎‘0)(𝑎‘1)(𝑎‘2)”⟩(cgrA‘𝐺)⟨“(𝑏‘0)(𝑏‘1)𝑥”⟩))}⟨“𝐷𝐸𝐹”⟩))
45		simpr 488	. . . . . . . . . 10 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → 𝑏 = ⟨“𝐷𝐸𝐹”⟩)
46	45	fveq1d 6647	. . . . . . . . 9 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → (𝑏‘0) = (⟨“𝐷𝐸𝐹”⟩‘0))
47	45	fveq1d 6647	. . . . . . . . 9 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → (𝑏‘1) = (⟨“𝐷𝐸𝐹”⟩‘1))
48	45	fveq1d 6647	. . . . . . . . 9 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → (𝑏‘2) = (⟨“𝐷𝐸𝐹”⟩‘2))
49	46, 47, 48	s3eqd 14217	. . . . . . . 8 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → ⟨“(𝑏‘0)(𝑏‘1)(𝑏‘2)”⟩ = ⟨“(⟨“𝐷𝐸𝐹”⟩‘0)(⟨“𝐷𝐸𝐹”⟩‘1)(⟨“𝐷𝐸𝐹”⟩‘2)”⟩)
50	49	breq2d 5042	. . . . . . 7 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → (𝑥(inA‘𝐺)⟨“(𝑏‘0)(𝑏‘1)(𝑏‘2)”⟩ ↔ 𝑥(inA‘𝐺)⟨“(⟨“𝐷𝐸𝐹”⟩‘0)(⟨“𝐷𝐸𝐹”⟩‘1)(⟨“𝐷𝐸𝐹”⟩‘2)”⟩))
51		simpl 486	. . . . . . . . . 10 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → 𝑎 = ⟨“𝐴𝐵𝐶”⟩)
52	51	fveq1d 6647	. . . . . . . . 9 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → (𝑎‘0) = (⟨“𝐴𝐵𝐶”⟩‘0))
53	51	fveq1d 6647	. . . . . . . . 9 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → (𝑎‘1) = (⟨“𝐴𝐵𝐶”⟩‘1))
54	51	fveq1d 6647	. . . . . . . . 9 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → (𝑎‘2) = (⟨“𝐴𝐵𝐶”⟩‘2))
55	52, 53, 54	s3eqd 14217	. . . . . . . 8 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → ⟨“(𝑎‘0)(𝑎‘1)(𝑎‘2)”⟩ = ⟨“(⟨“𝐴𝐵𝐶”⟩‘0)(⟨“𝐴𝐵𝐶”⟩‘1)(⟨“𝐴𝐵𝐶”⟩‘2)”⟩)
56		eqidd 2799	. . . . . . . . 9 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → 𝑥 = 𝑥)
57	46, 47, 56	s3eqd 14217	. . . . . . . 8 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → ⟨“(𝑏‘0)(𝑏‘1)𝑥”⟩ = ⟨“(⟨“𝐷𝐸𝐹”⟩‘0)(⟨“𝐷𝐸𝐹”⟩‘1)𝑥”⟩)
58	55, 57	breq12d 5043	. . . . . . 7 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → (⟨“(𝑎‘0)(𝑎‘1)(𝑎‘2)”⟩(cgrA‘𝐺)⟨“(𝑏‘0)(𝑏‘1)𝑥”⟩ ↔ ⟨“(⟨“𝐴𝐵𝐶”⟩‘0)(⟨“𝐴𝐵𝐶”⟩‘1)(⟨“𝐴𝐵𝐶”⟩‘2)”⟩(cgrA‘𝐺)⟨“(⟨“𝐷𝐸𝐹”⟩‘0)(⟨“𝐷𝐸𝐹”⟩‘1)𝑥”⟩))
59	50, 58	anbi12d 633	. . . . . 6 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → ((𝑥(inA‘𝐺)⟨“(𝑏‘0)(𝑏‘1)(𝑏‘2)”⟩ ∧ ⟨“(𝑎‘0)(𝑎‘1)(𝑎‘2)”⟩(cgrA‘𝐺)⟨“(𝑏‘0)(𝑏‘1)𝑥”⟩) ↔ (𝑥(inA‘𝐺)⟨“(⟨“𝐷𝐸𝐹”⟩‘0)(⟨“𝐷𝐸𝐹”⟩‘1)(⟨“𝐷𝐸𝐹”⟩‘2)”⟩ ∧ ⟨“(⟨“𝐴𝐵𝐶”⟩‘0)(⟨“𝐴𝐵𝐶”⟩‘1)(⟨“𝐴𝐵𝐶”⟩‘2)”⟩(cgrA‘𝐺)⟨“(⟨“𝐷𝐸𝐹”⟩‘0)(⟨“𝐷𝐸𝐹”⟩‘1)𝑥”⟩)))
60	59	rexbidv 3256	. . . . 5 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → (∃𝑥 ∈ 𝑃 (𝑥(inA‘𝐺)⟨“(𝑏‘0)(𝑏‘1)(𝑏‘2)”⟩ ∧ ⟨“(𝑎‘0)(𝑎‘1)(𝑎‘2)”⟩(cgrA‘𝐺)⟨“(𝑏‘0)(𝑏‘1)𝑥”⟩) ↔ ∃𝑥 ∈ 𝑃 (𝑥(inA‘𝐺)⟨“(⟨“𝐷𝐸𝐹”⟩‘0)(⟨“𝐷𝐸𝐹”⟩‘1)(⟨“𝐷𝐸𝐹”⟩‘2)”⟩ ∧ ⟨“(⟨“𝐴𝐵𝐶”⟩‘0)(⟨“𝐴𝐵𝐶”⟩‘1)(⟨“𝐴𝐵𝐶”⟩‘2)”⟩(cgrA‘𝐺)⟨“(⟨“𝐷𝐸𝐹”⟩‘0)(⟨“𝐷𝐸𝐹”⟩‘1)𝑥”⟩)))
61		eqid 2798	. . . . 5 ⊢ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ↑_m (0..^3)) ∧ 𝑏 ∈ (𝑃 ↑_m (0..^3))) ∧ ∃𝑥 ∈ 𝑃 (𝑥(inA‘𝐺)⟨“(𝑏‘0)(𝑏‘1)(𝑏‘2)”⟩ ∧ ⟨“(𝑎‘0)(𝑎‘1)(𝑎‘2)”⟩(cgrA‘𝐺)⟨“(𝑏‘0)(𝑏‘1)𝑥”⟩))} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ↑_m (0..^3)) ∧ 𝑏 ∈ (𝑃 ↑_m (0..^3))) ∧ ∃𝑥 ∈ 𝑃 (𝑥(inA‘𝐺)⟨“(𝑏‘0)(𝑏‘1)(𝑏‘2)”⟩ ∧ ⟨“(𝑎‘0)(𝑎‘1)(𝑎‘2)”⟩(cgrA‘𝐺)⟨“(𝑏‘0)(𝑏‘1)𝑥”⟩))}
62	60, 61	brab2a 5608	. . . 4 ⊢ (⟨“𝐴𝐵𝐶”⟩{⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ↑_m (0..^3)) ∧ 𝑏 ∈ (𝑃 ↑_m (0..^3))) ∧ ∃𝑥 ∈ 𝑃 (𝑥(inA‘𝐺)⟨“(𝑏‘0)(𝑏‘1)(𝑏‘2)”⟩ ∧ ⟨“(𝑎‘0)(𝑎‘1)(𝑎‘2)”⟩(cgrA‘𝐺)⟨“(𝑏‘0)(𝑏‘1)𝑥”⟩))}⟨“𝐷𝐸𝐹”⟩ ↔ ((⟨“𝐴𝐵𝐶”⟩ ∈ (𝑃 ↑_m (0..^3)) ∧ ⟨“𝐷𝐸𝐹”⟩ ∈ (𝑃 ↑_m (0..^3))) ∧ ∃𝑥 ∈ 𝑃 (𝑥(inA‘𝐺)⟨“(⟨“𝐷𝐸𝐹”⟩‘0)(⟨“𝐷𝐸𝐹”⟩‘1)(⟨“𝐷𝐸𝐹”⟩‘2)”⟩ ∧ ⟨“(⟨“𝐴𝐵𝐶”⟩‘0)(⟨“𝐴𝐵𝐶”⟩‘1)(⟨“𝐴𝐵𝐶”⟩‘2)”⟩(cgrA‘𝐺)⟨“(⟨“𝐷𝐸𝐹”⟩‘0)(⟨“𝐷𝐸𝐹”⟩‘1)𝑥”⟩)))
63	62	a1i 11	. . 3 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩{⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ↑_m (0..^3)) ∧ 𝑏 ∈ (𝑃 ↑_m (0..^3))) ∧ ∃𝑥 ∈ 𝑃 (𝑥(inA‘𝐺)⟨“(𝑏‘0)(𝑏‘1)(𝑏‘2)”⟩ ∧ ⟨“(𝑎‘0)(𝑎‘1)(𝑎‘2)”⟩(cgrA‘𝐺)⟨“(𝑏‘0)(𝑏‘1)𝑥”⟩))}⟨“𝐷𝐸𝐹”⟩ ↔ ((⟨“𝐴𝐵𝐶”⟩ ∈ (𝑃 ↑_m (0..^3)) ∧ ⟨“𝐷𝐸𝐹”⟩ ∈ (𝑃 ↑_m (0..^3))) ∧ ∃𝑥 ∈ 𝑃 (𝑥(inA‘𝐺)⟨“(⟨“𝐷𝐸𝐹”⟩‘0)(⟨“𝐷𝐸𝐹”⟩‘1)(⟨“𝐷𝐸𝐹”⟩‘2)”⟩ ∧ ⟨“(⟨“𝐴𝐵𝐶”⟩‘0)(⟨“𝐴𝐵𝐶”⟩‘1)(⟨“𝐴𝐵𝐶”⟩‘2)”⟩(cgrA‘𝐺)⟨“(⟨“𝐷𝐸𝐹”⟩‘0)(⟨“𝐷𝐸𝐹”⟩‘1)𝑥”⟩))))
64		s3fv0 14244	. . . . . . . . 9 ⊢ (𝐷 ∈ 𝑃 → (⟨“𝐷𝐸𝐹”⟩‘0) = 𝐷)
65	12, 64	syl 17	. . . . . . . 8 ⊢ (𝜑 → (⟨“𝐷𝐸𝐹”⟩‘0) = 𝐷)
66		s3fv1 14245	. . . . . . . . 9 ⊢ (𝐸 ∈ 𝑃 → (⟨“𝐷𝐸𝐹”⟩‘1) = 𝐸)
67	13, 66	syl 17	. . . . . . . 8 ⊢ (𝜑 → (⟨“𝐷𝐸𝐹”⟩‘1) = 𝐸)
68		s3fv2 14246	. . . . . . . . 9 ⊢ (𝐹 ∈ 𝑃 → (⟨“𝐷𝐸𝐹”⟩‘2) = 𝐹)
69	14, 68	syl 17	. . . . . . . 8 ⊢ (𝜑 → (⟨“𝐷𝐸𝐹”⟩‘2) = 𝐹)
70	65, 67, 69	s3eqd 14217	. . . . . . 7 ⊢ (𝜑 → ⟨“(⟨“𝐷𝐸𝐹”⟩‘0)(⟨“𝐷𝐸𝐹”⟩‘1)(⟨“𝐷𝐸𝐹”⟩‘2)”⟩ = ⟨“𝐷𝐸𝐹”⟩)
71	70	breq2d 5042	. . . . . 6 ⊢ (𝜑 → (𝑥(inA‘𝐺)⟨“(⟨“𝐷𝐸𝐹”⟩‘0)(⟨“𝐷𝐸𝐹”⟩‘1)(⟨“𝐷𝐸𝐹”⟩‘2)”⟩ ↔ 𝑥(inA‘𝐺)⟨“𝐷𝐸𝐹”⟩))
72		s3fv0 14244	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑃 → (⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴)
73	1, 72	syl 17	. . . . . . . 8 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴)
74		s3fv1 14245	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝑃 → (⟨“𝐴𝐵𝐶”⟩‘1) = 𝐵)
75	2, 74	syl 17	. . . . . . . 8 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩‘1) = 𝐵)
76		s3fv2 14246	. . . . . . . . 9 ⊢ (𝐶 ∈ 𝑃 → (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)
77	3, 76	syl 17	. . . . . . . 8 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)
78	73, 75, 77	s3eqd 14217	. . . . . . 7 ⊢ (𝜑 → ⟨“(⟨“𝐴𝐵𝐶”⟩‘0)(⟨“𝐴𝐵𝐶”⟩‘1)(⟨“𝐴𝐵𝐶”⟩‘2)”⟩ = ⟨“𝐴𝐵𝐶”⟩)
79		eqidd 2799	. . . . . . . 8 ⊢ (𝜑 → 𝑥 = 𝑥)
80	65, 67, 79	s3eqd 14217	. . . . . . 7 ⊢ (𝜑 → ⟨“(⟨“𝐷𝐸𝐹”⟩‘0)(⟨“𝐷𝐸𝐹”⟩‘1)𝑥”⟩ = ⟨“𝐷𝐸𝑥”⟩)
81	78, 80	breq12d 5043	. . . . . 6 ⊢ (𝜑 → (⟨“(⟨“𝐴𝐵𝐶”⟩‘0)(⟨“𝐴𝐵𝐶”⟩‘1)(⟨“𝐴𝐵𝐶”⟩‘2)”⟩(cgrA‘𝐺)⟨“(⟨“𝐷𝐸𝐹”⟩‘0)(⟨“𝐷𝐸𝐹”⟩‘1)𝑥”⟩ ↔ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑥”⟩))
82	71, 81	anbi12d 633	. . . . 5 ⊢ (𝜑 → ((𝑥(inA‘𝐺)⟨“(⟨“𝐷𝐸𝐹”⟩‘0)(⟨“𝐷𝐸𝐹”⟩‘1)(⟨“𝐷𝐸𝐹”⟩‘2)”⟩ ∧ ⟨“(⟨“𝐴𝐵𝐶”⟩‘0)(⟨“𝐴𝐵𝐶”⟩‘1)(⟨“𝐴𝐵𝐶”⟩‘2)”⟩(cgrA‘𝐺)⟨“(⟨“𝐷𝐸𝐹”⟩‘0)(⟨“𝐷𝐸𝐹”⟩‘1)𝑥”⟩) ↔ (𝑥(inA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑥”⟩)))
83	82	rexbidv 3256	. . . 4 ⊢ (𝜑 → (∃𝑥 ∈ 𝑃 (𝑥(inA‘𝐺)⟨“(⟨“𝐷𝐸𝐹”⟩‘0)(⟨“𝐷𝐸𝐹”⟩‘1)(⟨“𝐷𝐸𝐹”⟩‘2)”⟩ ∧ ⟨“(⟨“𝐴𝐵𝐶”⟩‘0)(⟨“𝐴𝐵𝐶”⟩‘1)(⟨“𝐴𝐵𝐶”⟩‘2)”⟩(cgrA‘𝐺)⟨“(⟨“𝐷𝐸𝐹”⟩‘0)(⟨“𝐷𝐸𝐹”⟩‘1)𝑥”⟩) ↔ ∃𝑥 ∈ 𝑃 (𝑥(inA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑥”⟩)))
84	83	anbi2d 631	. . 3 ⊢ (𝜑 → (((⟨“𝐴𝐵𝐶”⟩ ∈ (𝑃 ↑_m (0..^3)) ∧ ⟨“𝐷𝐸𝐹”⟩ ∈ (𝑃 ↑_m (0..^3))) ∧ ∃𝑥 ∈ 𝑃 (𝑥(inA‘𝐺)⟨“(⟨“𝐷𝐸𝐹”⟩‘0)(⟨“𝐷𝐸𝐹”⟩‘1)(⟨“𝐷𝐸𝐹”⟩‘2)”⟩ ∧ ⟨“(⟨“𝐴𝐵𝐶”⟩‘0)(⟨“𝐴𝐵𝐶”⟩‘1)(⟨“𝐴𝐵𝐶”⟩‘2)”⟩(cgrA‘𝐺)⟨“(⟨“𝐷𝐸𝐹”⟩‘0)(⟨“𝐷𝐸𝐹”⟩‘1)𝑥”⟩)) ↔ ((⟨“𝐴𝐵𝐶”⟩ ∈ (𝑃 ↑_m (0..^3)) ∧ ⟨“𝐷𝐸𝐹”⟩ ∈ (𝑃 ↑_m (0..^3))) ∧ ∃𝑥 ∈ 𝑃 (𝑥(inA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑥”⟩))))
85	44, 63, 84	3bitrd 308	. 2 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩(≤_∠‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ ((⟨“𝐴𝐵𝐶”⟩ ∈ (𝑃 ↑_m (0..^3)) ∧ ⟨“𝐷𝐸𝐹”⟩ ∈ (𝑃 ↑_m (0..^3))) ∧ ∃𝑥 ∈ 𝑃 (𝑥(inA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑥”⟩))))
86	20, 85	mpbirand 706	1 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩(≤_∠‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ ∃𝑥 ∈ 𝑃 (𝑥(inA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑥”⟩)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 Vcvv 3441 class class class wbr 5030 {copab 5092 × cxp 5517 ‘cfv 6324 (class class class)co 7135 ↑_m cmap 8389 0cc0 10526 1c1 10527 2c2 11680 3c3 11681 ℕ₀cn0 11885 ..^cfzo 13028 ♯chash 13686 Word cword 13857 ⟨“cs3 14195 Basecbs 16475 TarskiGcstrkg 26224 cgrAccgra 26601 inAcinag 26629 ≤_∠cleag 26630

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-hash 13687 df-word 13858 df-concat 13914 df-s1 13941 df-s2 14201 df-s3 14202 df-leag 26640

This theorem is referenced by: isleagd 26642 leagne1 26643 leagne2 26644 leagne3 26645 leagne4 26646

W3C validator