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Theorem isleag 26755

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 14295	. . . 4 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑃)
5		s3len 14317	. . . 4 ⊢ (♯‘⟨“𝐴𝐵𝐶”⟩) = 3
6		isleag.p	. . . . . 6 ⊢ 𝑃 = (Base‘𝐺)
7	6	fvexi 6678	. . . . 5 ⊢ 𝑃 ∈ V
8		3nn0 11966	. . . . 5 ⊢ 3 ∈ ℕ₀
9		wrdmap 13959	. . . . 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 14295	. . . 4 ⊢ (𝜑 → ⟨“𝐷𝐸𝐹”⟩ ∈ Word 𝑃)
16		s3len 14317	. . . 4 ⊢ (♯‘⟨“𝐷𝐸𝐹”⟩) = 3
17		wrdmap 13959	. . . . 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 3429	. . . . 5 ⊢ (𝐺 ∈ TarskiG → 𝐺 ∈ V)
23		fveq2 6664	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
24	23, 6	eqtr4di 2812	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝑃)
25	24	oveq1d 7172	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → ((Base‘𝑔) ↑_m (0..^3)) = (𝑃 ↑_m (0..^3)))
26	25	eleq2d 2838	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑎 ∈ ((Base‘𝑔) ↑_m (0..^3)) ↔ 𝑎 ∈ (𝑃 ↑_m (0..^3))))
27	25	eleq2d 2838	. . . . . . . . 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 6664	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (inA‘𝑔) = (inA‘𝐺))
30	29	breqd 5048	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑥(inA‘𝑔)⟨“(𝑏‘0)(𝑏‘1)(𝑏‘2)”⟩ ↔ 𝑥(inA‘𝐺)⟨“(𝑏‘0)(𝑏‘1)(𝑏‘2)”⟩))
31		fveq2 6664	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (cgrA‘𝑔) = (cgrA‘𝐺))
32	31	breqd 5048	. . . . . . . . . 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 3321	. . . . . . . 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 5103	. . . . . 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 26754	. . . . . 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 7190	. . . . . . . 8 ⊢ (𝑃 ↑_m (0..^3)) ∈ V
39	38, 38	xpex 7481	. . . . . . 7 ⊢ ((𝑃 ↑_m (0..^3)) × (𝑃 ↑_m (0..^3))) ∈ V
40		opabssxp 5618	. . . . . . 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 5197	. . . . . 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 6765	. . . . 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 5048	. . 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 6666	. . . . . . . . 9 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → (𝑏‘0) = (⟨“𝐷𝐸𝐹”⟩‘0))
47	45	fveq1d 6666	. . . . . . . . 9 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → (𝑏‘1) = (⟨“𝐷𝐸𝐹”⟩‘1))
48	45	fveq1d 6666	. . . . . . . . 9 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → (𝑏‘2) = (⟨“𝐷𝐸𝐹”⟩‘2))
49	46, 47, 48	s3eqd 14287	. . . . . . . 8 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → ⟨“(𝑏‘0)(𝑏‘1)(𝑏‘2)”⟩ = ⟨“(⟨“𝐷𝐸𝐹”⟩‘0)(⟨“𝐷𝐸𝐹”⟩‘1)(⟨“𝐷𝐸𝐹”⟩‘2)”⟩)
50	49	breq2d 5049	. . . . . . 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 6666	. . . . . . . . 9 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → (𝑎‘0) = (⟨“𝐴𝐵𝐶”⟩‘0))
53	51	fveq1d 6666	. . . . . . . . 9 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → (𝑎‘1) = (⟨“𝐴𝐵𝐶”⟩‘1))
54	51	fveq1d 6666	. . . . . . . . 9 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → (𝑎‘2) = (⟨“𝐴𝐵𝐶”⟩‘2))
55	52, 53, 54	s3eqd 14287	. . . . . . . 8 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → ⟨“(𝑎‘0)(𝑎‘1)(𝑎‘2)”⟩ = ⟨“(⟨“𝐴𝐵𝐶”⟩‘0)(⟨“𝐴𝐵𝐶”⟩‘1)(⟨“𝐴𝐵𝐶”⟩‘2)”⟩)
56		eqidd 2760	. . . . . . . . 9 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → 𝑥 = 𝑥)
57	46, 47, 56	s3eqd 14287	. . . . . . . 8 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → ⟨“(𝑏‘0)(𝑏‘1)𝑥”⟩ = ⟨“(⟨“𝐷𝐸𝐹”⟩‘0)(⟨“𝐷𝐸𝐹”⟩‘1)𝑥”⟩)
58	55, 57	breq12d 5050	. . . . . . 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 3222	. . . . 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 2759	. . . . 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 5619	. . . 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 14314	. . . . . . . . 9 ⊢ (𝐷 ∈ 𝑃 → (⟨“𝐷𝐸𝐹”⟩‘0) = 𝐷)
65	12, 64	syl 17	. . . . . . . 8 ⊢ (𝜑 → (⟨“𝐷𝐸𝐹”⟩‘0) = 𝐷)
66		s3fv1 14315	. . . . . . . . 9 ⊢ (𝐸 ∈ 𝑃 → (⟨“𝐷𝐸𝐹”⟩‘1) = 𝐸)
67	13, 66	syl 17	. . . . . . . 8 ⊢ (𝜑 → (⟨“𝐷𝐸𝐹”⟩‘1) = 𝐸)
68		s3fv2 14316	. . . . . . . . 9 ⊢ (𝐹 ∈ 𝑃 → (⟨“𝐷𝐸𝐹”⟩‘2) = 𝐹)
69	14, 68	syl 17	. . . . . . . 8 ⊢ (𝜑 → (⟨“𝐷𝐸𝐹”⟩‘2) = 𝐹)
70	65, 67, 69	s3eqd 14287	. . . . . . 7 ⊢ (𝜑 → ⟨“(⟨“𝐷𝐸𝐹”⟩‘0)(⟨“𝐷𝐸𝐹”⟩‘1)(⟨“𝐷𝐸𝐹”⟩‘2)”⟩ = ⟨“𝐷𝐸𝐹”⟩)
71	70	breq2d 5049	. . . . . 6 ⊢ (𝜑 → (𝑥(inA‘𝐺)⟨“(⟨“𝐷𝐸𝐹”⟩‘0)(⟨“𝐷𝐸𝐹”⟩‘1)(⟨“𝐷𝐸𝐹”⟩‘2)”⟩ ↔ 𝑥(inA‘𝐺)⟨“𝐷𝐸𝐹”⟩))
72		s3fv0 14314	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑃 → (⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴)
73	1, 72	syl 17	. . . . . . . 8 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴)
74		s3fv1 14315	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝑃 → (⟨“𝐴𝐵𝐶”⟩‘1) = 𝐵)
75	2, 74	syl 17	. . . . . . . 8 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩‘1) = 𝐵)
76		s3fv2 14316	. . . . . . . . 9 ⊢ (𝐶 ∈ 𝑃 → (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)
77	3, 76	syl 17	. . . . . . . 8 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)
78	73, 75, 77	s3eqd 14287	. . . . . . 7 ⊢ (𝜑 → ⟨“(⟨“𝐴𝐵𝐶”⟩‘0)(⟨“𝐴𝐵𝐶”⟩‘1)(⟨“𝐴𝐵𝐶”⟩‘2)”⟩ = ⟨“𝐴𝐵𝐶”⟩)
79		eqidd 2760	. . . . . . . 8 ⊢ (𝜑 → 𝑥 = 𝑥)
80	65, 67, 79	s3eqd 14287	. . . . . . 7 ⊢ (𝜑 → ⟨“(⟨“𝐷𝐸𝐹”⟩‘0)(⟨“𝐷𝐸𝐹”⟩‘1)𝑥”⟩ = ⟨“𝐷𝐸𝑥”⟩)
81	78, 80	breq12d 5050	. . . . . 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 3222	. . . 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 1539 ∈ wcel 2112 ∃wrex 3072 Vcvv 3410 class class class wbr 5037 {copab 5099 × cxp 5527 ‘cfv 6341 (class class class)co 7157 ↑_m cmap 8423 0cc0 10589 1c1 10590 2c2 11743 3c3 11744 ℕ₀cn0 11948 ..^cfzo 13096 ♯chash 13754 Word cword 13927 ⟨“cs3 14265 Basecbs 16556 TarskiGcstrkg 26338 cgrAccgra 26715 inAcinag 26743 ≤_∠cleag 26744

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5161 ax-sep 5174 ax-nul 5181 ax-pow 5239 ax-pr 5303 ax-un 7466 ax-cnex 10645 ax-resscn 10646 ax-1cn 10647 ax-icn 10648 ax-addcl 10649 ax-addrcl 10650 ax-mulcl 10651 ax-mulrcl 10652 ax-mulcom 10653 ax-addass 10654 ax-mulass 10655 ax-distr 10656 ax-i2m1 10657 ax-1ne0 10658 ax-1rid 10659 ax-rnegex 10660 ax-rrecex 10661 ax-cnre 10662 ax-pre-lttri 10663 ax-pre-lttrn 10664 ax-pre-ltadd 10665 ax-pre-mulgt0 10666

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2902 df-ne 2953 df-nel 3057 df-ral 3076 df-rex 3077 df-reu 3078 df-rab 3080 df-v 3412 df-sbc 3700 df-csb 3809 df-dif 3864 df-un 3866 df-in 3868 df-ss 3878 df-pss 3880 df-nul 4229 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4803 df-int 4843 df-iun 4889 df-br 5038 df-opab 5100 df-mpt 5118 df-tr 5144 df-id 5435 df-eprel 5440 df-po 5448 df-so 5449 df-fr 5488 df-we 5490 df-xp 5535 df-rel 5536 df-cnv 5537 df-co 5538 df-dm 5539 df-rn 5540 df-res 5541 df-ima 5542 df-pred 6132 df-ord 6178 df-on 6179 df-lim 6180 df-suc 6181 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-riota 7115 df-ov 7160 df-oprab 7161 df-mpo 7162 df-om 7587 df-1st 7700 df-2nd 7701 df-wrecs 7964 df-recs 8025 df-rdg 8063 df-1o 8119 df-er 8306 df-map 8425 df-en 8542 df-dom 8543 df-sdom 8544 df-fin 8545 df-card 9415 df-pnf 10729 df-mnf 10730 df-xr 10731 df-ltxr 10732 df-le 10733 df-sub 10924 df-neg 10925 df-nn 11689 df-2 11751 df-3 11752 df-n0 11949 df-z 12035 df-uz 12297 df-fz 12954 df-fzo 13097 df-hash 13755 df-word 13928 df-concat 13984 df-s1 14011 df-s2 14271 df-s3 14272 df-leag 26754

This theorem is referenced by: isleagd 26756 leagne1 26757 leagne2 26758 leagne3 26759 leagne4 26760

W3C validator