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Theorem isleag 27789

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 14761	. . . 4 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑃)
5		s3len 14783	. . . 4 ⊢ (♯‘⟨“𝐴𝐵𝐶”⟩) = 3
6		isleag.p	. . . . . 6 ⊢ 𝑃 = (Base‘𝐺)
7	6	fvexi 6856	. . . . 5 ⊢ 𝑃 ∈ V
8		3nn0 12431	. . . . 5 ⊢ 3 ∈ ℕ₀
9		wrdmap 14434	. . . . 5 ⊢ ((𝑃 ∈ V ∧ 3 ∈ ℕ₀) → ((⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑃 ∧ (♯‘⟨“𝐴𝐵𝐶”⟩) = 3) ↔ ⟨“𝐴𝐵𝐶”⟩ ∈ (𝑃 ↑_m (0..^3))))
10	7, 8, 9	mp2an 690	. . . 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 14761	. . . 4 ⊢ (𝜑 → ⟨“𝐷𝐸𝐹”⟩ ∈ Word 𝑃)
16		s3len 14783	. . . 4 ⊢ (♯‘⟨“𝐷𝐸𝐹”⟩) = 3
17		wrdmap 14434	. . . . 5 ⊢ ((𝑃 ∈ V ∧ 3 ∈ ℕ₀) → ((⟨“𝐷𝐸𝐹”⟩ ∈ Word 𝑃 ∧ (♯‘⟨“𝐷𝐸𝐹”⟩) = 3) ↔ ⟨“𝐷𝐸𝐹”⟩ ∈ (𝑃 ↑_m (0..^3))))
18	7, 8, 17	mp2an 690	. . . 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 3463	. . . . 5 ⊢ (𝐺 ∈ TarskiG → 𝐺 ∈ V)
23		fveq2 6842	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
24	23, 6	eqtr4di 2794	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝑃)
25	24	oveq1d 7372	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → ((Base‘𝑔) ↑_m (0..^3)) = (𝑃 ↑_m (0..^3)))
26	25	eleq2d 2823	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑎 ∈ ((Base‘𝑔) ↑_m (0..^3)) ↔ 𝑎 ∈ (𝑃 ↑_m (0..^3))))
27	25	eleq2d 2823	. . . . . . . . 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 6842	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (inA‘𝑔) = (inA‘𝐺))
30	29	breqd 5116	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑥(inA‘𝑔)⟨“(𝑏‘0)(𝑏‘1)(𝑏‘2)”⟩ ↔ 𝑥(inA‘𝐺)⟨“(𝑏‘0)(𝑏‘1)(𝑏‘2)”⟩))
31		fveq2 6842	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (cgrA‘𝑔) = (cgrA‘𝐺))
32	31	breqd 5116	. . . . . . . . . 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 3320	. . . . . . . 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 5171	. . . . . 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 27788	. . . . . 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 7390	. . . . . . . 8 ⊢ (𝑃 ↑_m (0..^3)) ∈ V
39	38, 38	xpex 7687	. . . . . . 7 ⊢ ((𝑃 ↑_m (0..^3)) × (𝑃 ↑_m (0..^3))) ∈ V
40		opabssxp 5724	. . . . . . 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 5279	. . . . . 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 6948	. . . . 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 5116	. . 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 6844	. . . . . . . . 9 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → (𝑏‘0) = (⟨“𝐷𝐸𝐹”⟩‘0))
47	45	fveq1d 6844	. . . . . . . . 9 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → (𝑏‘1) = (⟨“𝐷𝐸𝐹”⟩‘1))
48	45	fveq1d 6844	. . . . . . . . 9 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → (𝑏‘2) = (⟨“𝐷𝐸𝐹”⟩‘2))
49	46, 47, 48	s3eqd 14753	. . . . . . . 8 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → ⟨“(𝑏‘0)(𝑏‘1)(𝑏‘2)”⟩ = ⟨“(⟨“𝐷𝐸𝐹”⟩‘0)(⟨“𝐷𝐸𝐹”⟩‘1)(⟨“𝐷𝐸𝐹”⟩‘2)”⟩)
50	49	breq2d 5117	. . . . . . 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 6844	. . . . . . . . 9 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → (𝑎‘0) = (⟨“𝐴𝐵𝐶”⟩‘0))
53	51	fveq1d 6844	. . . . . . . . 9 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → (𝑎‘1) = (⟨“𝐴𝐵𝐶”⟩‘1))
54	51	fveq1d 6844	. . . . . . . . 9 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → (𝑎‘2) = (⟨“𝐴𝐵𝐶”⟩‘2))
55	52, 53, 54	s3eqd 14753	. . . . . . . 8 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → ⟨“(𝑎‘0)(𝑎‘1)(𝑎‘2)”⟩ = ⟨“(⟨“𝐴𝐵𝐶”⟩‘0)(⟨“𝐴𝐵𝐶”⟩‘1)(⟨“𝐴𝐵𝐶”⟩‘2)”⟩)
56		eqidd 2737	. . . . . . . . 9 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → 𝑥 = 𝑥)
57	46, 47, 56	s3eqd 14753	. . . . . . . 8 ⊢ ((𝑎 = ⟨“𝐴𝐵𝐶”⟩ ∧ 𝑏 = ⟨“𝐷𝐸𝐹”⟩) → ⟨“(𝑏‘0)(𝑏‘1)𝑥”⟩ = ⟨“(⟨“𝐷𝐸𝐹”⟩‘0)(⟨“𝐷𝐸𝐹”⟩‘1)𝑥”⟩)
58	55, 57	breq12d 5118	. . . . . . 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 3175	. . . . 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 2736	. . . . 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 5725	. . . 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 14780	. . . . . . . . 9 ⊢ (𝐷 ∈ 𝑃 → (⟨“𝐷𝐸𝐹”⟩‘0) = 𝐷)
65	12, 64	syl 17	. . . . . . . 8 ⊢ (𝜑 → (⟨“𝐷𝐸𝐹”⟩‘0) = 𝐷)
66		s3fv1 14781	. . . . . . . . 9 ⊢ (𝐸 ∈ 𝑃 → (⟨“𝐷𝐸𝐹”⟩‘1) = 𝐸)
67	13, 66	syl 17	. . . . . . . 8 ⊢ (𝜑 → (⟨“𝐷𝐸𝐹”⟩‘1) = 𝐸)
68		s3fv2 14782	. . . . . . . . 9 ⊢ (𝐹 ∈ 𝑃 → (⟨“𝐷𝐸𝐹”⟩‘2) = 𝐹)
69	14, 68	syl 17	. . . . . . . 8 ⊢ (𝜑 → (⟨“𝐷𝐸𝐹”⟩‘2) = 𝐹)
70	65, 67, 69	s3eqd 14753	. . . . . . 7 ⊢ (𝜑 → ⟨“(⟨“𝐷𝐸𝐹”⟩‘0)(⟨“𝐷𝐸𝐹”⟩‘1)(⟨“𝐷𝐸𝐹”⟩‘2)”⟩ = ⟨“𝐷𝐸𝐹”⟩)
71	70	breq2d 5117	. . . . . 6 ⊢ (𝜑 → (𝑥(inA‘𝐺)⟨“(⟨“𝐷𝐸𝐹”⟩‘0)(⟨“𝐷𝐸𝐹”⟩‘1)(⟨“𝐷𝐸𝐹”⟩‘2)”⟩ ↔ 𝑥(inA‘𝐺)⟨“𝐷𝐸𝐹”⟩))
72		s3fv0 14780	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑃 → (⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴)
73	1, 72	syl 17	. . . . . . . 8 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴)
74		s3fv1 14781	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝑃 → (⟨“𝐴𝐵𝐶”⟩‘1) = 𝐵)
75	2, 74	syl 17	. . . . . . . 8 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩‘1) = 𝐵)
76		s3fv2 14782	. . . . . . . . 9 ⊢ (𝐶 ∈ 𝑃 → (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)
77	3, 76	syl 17	. . . . . . . 8 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)
78	73, 75, 77	s3eqd 14753	. . . . . . 7 ⊢ (𝜑 → ⟨“(⟨“𝐴𝐵𝐶”⟩‘0)(⟨“𝐴𝐵𝐶”⟩‘1)(⟨“𝐴𝐵𝐶”⟩‘2)”⟩ = ⟨“𝐴𝐵𝐶”⟩)
79		eqidd 2737	. . . . . . . 8 ⊢ (𝜑 → 𝑥 = 𝑥)
80	65, 67, 79	s3eqd 14753	. . . . . . 7 ⊢ (𝜑 → ⟨“(⟨“𝐷𝐸𝐹”⟩‘0)(⟨“𝐷𝐸𝐹”⟩‘1)𝑥”⟩ = ⟨“𝐷𝐸𝑥”⟩)
81	78, 80	breq12d 5118	. . . . . 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 3175	. . . 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 304	. 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 705	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 1541 ∈ wcel 2106 ∃wrex 3073 Vcvv 3445 class class class wbr 5105 {copab 5167 × cxp 5631 ‘cfv 6496 (class class class)co 7357 ↑_m cmap 8765 0cc0 11051 1c1 11052 2c2 12208 3c3 12209 ℕ₀cn0 12413 ..^cfzo 13567 ♯chash 14230 Word cword 14402 ⟨“cs3 14731 Basecbs 17083 TarskiGcstrkg 27369 cgrAccgra 27749 inAcinag 27777 ≤_∠cleag 27778

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-n0 12414 df-z 12500 df-uz 12764 df-fz 13425 df-fzo 13568 df-hash 14231 df-word 14403 df-concat 14459 df-s1 14484 df-s2 14737 df-s3 14738 df-leag 27788

This theorem is referenced by: isleagd 27790 leagne1 27791 leagne2 27792 leagne3 27793 leagne4 27794

W3C validator