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Theorem isinag 28069

Description: Property for point 𝑋 to lie in the angle ⟨“𝐴𝐵𝐶”⟩. Definition 11.23 of [Schwabhauser] p. 101. (Contributed by Thierry Arnoux, 15-Aug-2020.)

**Hypotheses**
Ref	Expression
isinag.p	⊢ 𝑃 = (Base‘𝐺)
isinag.i	⊢ 𝐼 = (Itv‘𝐺)
isinag.k	⊢ 𝐾 = (hlG‘𝐺)
isinag.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
isinag.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
isinag.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
isinag.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
isinag.g	⊢ (𝜑 → 𝐺 ∈ 𝑉)

**Assertion**
Ref	Expression
isinag	⊢ (𝜑 → (𝑋(inA‘𝐺)⟨“𝐴𝐵𝐶”⟩ ↔ ((𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐵 ∧ 𝑋 ≠ 𝐵) ∧ ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐶) ∧ (𝑥 = 𝐵 ∨ 𝑥(𝐾‘𝐵)𝑋)))))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 𝑥,𝐶 𝑥,𝐺 𝑥,𝑃 𝑥,𝑋 𝜑,𝑥 Allowed substitution hints: 𝐼(𝑥) 𝐾(𝑥) 𝑉(𝑥)

Proof of Theorem isinag Dummy variables 𝑝 𝑡 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 486	. . . . . . . 8 ⊢ ((𝑝 = 𝑋 ∧ 𝑡 = ⟨“𝐴𝐵𝐶”⟩) → 𝑡 = ⟨“𝐴𝐵𝐶”⟩)
2	1	fveq1d 6890	. . . . . . 7 ⊢ ((𝑝 = 𝑋 ∧ 𝑡 = ⟨“𝐴𝐵𝐶”⟩) → (𝑡‘0) = (⟨“𝐴𝐵𝐶”⟩‘0))
3	1	fveq1d 6890	. . . . . . 7 ⊢ ((𝑝 = 𝑋 ∧ 𝑡 = ⟨“𝐴𝐵𝐶”⟩) → (𝑡‘1) = (⟨“𝐴𝐵𝐶”⟩‘1))
4	2, 3	neeq12d 3003	. . . . . 6 ⊢ ((𝑝 = 𝑋 ∧ 𝑡 = ⟨“𝐴𝐵𝐶”⟩) → ((𝑡‘0) ≠ (𝑡‘1) ↔ (⟨“𝐴𝐵𝐶”⟩‘0) ≠ (⟨“𝐴𝐵𝐶”⟩‘1)))
5	1	fveq1d 6890	. . . . . . 7 ⊢ ((𝑝 = 𝑋 ∧ 𝑡 = ⟨“𝐴𝐵𝐶”⟩) → (𝑡‘2) = (⟨“𝐴𝐵𝐶”⟩‘2))
6	5, 3	neeq12d 3003	. . . . . 6 ⊢ ((𝑝 = 𝑋 ∧ 𝑡 = ⟨“𝐴𝐵𝐶”⟩) → ((𝑡‘2) ≠ (𝑡‘1) ↔ (⟨“𝐴𝐵𝐶”⟩‘2) ≠ (⟨“𝐴𝐵𝐶”⟩‘1)))
7		simpl 484	. . . . . . 7 ⊢ ((𝑝 = 𝑋 ∧ 𝑡 = ⟨“𝐴𝐵𝐶”⟩) → 𝑝 = 𝑋)
8	7, 3	neeq12d 3003	. . . . . 6 ⊢ ((𝑝 = 𝑋 ∧ 𝑡 = ⟨“𝐴𝐵𝐶”⟩) → (𝑝 ≠ (𝑡‘1) ↔ 𝑋 ≠ (⟨“𝐴𝐵𝐶”⟩‘1)))
9	4, 6, 8	3anbi123d 1437	. . . . 5 ⊢ ((𝑝 = 𝑋 ∧ 𝑡 = ⟨“𝐴𝐵𝐶”⟩) → (((𝑡‘0) ≠ (𝑡‘1) ∧ (𝑡‘2) ≠ (𝑡‘1) ∧ 𝑝 ≠ (𝑡‘1)) ↔ ((⟨“𝐴𝐵𝐶”⟩‘0) ≠ (⟨“𝐴𝐵𝐶”⟩‘1) ∧ (⟨“𝐴𝐵𝐶”⟩‘2) ≠ (⟨“𝐴𝐵𝐶”⟩‘1) ∧ 𝑋 ≠ (⟨“𝐴𝐵𝐶”⟩‘1))))
10	2, 5	oveq12d 7422	. . . . . . . 8 ⊢ ((𝑝 = 𝑋 ∧ 𝑡 = ⟨“𝐴𝐵𝐶”⟩) → ((𝑡‘0)𝐼(𝑡‘2)) = ((⟨“𝐴𝐵𝐶”⟩‘0)𝐼(⟨“𝐴𝐵𝐶”⟩‘2)))
11	10	eleq2d 2820	. . . . . . 7 ⊢ ((𝑝 = 𝑋 ∧ 𝑡 = ⟨“𝐴𝐵𝐶”⟩) → (𝑥 ∈ ((𝑡‘0)𝐼(𝑡‘2)) ↔ 𝑥 ∈ ((⟨“𝐴𝐵𝐶”⟩‘0)𝐼(⟨“𝐴𝐵𝐶”⟩‘2))))
12	3	eqeq2d 2744	. . . . . . . 8 ⊢ ((𝑝 = 𝑋 ∧ 𝑡 = ⟨“𝐴𝐵𝐶”⟩) → (𝑥 = (𝑡‘1) ↔ 𝑥 = (⟨“𝐴𝐵𝐶”⟩‘1)))
13		eqidd 2734	. . . . . . . . 9 ⊢ ((𝑝 = 𝑋 ∧ 𝑡 = ⟨“𝐴𝐵𝐶”⟩) → 𝑥 = 𝑥)
14	3	fveq2d 6892	. . . . . . . . 9 ⊢ ((𝑝 = 𝑋 ∧ 𝑡 = ⟨“𝐴𝐵𝐶”⟩) → (𝐾‘(𝑡‘1)) = (𝐾‘(⟨“𝐴𝐵𝐶”⟩‘1)))
15	13, 14, 7	breq123d 5161	. . . . . . . 8 ⊢ ((𝑝 = 𝑋 ∧ 𝑡 = ⟨“𝐴𝐵𝐶”⟩) → (𝑥(𝐾‘(𝑡‘1))𝑝 ↔ 𝑥(𝐾‘(⟨“𝐴𝐵𝐶”⟩‘1))𝑋))
16	12, 15	orbi12d 918	. . . . . . 7 ⊢ ((𝑝 = 𝑋 ∧ 𝑡 = ⟨“𝐴𝐵𝐶”⟩) → ((𝑥 = (𝑡‘1) ∨ 𝑥(𝐾‘(𝑡‘1))𝑝) ↔ (𝑥 = (⟨“𝐴𝐵𝐶”⟩‘1) ∨ 𝑥(𝐾‘(⟨“𝐴𝐵𝐶”⟩‘1))𝑋)))
17	11, 16	anbi12d 632	. . . . . 6 ⊢ ((𝑝 = 𝑋 ∧ 𝑡 = ⟨“𝐴𝐵𝐶”⟩) → ((𝑥 ∈ ((𝑡‘0)𝐼(𝑡‘2)) ∧ (𝑥 = (𝑡‘1) ∨ 𝑥(𝐾‘(𝑡‘1))𝑝)) ↔ (𝑥 ∈ ((⟨“𝐴𝐵𝐶”⟩‘0)𝐼(⟨“𝐴𝐵𝐶”⟩‘2)) ∧ (𝑥 = (⟨“𝐴𝐵𝐶”⟩‘1) ∨ 𝑥(𝐾‘(⟨“𝐴𝐵𝐶”⟩‘1))𝑋))))
18	17	rexbidv 3179	. . . . 5 ⊢ ((𝑝 = 𝑋 ∧ 𝑡 = ⟨“𝐴𝐵𝐶”⟩) → (∃𝑥 ∈ 𝑃 (𝑥 ∈ ((𝑡‘0)𝐼(𝑡‘2)) ∧ (𝑥 = (𝑡‘1) ∨ 𝑥(𝐾‘(𝑡‘1))𝑝)) ↔ ∃𝑥 ∈ 𝑃 (𝑥 ∈ ((⟨“𝐴𝐵𝐶”⟩‘0)𝐼(⟨“𝐴𝐵𝐶”⟩‘2)) ∧ (𝑥 = (⟨“𝐴𝐵𝐶”⟩‘1) ∨ 𝑥(𝐾‘(⟨“𝐴𝐵𝐶”⟩‘1))𝑋))))
19	9, 18	anbi12d 632	. . . 4 ⊢ ((𝑝 = 𝑋 ∧ 𝑡 = ⟨“𝐴𝐵𝐶”⟩) → ((((𝑡‘0) ≠ (𝑡‘1) ∧ (𝑡‘2) ≠ (𝑡‘1) ∧ 𝑝 ≠ (𝑡‘1)) ∧ ∃𝑥 ∈ 𝑃 (𝑥 ∈ ((𝑡‘0)𝐼(𝑡‘2)) ∧ (𝑥 = (𝑡‘1) ∨ 𝑥(𝐾‘(𝑡‘1))𝑝))) ↔ (((⟨“𝐴𝐵𝐶”⟩‘0) ≠ (⟨“𝐴𝐵𝐶”⟩‘1) ∧ (⟨“𝐴𝐵𝐶”⟩‘2) ≠ (⟨“𝐴𝐵𝐶”⟩‘1) ∧ 𝑋 ≠ (⟨“𝐴𝐵𝐶”⟩‘1)) ∧ ∃𝑥 ∈ 𝑃 (𝑥 ∈ ((⟨“𝐴𝐵𝐶”⟩‘0)𝐼(⟨“𝐴𝐵𝐶”⟩‘2)) ∧ (𝑥 = (⟨“𝐴𝐵𝐶”⟩‘1) ∨ 𝑥(𝐾‘(⟨“𝐴𝐵𝐶”⟩‘1))𝑋)))))
20		eqid 2733	. . . 4 ⊢ {⟨𝑝, 𝑡⟩ ∣ ((𝑝 ∈ 𝑃 ∧ 𝑡 ∈ (𝑃 ↑_m (0..^3))) ∧ (((𝑡‘0) ≠ (𝑡‘1) ∧ (𝑡‘2) ≠ (𝑡‘1) ∧ 𝑝 ≠ (𝑡‘1)) ∧ ∃𝑥 ∈ 𝑃 (𝑥 ∈ ((𝑡‘0)𝐼(𝑡‘2)) ∧ (𝑥 = (𝑡‘1) ∨ 𝑥(𝐾‘(𝑡‘1))𝑝))))} = {⟨𝑝, 𝑡⟩ ∣ ((𝑝 ∈ 𝑃 ∧ 𝑡 ∈ (𝑃 ↑_m (0..^3))) ∧ (((𝑡‘0) ≠ (𝑡‘1) ∧ (𝑡‘2) ≠ (𝑡‘1) ∧ 𝑝 ≠ (𝑡‘1)) ∧ ∃𝑥 ∈ 𝑃 (𝑥 ∈ ((𝑡‘0)𝐼(𝑡‘2)) ∧ (𝑥 = (𝑡‘1) ∨ 𝑥(𝐾‘(𝑡‘1))𝑝))))}
21	19, 20	brab2a 5767	. . 3 ⊢ (𝑋{⟨𝑝, 𝑡⟩ ∣ ((𝑝 ∈ 𝑃 ∧ 𝑡 ∈ (𝑃 ↑_m (0..^3))) ∧ (((𝑡‘0) ≠ (𝑡‘1) ∧ (𝑡‘2) ≠ (𝑡‘1) ∧ 𝑝 ≠ (𝑡‘1)) ∧ ∃𝑥 ∈ 𝑃 (𝑥 ∈ ((𝑡‘0)𝐼(𝑡‘2)) ∧ (𝑥 = (𝑡‘1) ∨ 𝑥(𝐾‘(𝑡‘1))𝑝))))}⟨“𝐴𝐵𝐶”⟩ ↔ ((𝑋 ∈ 𝑃 ∧ ⟨“𝐴𝐵𝐶”⟩ ∈ (𝑃 ↑_m (0..^3))) ∧ (((⟨“𝐴𝐵𝐶”⟩‘0) ≠ (⟨“𝐴𝐵𝐶”⟩‘1) ∧ (⟨“𝐴𝐵𝐶”⟩‘2) ≠ (⟨“𝐴𝐵𝐶”⟩‘1) ∧ 𝑋 ≠ (⟨“𝐴𝐵𝐶”⟩‘1)) ∧ ∃𝑥 ∈ 𝑃 (𝑥 ∈ ((⟨“𝐴𝐵𝐶”⟩‘0)𝐼(⟨“𝐴𝐵𝐶”⟩‘2)) ∧ (𝑥 = (⟨“𝐴𝐵𝐶”⟩‘1) ∨ 𝑥(𝐾‘(⟨“𝐴𝐵𝐶”⟩‘1))𝑋)))))
22		isinag.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
23		s3fv0 14838	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑃 → (⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴)
24	22, 23	syl 17	. . . . . . 7 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴)
25		isinag.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
26		s3fv1 14839	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑃 → (⟨“𝐴𝐵𝐶”⟩‘1) = 𝐵)
27	25, 26	syl 17	. . . . . . 7 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩‘1) = 𝐵)
28	24, 27	neeq12d 3003	. . . . . 6 ⊢ (𝜑 → ((⟨“𝐴𝐵𝐶”⟩‘0) ≠ (⟨“𝐴𝐵𝐶”⟩‘1) ↔ 𝐴 ≠ 𝐵))
29		isinag.c	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
30		s3fv2 14840	. . . . . . . 8 ⊢ (𝐶 ∈ 𝑃 → (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)
31	29, 30	syl 17	. . . . . . 7 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)
32	31, 27	neeq12d 3003	. . . . . 6 ⊢ (𝜑 → ((⟨“𝐴𝐵𝐶”⟩‘2) ≠ (⟨“𝐴𝐵𝐶”⟩‘1) ↔ 𝐶 ≠ 𝐵))
33	27	neeq2d 3002	. . . . . 6 ⊢ (𝜑 → (𝑋 ≠ (⟨“𝐴𝐵𝐶”⟩‘1) ↔ 𝑋 ≠ 𝐵))
34	28, 32, 33	3anbi123d 1437	. . . . 5 ⊢ (𝜑 → (((⟨“𝐴𝐵𝐶”⟩‘0) ≠ (⟨“𝐴𝐵𝐶”⟩‘1) ∧ (⟨“𝐴𝐵𝐶”⟩‘2) ≠ (⟨“𝐴𝐵𝐶”⟩‘1) ∧ 𝑋 ≠ (⟨“𝐴𝐵𝐶”⟩‘1)) ↔ (𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐵 ∧ 𝑋 ≠ 𝐵)))
35	24, 31	oveq12d 7422	. . . . . . . 8 ⊢ (𝜑 → ((⟨“𝐴𝐵𝐶”⟩‘0)𝐼(⟨“𝐴𝐵𝐶”⟩‘2)) = (𝐴𝐼𝐶))
36	35	eleq2d 2820	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ((⟨“𝐴𝐵𝐶”⟩‘0)𝐼(⟨“𝐴𝐵𝐶”⟩‘2)) ↔ 𝑥 ∈ (𝐴𝐼𝐶)))
37	27	eqeq2d 2744	. . . . . . . 8 ⊢ (𝜑 → (𝑥 = (⟨“𝐴𝐵𝐶”⟩‘1) ↔ 𝑥 = 𝐵))
38	27	fveq2d 6892	. . . . . . . . 9 ⊢ (𝜑 → (𝐾‘(⟨“𝐴𝐵𝐶”⟩‘1)) = (𝐾‘𝐵))
39	38	breqd 5158	. . . . . . . 8 ⊢ (𝜑 → (𝑥(𝐾‘(⟨“𝐴𝐵𝐶”⟩‘1))𝑋 ↔ 𝑥(𝐾‘𝐵)𝑋))
40	37, 39	orbi12d 918	. . . . . . 7 ⊢ (𝜑 → ((𝑥 = (⟨“𝐴𝐵𝐶”⟩‘1) ∨ 𝑥(𝐾‘(⟨“𝐴𝐵𝐶”⟩‘1))𝑋) ↔ (𝑥 = 𝐵 ∨ 𝑥(𝐾‘𝐵)𝑋)))
41	36, 40	anbi12d 632	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ ((⟨“𝐴𝐵𝐶”⟩‘0)𝐼(⟨“𝐴𝐵𝐶”⟩‘2)) ∧ (𝑥 = (⟨“𝐴𝐵𝐶”⟩‘1) ∨ 𝑥(𝐾‘(⟨“𝐴𝐵𝐶”⟩‘1))𝑋)) ↔ (𝑥 ∈ (𝐴𝐼𝐶) ∧ (𝑥 = 𝐵 ∨ 𝑥(𝐾‘𝐵)𝑋))))
42	41	rexbidv 3179	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝑃 (𝑥 ∈ ((⟨“𝐴𝐵𝐶”⟩‘0)𝐼(⟨“𝐴𝐵𝐶”⟩‘2)) ∧ (𝑥 = (⟨“𝐴𝐵𝐶”⟩‘1) ∨ 𝑥(𝐾‘(⟨“𝐴𝐵𝐶”⟩‘1))𝑋)) ↔ ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐶) ∧ (𝑥 = 𝐵 ∨ 𝑥(𝐾‘𝐵)𝑋))))
43	34, 42	anbi12d 632	. . . 4 ⊢ (𝜑 → ((((⟨“𝐴𝐵𝐶”⟩‘0) ≠ (⟨“𝐴𝐵𝐶”⟩‘1) ∧ (⟨“𝐴𝐵𝐶”⟩‘2) ≠ (⟨“𝐴𝐵𝐶”⟩‘1) ∧ 𝑋 ≠ (⟨“𝐴𝐵𝐶”⟩‘1)) ∧ ∃𝑥 ∈ 𝑃 (𝑥 ∈ ((⟨“𝐴𝐵𝐶”⟩‘0)𝐼(⟨“𝐴𝐵𝐶”⟩‘2)) ∧ (𝑥 = (⟨“𝐴𝐵𝐶”⟩‘1) ∨ 𝑥(𝐾‘(⟨“𝐴𝐵𝐶”⟩‘1))𝑋))) ↔ ((𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐵 ∧ 𝑋 ≠ 𝐵) ∧ ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐶) ∧ (𝑥 = 𝐵 ∨ 𝑥(𝐾‘𝐵)𝑋)))))
44	43	anbi2d 630	. . 3 ⊢ (𝜑 → (((𝑋 ∈ 𝑃 ∧ ⟨“𝐴𝐵𝐶”⟩ ∈ (𝑃 ↑_m (0..^3))) ∧ (((⟨“𝐴𝐵𝐶”⟩‘0) ≠ (⟨“𝐴𝐵𝐶”⟩‘1) ∧ (⟨“𝐴𝐵𝐶”⟩‘2) ≠ (⟨“𝐴𝐵𝐶”⟩‘1) ∧ 𝑋 ≠ (⟨“𝐴𝐵𝐶”⟩‘1)) ∧ ∃𝑥 ∈ 𝑃 (𝑥 ∈ ((⟨“𝐴𝐵𝐶”⟩‘0)𝐼(⟨“𝐴𝐵𝐶”⟩‘2)) ∧ (𝑥 = (⟨“𝐴𝐵𝐶”⟩‘1) ∨ 𝑥(𝐾‘(⟨“𝐴𝐵𝐶”⟩‘1))𝑋)))) ↔ ((𝑋 ∈ 𝑃 ∧ ⟨“𝐴𝐵𝐶”⟩ ∈ (𝑃 ↑_m (0..^3))) ∧ ((𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐵 ∧ 𝑋 ≠ 𝐵) ∧ ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐶) ∧ (𝑥 = 𝐵 ∨ 𝑥(𝐾‘𝐵)𝑋))))))
45	21, 44	bitrid 283	. 2 ⊢ (𝜑 → (𝑋{⟨𝑝, 𝑡⟩ ∣ ((𝑝 ∈ 𝑃 ∧ 𝑡 ∈ (𝑃 ↑_m (0..^3))) ∧ (((𝑡‘0) ≠ (𝑡‘1) ∧ (𝑡‘2) ≠ (𝑡‘1) ∧ 𝑝 ≠ (𝑡‘1)) ∧ ∃𝑥 ∈ 𝑃 (𝑥 ∈ ((𝑡‘0)𝐼(𝑡‘2)) ∧ (𝑥 = (𝑡‘1) ∨ 𝑥(𝐾‘(𝑡‘1))𝑝))))}⟨“𝐴𝐵𝐶”⟩ ↔ ((𝑋 ∈ 𝑃 ∧ ⟨“𝐴𝐵𝐶”⟩ ∈ (𝑃 ↑_m (0..^3))) ∧ ((𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐵 ∧ 𝑋 ≠ 𝐵) ∧ ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐶) ∧ (𝑥 = 𝐵 ∨ 𝑥(𝐾‘𝐵)𝑋))))))
46		isinag.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
47		elex 3493	. . . 4 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
48		fveq2 6888	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
49		isinag.p	. . . . . . . . . 10 ⊢ 𝑃 = (Base‘𝐺)
50	48, 49	eqtr4di 2791	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝑃)
51	50	eleq2d 2820	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑝 ∈ (Base‘𝑔) ↔ 𝑝 ∈ 𝑃))
52	50	oveq1d 7419	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ((Base‘𝑔) ↑_m (0..^3)) = (𝑃 ↑_m (0..^3)))
53	52	eleq2d 2820	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑡 ∈ ((Base‘𝑔) ↑_m (0..^3)) ↔ 𝑡 ∈ (𝑃 ↑_m (0..^3))))
54	51, 53	anbi12d 632	. . . . . . 7 ⊢ (𝑔 = 𝐺 → ((𝑝 ∈ (Base‘𝑔) ∧ 𝑡 ∈ ((Base‘𝑔) ↑_m (0..^3))) ↔ (𝑝 ∈ 𝑃 ∧ 𝑡 ∈ (𝑃 ↑_m (0..^3)))))
55		fveq2 6888	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝐺 → (Itv‘𝑔) = (Itv‘𝐺))
56		isinag.i	. . . . . . . . . . . . 13 ⊢ 𝐼 = (Itv‘𝐺)
57	55, 56	eqtr4di 2791	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝐺 → (Itv‘𝑔) = 𝐼)
58	57	oveqd 7421	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → ((𝑡‘0)(Itv‘𝑔)(𝑡‘2)) = ((𝑡‘0)𝐼(𝑡‘2)))
59	58	eleq2d 2820	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ ((𝑡‘0)(Itv‘𝑔)(𝑡‘2)) ↔ 𝑥 ∈ ((𝑡‘0)𝐼(𝑡‘2))))
60		fveq2 6888	. . . . . . . . . . . . . 14 ⊢ (𝑔 = 𝐺 → (hlG‘𝑔) = (hlG‘𝐺))
61		isinag.k	. . . . . . . . . . . . . 14 ⊢ 𝐾 = (hlG‘𝐺)
62	60, 61	eqtr4di 2791	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝐺 → (hlG‘𝑔) = 𝐾)
63	62	fveq1d 6890	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝐺 → ((hlG‘𝑔)‘(𝑡‘1)) = (𝐾‘(𝑡‘1)))
64	63	breqd 5158	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (𝑥((hlG‘𝑔)‘(𝑡‘1))𝑝 ↔ 𝑥(𝐾‘(𝑡‘1))𝑝))
65	64	orbi2d 915	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → ((𝑥 = (𝑡‘1) ∨ 𝑥((hlG‘𝑔)‘(𝑡‘1))𝑝) ↔ (𝑥 = (𝑡‘1) ∨ 𝑥(𝐾‘(𝑡‘1))𝑝)))
66	59, 65	anbi12d 632	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ((𝑥 ∈ ((𝑡‘0)(Itv‘𝑔)(𝑡‘2)) ∧ (𝑥 = (𝑡‘1) ∨ 𝑥((hlG‘𝑔)‘(𝑡‘1))𝑝)) ↔ (𝑥 ∈ ((𝑡‘0)𝐼(𝑡‘2)) ∧ (𝑥 = (𝑡‘1) ∨ 𝑥(𝐾‘(𝑡‘1))𝑝))))
67	50, 66	rexeqbidv 3344	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (∃𝑥 ∈ (Base‘𝑔)(𝑥 ∈ ((𝑡‘0)(Itv‘𝑔)(𝑡‘2)) ∧ (𝑥 = (𝑡‘1) ∨ 𝑥((hlG‘𝑔)‘(𝑡‘1))𝑝)) ↔ ∃𝑥 ∈ 𝑃 (𝑥 ∈ ((𝑡‘0)𝐼(𝑡‘2)) ∧ (𝑥 = (𝑡‘1) ∨ 𝑥(𝐾‘(𝑡‘1))𝑝))))
68	67	anbi2d 630	. . . . . . 7 ⊢ (𝑔 = 𝐺 → ((((𝑡‘0) ≠ (𝑡‘1) ∧ (𝑡‘2) ≠ (𝑡‘1) ∧ 𝑝 ≠ (𝑡‘1)) ∧ ∃𝑥 ∈ (Base‘𝑔)(𝑥 ∈ ((𝑡‘0)(Itv‘𝑔)(𝑡‘2)) ∧ (𝑥 = (𝑡‘1) ∨ 𝑥((hlG‘𝑔)‘(𝑡‘1))𝑝))) ↔ (((𝑡‘0) ≠ (𝑡‘1) ∧ (𝑡‘2) ≠ (𝑡‘1) ∧ 𝑝 ≠ (𝑡‘1)) ∧ ∃𝑥 ∈ 𝑃 (𝑥 ∈ ((𝑡‘0)𝐼(𝑡‘2)) ∧ (𝑥 = (𝑡‘1) ∨ 𝑥(𝐾‘(𝑡‘1))𝑝)))))
69	54, 68	anbi12d 632	. . . . . 6 ⊢ (𝑔 = 𝐺 → (((𝑝 ∈ (Base‘𝑔) ∧ 𝑡 ∈ ((Base‘𝑔) ↑_m (0..^3))) ∧ (((𝑡‘0) ≠ (𝑡‘1) ∧ (𝑡‘2) ≠ (𝑡‘1) ∧ 𝑝 ≠ (𝑡‘1)) ∧ ∃𝑥 ∈ (Base‘𝑔)(𝑥 ∈ ((𝑡‘0)(Itv‘𝑔)(𝑡‘2)) ∧ (𝑥 = (𝑡‘1) ∨ 𝑥((hlG‘𝑔)‘(𝑡‘1))𝑝)))) ↔ ((𝑝 ∈ 𝑃 ∧ 𝑡 ∈ (𝑃 ↑_m (0..^3))) ∧ (((𝑡‘0) ≠ (𝑡‘1) ∧ (𝑡‘2) ≠ (𝑡‘1) ∧ 𝑝 ≠ (𝑡‘1)) ∧ ∃𝑥 ∈ 𝑃 (𝑥 ∈ ((𝑡‘0)𝐼(𝑡‘2)) ∧ (𝑥 = (𝑡‘1) ∨ 𝑥(𝐾‘(𝑡‘1))𝑝))))))
70	69	opabbidv 5213	. . . . 5 ⊢ (𝑔 = 𝐺 → {⟨𝑝, 𝑡⟩ ∣ ((𝑝 ∈ (Base‘𝑔) ∧ 𝑡 ∈ ((Base‘𝑔) ↑_m (0..^3))) ∧ (((𝑡‘0) ≠ (𝑡‘1) ∧ (𝑡‘2) ≠ (𝑡‘1) ∧ 𝑝 ≠ (𝑡‘1)) ∧ ∃𝑥 ∈ (Base‘𝑔)(𝑥 ∈ ((𝑡‘0)(Itv‘𝑔)(𝑡‘2)) ∧ (𝑥 = (𝑡‘1) ∨ 𝑥((hlG‘𝑔)‘(𝑡‘1))𝑝))))} = {⟨𝑝, 𝑡⟩ ∣ ((𝑝 ∈ 𝑃 ∧ 𝑡 ∈ (𝑃 ↑_m (0..^3))) ∧ (((𝑡‘0) ≠ (𝑡‘1) ∧ (𝑡‘2) ≠ (𝑡‘1) ∧ 𝑝 ≠ (𝑡‘1)) ∧ ∃𝑥 ∈ 𝑃 (𝑥 ∈ ((𝑡‘0)𝐼(𝑡‘2)) ∧ (𝑥 = (𝑡‘1) ∨ 𝑥(𝐾‘(𝑡‘1))𝑝))))})
71		df-inag 28068	. . . . 5 ⊢ inA = (𝑔 ∈ V ↦ {⟨𝑝, 𝑡⟩ ∣ ((𝑝 ∈ (Base‘𝑔) ∧ 𝑡 ∈ ((Base‘𝑔) ↑_m (0..^3))) ∧ (((𝑡‘0) ≠ (𝑡‘1) ∧ (𝑡‘2) ≠ (𝑡‘1) ∧ 𝑝 ≠ (𝑡‘1)) ∧ ∃𝑥 ∈ (Base‘𝑔)(𝑥 ∈ ((𝑡‘0)(Itv‘𝑔)(𝑡‘2)) ∧ (𝑥 = (𝑡‘1) ∨ 𝑥((hlG‘𝑔)‘(𝑡‘1))𝑝))))})
72	49	fvexi 6902	. . . . . . 7 ⊢ 𝑃 ∈ V
73		ovex 7437	. . . . . . 7 ⊢ (𝑃 ↑_m (0..^3)) ∈ V
74	72, 73	xpex 7735	. . . . . 6 ⊢ (𝑃 × (𝑃 ↑_m (0..^3))) ∈ V
75		opabssxp 5766	. . . . . 6 ⊢ {⟨𝑝, 𝑡⟩ ∣ ((𝑝 ∈ 𝑃 ∧ 𝑡 ∈ (𝑃 ↑_m (0..^3))) ∧ (((𝑡‘0) ≠ (𝑡‘1) ∧ (𝑡‘2) ≠ (𝑡‘1) ∧ 𝑝 ≠ (𝑡‘1)) ∧ ∃𝑥 ∈ 𝑃 (𝑥 ∈ ((𝑡‘0)𝐼(𝑡‘2)) ∧ (𝑥 = (𝑡‘1) ∨ 𝑥(𝐾‘(𝑡‘1))𝑝))))} ⊆ (𝑃 × (𝑃 ↑_m (0..^3)))
76	74, 75	ssexi 5321	. . . . 5 ⊢ {⟨𝑝, 𝑡⟩ ∣ ((𝑝 ∈ 𝑃 ∧ 𝑡 ∈ (𝑃 ↑_m (0..^3))) ∧ (((𝑡‘0) ≠ (𝑡‘1) ∧ (𝑡‘2) ≠ (𝑡‘1) ∧ 𝑝 ≠ (𝑡‘1)) ∧ ∃𝑥 ∈ 𝑃 (𝑥 ∈ ((𝑡‘0)𝐼(𝑡‘2)) ∧ (𝑥 = (𝑡‘1) ∨ 𝑥(𝐾‘(𝑡‘1))𝑝))))} ∈ V
77	70, 71, 76	fvmpt 6994	. . . 4 ⊢ (𝐺 ∈ V → (inA‘𝐺) = {⟨𝑝, 𝑡⟩ ∣ ((𝑝 ∈ 𝑃 ∧ 𝑡 ∈ (𝑃 ↑_m (0..^3))) ∧ (((𝑡‘0) ≠ (𝑡‘1) ∧ (𝑡‘2) ≠ (𝑡‘1) ∧ 𝑝 ≠ (𝑡‘1)) ∧ ∃𝑥 ∈ 𝑃 (𝑥 ∈ ((𝑡‘0)𝐼(𝑡‘2)) ∧ (𝑥 = (𝑡‘1) ∨ 𝑥(𝐾‘(𝑡‘1))𝑝))))})
78	46, 47, 77	3syl 18	. . 3 ⊢ (𝜑 → (inA‘𝐺) = {⟨𝑝, 𝑡⟩ ∣ ((𝑝 ∈ 𝑃 ∧ 𝑡 ∈ (𝑃 ↑_m (0..^3))) ∧ (((𝑡‘0) ≠ (𝑡‘1) ∧ (𝑡‘2) ≠ (𝑡‘1) ∧ 𝑝 ≠ (𝑡‘1)) ∧ ∃𝑥 ∈ 𝑃 (𝑥 ∈ ((𝑡‘0)𝐼(𝑡‘2)) ∧ (𝑥 = (𝑡‘1) ∨ 𝑥(𝐾‘(𝑡‘1))𝑝))))})
79	78	breqd 5158	. 2 ⊢ (𝜑 → (𝑋(inA‘𝐺)⟨“𝐴𝐵𝐶”⟩ ↔ 𝑋{⟨𝑝, 𝑡⟩ ∣ ((𝑝 ∈ 𝑃 ∧ 𝑡 ∈ (𝑃 ↑_m (0..^3))) ∧ (((𝑡‘0) ≠ (𝑡‘1) ∧ (𝑡‘2) ≠ (𝑡‘1) ∧ 𝑝 ≠ (𝑡‘1)) ∧ ∃𝑥 ∈ 𝑃 (𝑥 ∈ ((𝑡‘0)𝐼(𝑡‘2)) ∧ (𝑥 = (𝑡‘1) ∨ 𝑥(𝐾‘(𝑡‘1))𝑝))))}⟨“𝐴𝐵𝐶”⟩))
80		isinag.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
81	22, 25, 29	s3cld 14819	. . . . 5 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑃)
82		s3len 14841	. . . . 5 ⊢ (♯‘⟨“𝐴𝐵𝐶”⟩) = 3
83		3nn0 12486	. . . . . 6 ⊢ 3 ∈ ℕ₀
84		wrdmap 14492	. . . . . 6 ⊢ ((𝑃 ∈ V ∧ 3 ∈ ℕ₀) → ((⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑃 ∧ (♯‘⟨“𝐴𝐵𝐶”⟩) = 3) ↔ ⟨“𝐴𝐵𝐶”⟩ ∈ (𝑃 ↑_m (0..^3))))
85	72, 83, 84	mp2an 691	. . . . 5 ⊢ ((⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑃 ∧ (♯‘⟨“𝐴𝐵𝐶”⟩) = 3) ↔ ⟨“𝐴𝐵𝐶”⟩ ∈ (𝑃 ↑_m (0..^3)))
86	81, 82, 85	sylanblc 590	. . . 4 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (𝑃 ↑_m (0..^3)))
87	80, 86	jca 513	. . 3 ⊢ (𝜑 → (𝑋 ∈ 𝑃 ∧ ⟨“𝐴𝐵𝐶”⟩ ∈ (𝑃 ↑_m (0..^3))))
88	87	biantrurd 534	. 2 ⊢ (𝜑 → (((𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐵 ∧ 𝑋 ≠ 𝐵) ∧ ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐶) ∧ (𝑥 = 𝐵 ∨ 𝑥(𝐾‘𝐵)𝑋))) ↔ ((𝑋 ∈ 𝑃 ∧ ⟨“𝐴𝐵𝐶”⟩ ∈ (𝑃 ↑_m (0..^3))) ∧ ((𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐵 ∧ 𝑋 ≠ 𝐵) ∧ ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐶) ∧ (𝑥 = 𝐵 ∨ 𝑥(𝐾‘𝐵)𝑋))))))
89	45, 79, 88	3bitr4d 311	1 ⊢ (𝜑 → (𝑋(inA‘𝐺)⟨“𝐴𝐵𝐶”⟩ ↔ ((𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐵 ∧ 𝑋 ≠ 𝐵) ∧ ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐶) ∧ (𝑥 = 𝐵 ∨ 𝑥(𝐾‘𝐵)𝑋)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∃wrex 3071 Vcvv 3475 class class class wbr 5147 {copab 5209 × cxp 5673 ‘cfv 6540 (class class class)co 7404 ↑_m cmap 8816 0cc0 11106 1c1 11107 2c2 12263 3c3 12264 ℕ₀cn0 12468 ..^cfzo 13623 ♯chash 14286 Word cword 14460 ⟨“cs3 14789 Basecbs 17140 Itvcitv 27664 hlGchlg 27831 inAcinag 28066

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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-1st 7970 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-fzo 13624 df-hash 14287 df-word 14461 df-concat 14517 df-s1 14542 df-s2 14795 df-s3 14796 df-inag 28068

This theorem is referenced by: isinagd 28070 inagswap 28072 inagne1 28073 inagne2 28074 inagne3 28075 inaghl 28076

W3C validator