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Theorem isleag 27208

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 14585	. . . 4 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑃)
5		s3len 14607	. . . 4 ⊢ (♯‘⟨“𝐴𝐵𝐶”⟩) = 3
6		isleag.p	. . . . . 6 ⊢ 𝑃 = (Base‘𝐺)
7	6	fvexi 6788	. . . . 5 ⊢ 𝑃 ∈ V
8		3nn0 12251	. . . . 5 ⊢ 3 ∈ ℕ₀
9		wrdmap 14249	. . . . 5 ⊢ ((𝑃 ∈ V ∧ 3 ∈ ℕ₀) → ((⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑃 ∧ (♯‘⟨“𝐴𝐵𝐶”⟩) = 3) ↔ ⟨“𝐴𝐵𝐶”⟩ ∈ (𝑃 ↑_m (0..^3))))
10	7, 8, 9	mp2an 689	. . . 4 ⊢ ((⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑃 ∧ (♯‘⟨“𝐴𝐵𝐶”⟩) = 3) ↔ ⟨“𝐴𝐵𝐶”⟩ ∈ (𝑃 ↑_m (0..^3)))
11	4, 5, 10	sylanblc 589	. . 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 14585	. . . 4 ⊢ (𝜑 → ⟨“𝐷𝐸𝐹”⟩ ∈ Word 𝑃)
16		s3len 14607	. . . 4 ⊢ (♯‘⟨“𝐷𝐸𝐹”⟩) = 3
17		wrdmap 14249	. . . . 5 ⊢ ((𝑃 ∈ V ∧ 3 ∈ ℕ₀) → ((⟨“𝐷𝐸𝐹”⟩ ∈ Word 𝑃 ∧ (♯‘⟨“𝐷𝐸𝐹”⟩) = 3) ↔ ⟨“𝐷𝐸𝐹”⟩ ∈ (𝑃 ↑_m (0..^3))))
18	7, 8, 17	mp2an 689	. . . 4 ⊢ ((⟨“𝐷𝐸𝐹”⟩ ∈ Word 𝑃 ∧ (♯‘⟨“𝐷𝐸𝐹”⟩) = 3) ↔ ⟨“𝐷𝐸𝐹”⟩ ∈ (𝑃 ↑_m (0..^3)))
19	15, 16, 18	sylanblc 589	. . 3 ⊢ (𝜑 → ⟨“𝐷𝐸𝐹”⟩ ∈ (𝑃 ↑_m (0..^3)))
20	11, 19	jca 512	. 2 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝑃 ↑_m (0..^3)) ∧ ⟨“𝐷𝐸𝐹”⟩ ∈ (𝑃 ↑_m (0..^3))))
21		isleag.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
22		elex 3450	. . . . 5 ⊢ (𝐺 ∈ TarskiG → 𝐺 ∈ V)
23		fveq2 6774	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
24	23, 6	eqtr4di 2796	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝑃)
25	24	oveq1d 7290	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → ((Base‘𝑔) ↑_m (0..^3)) = (𝑃 ↑_m (0..^3)))
26	25	eleq2d 2824	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑎 ∈ ((Base‘𝑔) ↑_m (0..^3)) ↔ 𝑎 ∈ (𝑃 ↑_m (0..^3))))
27	25	eleq2d 2824	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑏 ∈ ((Base‘𝑔) ↑_m (0..^3)) ↔ 𝑏 ∈ (𝑃 ↑_m (0..^3))))
28	26, 27	anbi12d 631	. . . . . . . 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 6774	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (inA‘𝑔) = (inA‘𝐺))
30	29	breqd 5085	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑥(inA‘𝑔)⟨“(𝑏‘0)(𝑏‘1)(𝑏‘2)”⟩ ↔ 𝑥(inA‘𝐺)⟨“(𝑏‘0)(𝑏‘1)(𝑏‘2)”⟩))
31		fveq2 6774	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (cgrA‘𝑔) = (cgrA‘𝐺))
32	31	breqd 5085	. . . . . . . . . 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 631	. . . . . . . . 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 3337	. . . . . . . 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 631	. . . . . . 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 5140	. . . . . 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 27207	. . . . . 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 7308	. . . . . . . 8 ⊢ (𝑃 ↑_m (0..^3)) ∈ V
39	38, 38	xpex 7603	. . . . . . 7 ⊢ ((𝑃 ↑_m (0..^3)) × (𝑃 ↑_m (0..^3))) ∈ V
40		opabssxp 5679	. . . . . . 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 5246	. . . . . 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 6875	. . . . 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 5085	. . 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 485	. . . . . . . . . 10 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → 𝑏 = ⟨“𝐷𝐸𝐹”⟩)
46	45	fveq1d 6776	. . . . . . . . 9 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → (𝑏‘0) = (⟨“𝐷𝐸𝐹”⟩‘0))
47	45	fveq1d 6776	. . . . . . . . 9 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → (𝑏‘1) = (⟨“𝐷𝐸𝐹”⟩‘1))
48	45	fveq1d 6776	. . . . . . . . 9 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → (𝑏‘2) = (⟨“𝐷𝐸𝐹”⟩‘2))
49	46, 47, 48	s3eqd 14577	. . . . . . . 8 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → ⟨“(𝑏‘0)(𝑏‘1)(𝑏‘2)”⟩ = ⟨“(⟨“𝐷𝐸𝐹”⟩‘0)(⟨“𝐷𝐸𝐹”⟩‘1)(⟨“𝐷𝐸𝐹”⟩‘2)”⟩)
50	49	breq2d 5086	. . . . . . 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 483	. . . . . . . . . 10 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → 𝑎 = ⟨“𝐴𝐵𝐶”⟩)
52	51	fveq1d 6776	. . . . . . . . 9 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → (𝑎‘0) = (⟨“𝐴𝐵𝐶”⟩‘0))
53	51	fveq1d 6776	. . . . . . . . 9 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → (𝑎‘1) = (⟨“𝐴𝐵𝐶”⟩‘1))
54	51	fveq1d 6776	. . . . . . . . 9 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → (𝑎‘2) = (⟨“𝐴𝐵𝐶”⟩‘2))
55	52, 53, 54	s3eqd 14577	. . . . . . . 8 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → ⟨“(𝑎‘0)(𝑎‘1)(𝑎‘2)”⟩ = ⟨“(⟨“𝐴𝐵𝐶”⟩‘0)(⟨“𝐴𝐵𝐶”⟩‘1)(⟨“𝐴𝐵𝐶”⟩‘2)”⟩)
56		eqidd 2739	. . . . . . . . 9 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → 𝑥 = 𝑥)
57	46, 47, 56	s3eqd 14577	. . . . . . . 8 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → ⟨“(𝑏‘0)(𝑏‘1)𝑥”⟩ = ⟨“(⟨“𝐷𝐸𝐹”⟩‘0)(⟨“𝐷𝐸𝐹”⟩‘1)𝑥”⟩)
58	55, 57	breq12d 5087	. . . . . . 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 631	. . . . . 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 3226	. . . . 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 2738	. . . . 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 5680	. . . 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 14604	. . . . . . . . 9 ⊢ (𝐷 ∈ 𝑃 → (⟨“𝐷𝐸𝐹”⟩‘0) = 𝐷)
65	12, 64	syl 17	. . . . . . . 8 ⊢ (𝜑 → (⟨“𝐷𝐸𝐹”⟩‘0) = 𝐷)
66		s3fv1 14605	. . . . . . . . 9 ⊢ (𝐸 ∈ 𝑃 → (⟨“𝐷𝐸𝐹”⟩‘1) = 𝐸)
67	13, 66	syl 17	. . . . . . . 8 ⊢ (𝜑 → (⟨“𝐷𝐸𝐹”⟩‘1) = 𝐸)
68		s3fv2 14606	. . . . . . . . 9 ⊢ (𝐹 ∈ 𝑃 → (⟨“𝐷𝐸𝐹”⟩‘2) = 𝐹)
69	14, 68	syl 17	. . . . . . . 8 ⊢ (𝜑 → (⟨“𝐷𝐸𝐹”⟩‘2) = 𝐹)
70	65, 67, 69	s3eqd 14577	. . . . . . 7 ⊢ (𝜑 → ⟨“(⟨“𝐷𝐸𝐹”⟩‘0)(⟨“𝐷𝐸𝐹”⟩‘1)(⟨“𝐷𝐸𝐹”⟩‘2)”⟩ = ⟨“𝐷𝐸𝐹”⟩)
71	70	breq2d 5086	. . . . . 6 ⊢ (𝜑 → (𝑥(inA‘𝐺)⟨“(⟨“𝐷𝐸𝐹”⟩‘0)(⟨“𝐷𝐸𝐹”⟩‘1)(⟨“𝐷𝐸𝐹”⟩‘2)”⟩ ↔ 𝑥(inA‘𝐺)⟨“𝐷𝐸𝐹”⟩))
72		s3fv0 14604	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑃 → (⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴)
73	1, 72	syl 17	. . . . . . . 8 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴)
74		s3fv1 14605	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝑃 → (⟨“𝐴𝐵𝐶”⟩‘1) = 𝐵)
75	2, 74	syl 17	. . . . . . . 8 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩‘1) = 𝐵)
76		s3fv2 14606	. . . . . . . . 9 ⊢ (𝐶 ∈ 𝑃 → (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)
77	3, 76	syl 17	. . . . . . . 8 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)
78	73, 75, 77	s3eqd 14577	. . . . . . 7 ⊢ (𝜑 → ⟨“(⟨“𝐴𝐵𝐶”⟩‘0)(⟨“𝐴𝐵𝐶”⟩‘1)(⟨“𝐴𝐵𝐶”⟩‘2)”⟩ = ⟨“𝐴𝐵𝐶”⟩)
79		eqidd 2739	. . . . . . . 8 ⊢ (𝜑 → 𝑥 = 𝑥)
80	65, 67, 79	s3eqd 14577	. . . . . . 7 ⊢ (𝜑 → ⟨“(⟨“𝐷𝐸𝐹”⟩‘0)(⟨“𝐷𝐸𝐹”⟩‘1)𝑥”⟩ = ⟨“𝐷𝐸𝑥”⟩)
81	78, 80	breq12d 5087	. . . . . 6 ⊢ (𝜑 → (⟨“(⟨“𝐴𝐵𝐶”⟩‘0)(⟨“𝐴𝐵𝐶”⟩‘1)(⟨“𝐴𝐵𝐶”⟩‘2)”⟩(cgrA‘𝐺)⟨“(⟨“𝐷𝐸𝐹”⟩‘0)(⟨“𝐷𝐸𝐹”⟩‘1)𝑥”⟩ ↔ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑥”⟩))
82	71, 81	anbi12d 631	. . . . 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 3226	. . . 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 629	. . 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 305	. 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 704	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 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∃wrex 3065 Vcvv 3432 class class class wbr 5074 {copab 5136 × cxp 5587 ‘cfv 6433 (class class class)co 7275 ↑_m cmap 8615 0cc0 10871 1c1 10872 2c2 12028 3c3 12029 ℕ₀cn0 12233 ..^cfzo 13382 ♯chash 14044 Word cword 14217 ⟨“cs3 14555 Basecbs 16912 TarskiGcstrkg 26788 cgrAccgra 27168 inAcinag 27196 ≤_∠cleag 27197

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-fzo 13383 df-hash 14045 df-word 14218 df-concat 14274 df-s1 14301 df-s2 14561 df-s3 14562 df-leag 27207

This theorem is referenced by: isleagd 27209 leagne1 27210 leagne2 27211 leagne3 27212 leagne4 27213

W3C validator