Theorem constrlccllem 34378

Description: Constructible numbers are closed under line-circle intersections. (Contributed by Thierry Arnoux, 2-Nov-2025.)

Hypotheses

Ref	Expression
constr0.1	⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})
constrlccllem.a	⊢ (𝜑 → 𝐴 ∈ Constr)
constrlccllem.b	⊢ (𝜑 → 𝐵 ∈ Constr)
constrlccllem.c	⊢ (𝜑 → 𝐺 ∈ Constr)
constrlccllem.e	⊢ (𝜑 → 𝐸 ∈ Constr)
constrlccllem.f	⊢ (𝜑 → 𝐹 ∈ Constr)
constrlccllem.t	⊢ (𝜑 → 𝑇 ∈ ℝ)
constrlccllem.x	⊢ (𝜑 → 𝑋 ∈ ℂ)
constrlccllem.1	⊢ (𝜑 → 𝑋 = (𝐴 + (𝑇 · (𝐵 − 𝐴))))
constrlccllem.2	⊢ (𝜑 → (abs‘(𝑋 − 𝐺)) = (abs‘(𝐸 − 𝐹)))

Assertion

Ref	Expression
constrlccllem	⊢ (𝜑 → 𝑋 ∈ Constr)

Distinct variable groups:   𝐴,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓,𝑠,𝑡,𝑥   𝐵,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓,𝑠,𝑡,𝑥   𝐶,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓,𝑠,𝑡,𝑥   𝐸,𝑎,𝑏,𝑐,𝑒,𝑓,𝑠,𝑡,𝑥   𝐹,𝑎,𝑏,𝑐,𝑒,𝑓,𝑠,𝑡,𝑥   𝐺,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓,𝑠,𝑡,𝑥   𝑡,𝑇   𝑋,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓,𝑟,𝑡   𝑠,𝑟,𝑥   𝜑,𝑎,𝑏,𝑐,𝑒,𝑓,𝑠,𝑡,𝑥

Allowed substitution hints:   𝜑(𝑟, 𝑑)   𝐴(𝑟)   𝐵(𝑟)   𝐶(𝑟)   𝑇(𝑥, 𝑒, 𝑓, 𝑠, 𝑟, 𝑎, 𝑏, 𝑐, 𝑑)   𝐸(𝑟, 𝑑)   𝐹(𝑟, 𝑑)   𝐺(𝑟)   𝑋(𝑥, 𝑠)

Proof of Theorem constrlccllem

Dummy variables 𝑛 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		peano2b 7892	. . . . . 6 ⊢ (𝑛 ∈ ω ↔ suc 𝑛 ∈ ω)
2	1	biimpi 219	. . . . 5 ⊢ (𝑛 ∈ ω → suc 𝑛 ∈ ω)
3	2	ad2antlr 740	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → suc 𝑛 ∈ ω)
4		fveq2 6883	. . . . . 6 ⊢ (𝑚 = suc 𝑛 → (𝐶‘𝑚) = (𝐶‘suc 𝑛))
5	4	eleq2d 2847	. . . . 5 ⊢ (𝑚 = suc 𝑛 → (𝑋 ∈ (𝐶‘𝑚) ↔ 𝑋 ∈ (𝐶‘suc 𝑛)))
6	5	adantl 487	. . . 4 ⊢ ((((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) ∧ 𝑚 = suc 𝑛) → (𝑋 ∈ (𝐶‘𝑚) ↔ 𝑋 ∈ (𝐶‘suc 𝑛)))
7		constrlccllem.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ ℂ)
8	7	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝑋 ∈ ℂ)
9		id 23	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐴 → 𝑎 = 𝐴)
10		oveq2 7426	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝐴 → (𝑏 − 𝑎) = (𝑏 − 𝐴))
11	10	oveq2d 7434	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐴 → (𝑡 · (𝑏 − 𝑎)) = (𝑡 · (𝑏 − 𝐴)))
12	9, 11	oveq12d 7436	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (𝑎 + (𝑡 · (𝑏 − 𝑎))) = (𝐴 + (𝑡 · (𝑏 − 𝐴))))
13	12	eqeq2d 2772	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → (𝑋 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ↔ 𝑋 = (𝐴 + (𝑡 · (𝑏 − 𝐴)))))
14	13	anbi1d 643	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → ((𝑋 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑋 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ↔ (𝑋 = (𝐴 + (𝑡 · (𝑏 − 𝐴))) ∧ (abs‘(𝑋 − 𝑐)) = (abs‘(𝑒 − 𝑓)))))
15	14	rexbidv 3187	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (∃𝑡 ∈ ℝ (𝑋 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑋 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ↔ ∃𝑡 ∈ ℝ (𝑋 = (𝐴 + (𝑡 · (𝑏 − 𝐴))) ∧ (abs‘(𝑋 − 𝑐)) = (abs‘(𝑒 − 𝑓)))))
16	15	2rexbidv 3228	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (∃𝑒 ∈ (𝐶‘𝑛)∃𝑓 ∈ (𝐶‘𝑛)∃𝑡 ∈ ℝ (𝑋 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑋 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ↔ ∃𝑒 ∈ (𝐶‘𝑛)∃𝑓 ∈ (𝐶‘𝑛)∃𝑡 ∈ ℝ (𝑋 = (𝐴 + (𝑡 · (𝑏 − 𝐴))) ∧ (abs‘(𝑋 − 𝑐)) = (abs‘(𝑒 − 𝑓)))))
17		oveq1 7425	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝐵 → (𝑏 − 𝐴) = (𝐵 − 𝐴))
18	17	oveq2d 7434	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝐵 → (𝑡 · (𝑏 − 𝐴)) = (𝑡 · (𝐵 − 𝐴)))
19	18	oveq2d 7434	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐵 → (𝐴 + (𝑡 · (𝑏 − 𝐴))) = (𝐴 + (𝑡 · (𝐵 − 𝐴))))
20	19	eqeq2d 2772	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → (𝑋 = (𝐴 + (𝑡 · (𝑏 − 𝐴))) ↔ 𝑋 = (𝐴 + (𝑡 · (𝐵 − 𝐴)))))
21	20	anbi1d 643	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → ((𝑋 = (𝐴 + (𝑡 · (𝑏 − 𝐴))) ∧ (abs‘(𝑋 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ↔ (𝑋 = (𝐴 + (𝑡 · (𝐵 − 𝐴))) ∧ (abs‘(𝑋 − 𝑐)) = (abs‘(𝑒 − 𝑓)))))
22	21	rexbidv 3187	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → (∃𝑡 ∈ ℝ (𝑋 = (𝐴 + (𝑡 · (𝑏 − 𝐴))) ∧ (abs‘(𝑋 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ↔ ∃𝑡 ∈ ℝ (𝑋 = (𝐴 + (𝑡 · (𝐵 − 𝐴))) ∧ (abs‘(𝑋 − 𝑐)) = (abs‘(𝑒 − 𝑓)))))
23	22	2rexbidv 3228	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (∃𝑒 ∈ (𝐶‘𝑛)∃𝑓 ∈ (𝐶‘𝑛)∃𝑡 ∈ ℝ (𝑋 = (𝐴 + (𝑡 · (𝑏 − 𝐴))) ∧ (abs‘(𝑋 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ↔ ∃𝑒 ∈ (𝐶‘𝑛)∃𝑓 ∈ (𝐶‘𝑛)∃𝑡 ∈ ℝ (𝑋 = (𝐴 + (𝑡 · (𝐵 − 𝐴))) ∧ (abs‘(𝑋 − 𝑐)) = (abs‘(𝑒 − 𝑓)))))
24		oveq2 7426	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝐺 → (𝑋 − 𝑐) = (𝑋 − 𝐺))
25	24	fveq2d 6887	. . . . . . . . . . 11 ⊢ (𝑐 = 𝐺 → (abs‘(𝑋 − 𝑐)) = (abs‘(𝑋 − 𝐺)))
26	25	eqeq1d 2763	. . . . . . . . . 10 ⊢ (𝑐 = 𝐺 → ((abs‘(𝑋 − 𝑐)) = (abs‘(𝑒 − 𝑓)) ↔ (abs‘(𝑋 − 𝐺)) = (abs‘(𝑒 − 𝑓))))
27	26	anbi2d 642	. . . . . . . . 9 ⊢ (𝑐 = 𝐺 → ((𝑋 = (𝐴 + (𝑡 · (𝐵 − 𝐴))) ∧ (abs‘(𝑋 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ↔ (𝑋 = (𝐴 + (𝑡 · (𝐵 − 𝐴))) ∧ (abs‘(𝑋 − 𝐺)) = (abs‘(𝑒 − 𝑓)))))
28	27	rexbidv 3187	. . . . . . . 8 ⊢ (𝑐 = 𝐺 → (∃𝑡 ∈ ℝ (𝑋 = (𝐴 + (𝑡 · (𝐵 − 𝐴))) ∧ (abs‘(𝑋 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ↔ ∃𝑡 ∈ ℝ (𝑋 = (𝐴 + (𝑡 · (𝐵 − 𝐴))) ∧ (abs‘(𝑋 − 𝐺)) = (abs‘(𝑒 − 𝑓)))))
29	28	2rexbidv 3228	. . . . . . 7 ⊢ (𝑐 = 𝐺 → (∃𝑒 ∈ (𝐶‘𝑛)∃𝑓 ∈ (𝐶‘𝑛)∃𝑡 ∈ ℝ (𝑋 = (𝐴 + (𝑡 · (𝐵 − 𝐴))) ∧ (abs‘(𝑋 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ↔ ∃𝑒 ∈ (𝐶‘𝑛)∃𝑓 ∈ (𝐶‘𝑛)∃𝑡 ∈ ℝ (𝑋 = (𝐴 + (𝑡 · (𝐵 − 𝐴))) ∧ (abs‘(𝑋 − 𝐺)) = (abs‘(𝑒 − 𝑓)))))
30		constrlccllem.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ Constr)
31	30	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝐴 ∈ Constr)
32		simpr 490	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛))
33	32	unssad 4139	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → {𝐴, 𝐵, 𝐺} ⊆ (𝐶‘𝑛))
34	31, 33	tpssad 33128	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝐴 ∈ (𝐶‘𝑛))
35		constrlccllem.b	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ Constr)
36	35	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝐵 ∈ Constr)
37	36, 33	tpssbd 33129	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝐵 ∈ (𝐶‘𝑛))
38		constrlccllem.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ Constr)
39	38	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝐺 ∈ Constr)
40	39, 33	tpsscd 33130	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝐺 ∈ (𝐶‘𝑛))
41		oveq1 7425	. . . . . . . . . . 11 ⊢ (𝑒 = 𝐸 → (𝑒 − 𝑓) = (𝐸 − 𝑓))
42	41	fveq2d 6887	. . . . . . . . . 10 ⊢ (𝑒 = 𝐸 → (abs‘(𝑒 − 𝑓)) = (abs‘(𝐸 − 𝑓)))
43	42	eqeq2d 2772	. . . . . . . . 9 ⊢ (𝑒 = 𝐸 → ((abs‘(𝑋 − 𝐺)) = (abs‘(𝑒 − 𝑓)) ↔ (abs‘(𝑋 − 𝐺)) = (abs‘(𝐸 − 𝑓))))
44	43	anbi2d 642	. . . . . . . 8 ⊢ (𝑒 = 𝐸 → ((𝑋 = (𝐴 + (𝑡 · (𝐵 − 𝐴))) ∧ (abs‘(𝑋 − 𝐺)) = (abs‘(𝑒 − 𝑓))) ↔ (𝑋 = (𝐴 + (𝑡 · (𝐵 − 𝐴))) ∧ (abs‘(𝑋 − 𝐺)) = (abs‘(𝐸 − 𝑓)))))
45		oveq2 7426	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → (𝐸 − 𝑓) = (𝐸 − 𝐹))
46	45	fveq2d 6887	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (abs‘(𝐸 − 𝑓)) = (abs‘(𝐸 − 𝐹)))
47	46	eqeq2d 2772	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → ((abs‘(𝑋 − 𝐺)) = (abs‘(𝐸 − 𝑓)) ↔ (abs‘(𝑋 − 𝐺)) = (abs‘(𝐸 − 𝐹))))
48	47	anbi2d 642	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((𝑋 = (𝐴 + (𝑡 · (𝐵 − 𝐴))) ∧ (abs‘(𝑋 − 𝐺)) = (abs‘(𝐸 − 𝑓))) ↔ (𝑋 = (𝐴 + (𝑡 · (𝐵 − 𝐴))) ∧ (abs‘(𝑋 − 𝐺)) = (abs‘(𝐸 − 𝐹)))))
49		oveq1 7425	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → (𝑡 · (𝐵 − 𝐴)) = (𝑇 · (𝐵 − 𝐴)))
50	49	oveq2d 7434	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → (𝐴 + (𝑡 · (𝐵 − 𝐴))) = (𝐴 + (𝑇 · (𝐵 − 𝐴))))
51	50	eqeq2d 2772	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (𝑋 = (𝐴 + (𝑡 · (𝐵 − 𝐴))) ↔ 𝑋 = (𝐴 + (𝑇 · (𝐵 − 𝐴)))))
52	51	anbi1d 643	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → ((𝑋 = (𝐴 + (𝑡 · (𝐵 − 𝐴))) ∧ (abs‘(𝑋 − 𝐺)) = (abs‘(𝐸 − 𝐹))) ↔ (𝑋 = (𝐴 + (𝑇 · (𝐵 − 𝐴))) ∧ (abs‘(𝑋 − 𝐺)) = (abs‘(𝐸 − 𝐹)))))
53		constrlccllem.e	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ Constr)
54	53	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝐸 ∈ Constr)
55	32	unssbd 4140	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → {𝐸, 𝐹} ⊆ (𝐶‘𝑛))
56	54, 55	prssad 33118	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝐸 ∈ (𝐶‘𝑛))
57		constrlccllem.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ Constr)
58	57	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝐹 ∈ Constr)
59	58, 55	prssbd 33119	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝐹 ∈ (𝐶‘𝑛))
60		constrlccllem.t	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ ℝ)
61	60	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝑇 ∈ ℝ)
62		constrlccllem.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 = (𝐴 + (𝑇 · (𝐵 − 𝐴))))
63	62	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝑋 = (𝐴 + (𝑇 · (𝐵 − 𝐴))))
64		constrlccllem.2	. . . . . . . . . 10 ⊢ (𝜑 → (abs‘(𝑋 − 𝐺)) = (abs‘(𝐸 − 𝐹)))
65	64	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → (abs‘(𝑋 − 𝐺)) = (abs‘(𝐸 − 𝐹)))
66	63, 65	jca 521	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → (𝑋 = (𝐴 + (𝑇 · (𝐵 − 𝐴))) ∧ (abs‘(𝑋 − 𝐺)) = (abs‘(𝐸 − 𝐹))))
67	44, 48, 52, 56, 59, 61, 66	3rspcedvdw 3594	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → ∃𝑒 ∈ (𝐶‘𝑛)∃𝑓 ∈ (𝐶‘𝑛)∃𝑡 ∈ ℝ (𝑋 = (𝐴 + (𝑡 · (𝐵 − 𝐴))) ∧ (abs‘(𝑋 − 𝐺)) = (abs‘(𝑒 − 𝑓))))
68	16, 23, 29, 34, 37, 40, 67	3rspcedvdw 3594	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → ∃𝑎 ∈ (𝐶‘𝑛)∃𝑏 ∈ (𝐶‘𝑛)∃𝑐 ∈ (𝐶‘𝑛)∃𝑒 ∈ (𝐶‘𝑛)∃𝑓 ∈ (𝐶‘𝑛)∃𝑡 ∈ ℝ (𝑋 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑋 − 𝑐)) = (abs‘(𝑒 − 𝑓))))
69	68	3mix2d 1356	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → (∃𝑎 ∈ (𝐶‘𝑛)∃𝑏 ∈ (𝐶‘𝑛)∃𝑐 ∈ (𝐶‘𝑛)∃𝑑 ∈ (𝐶‘𝑛)∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑋 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑋 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ (𝐶‘𝑛)∃𝑏 ∈ (𝐶‘𝑛)∃𝑐 ∈ (𝐶‘𝑛)∃𝑒 ∈ (𝐶‘𝑛)∃𝑓 ∈ (𝐶‘𝑛)∃𝑡 ∈ ℝ (𝑋 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑋 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ (𝐶‘𝑛)∃𝑏 ∈ (𝐶‘𝑛)∃𝑐 ∈ (𝐶‘𝑛)∃𝑑 ∈ (𝐶‘𝑛)∃𝑒 ∈ (𝐶‘𝑛)∃𝑓 ∈ (𝐶‘𝑛)(𝑎 ≠ 𝑑 ∧ (abs‘(𝑋 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑋 − 𝑑)) = (abs‘(𝑒 − 𝑓)))))
70		constr0.1	. . . . . 6 ⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})
71		nnon 7881	. . . . . . 7 ⊢ (𝑛 ∈ ω → 𝑛 ∈ On)
72	71	ad2antlr 740	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝑛 ∈ On)
73		eqid 2761	. . . . . 6 ⊢ (𝐶‘𝑛) = (𝐶‘𝑛)
74	70, 72, 73	constrsuc 34363	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → (𝑋 ∈ (𝐶‘suc 𝑛) ↔ (𝑋 ∈ ℂ ∧ (∃𝑎 ∈ (𝐶‘𝑛)∃𝑏 ∈ (𝐶‘𝑛)∃𝑐 ∈ (𝐶‘𝑛)∃𝑑 ∈ (𝐶‘𝑛)∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑋 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑋 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ (𝐶‘𝑛)∃𝑏 ∈ (𝐶‘𝑛)∃𝑐 ∈ (𝐶‘𝑛)∃𝑒 ∈ (𝐶‘𝑛)∃𝑓 ∈ (𝐶‘𝑛)∃𝑡 ∈ ℝ (𝑋 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑋 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ (𝐶‘𝑛)∃𝑏 ∈ (𝐶‘𝑛)∃𝑐 ∈ (𝐶‘𝑛)∃𝑑 ∈ (𝐶‘𝑛)∃𝑒 ∈ (𝐶‘𝑛)∃𝑓 ∈ (𝐶‘𝑛)(𝑎 ≠ 𝑑 ∧ (abs‘(𝑋 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑋 − 𝑑)) = (abs‘(𝑒 − 𝑓)))))))
75	8, 69, 74	mpbir2and 726	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝑋 ∈ (𝐶‘suc 𝑛))
76	3, 6, 75	rspcedvd 3579	. . 3 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → ∃𝑚 ∈ ω 𝑋 ∈ (𝐶‘𝑚))
77	70	isconstr 34361	. . 3 ⊢ (𝑋 ∈ Constr ↔ ∃𝑚 ∈ ω 𝑋 ∈ (𝐶‘𝑚))
78	76, 77	sylibr 237	. 2 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝑋 ∈ Constr)
79	30, 35, 38	tpssd 33127	. . . 4 ⊢ (𝜑 → {𝐴, 𝐵, 𝐺} ⊆ Constr)
80	53, 57	prssd 4783	. . . 4 ⊢ (𝜑 → {𝐸, 𝐹} ⊆ Constr)
81	79, 80	unssd 4138	. . 3 ⊢ (𝜑 → ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ Constr)
82		tpfi 9310	. . . . 5 ⊢ {𝐴, 𝐵, 𝐺} ∈ Fin
83	82	a1i 11	. . . 4 ⊢ (𝜑 → {𝐴, 𝐵, 𝐺} ∈ Fin)
84		prfi 9308	. . . . 5 ⊢ {𝐸, 𝐹} ∈ Fin
85	84	a1i 11	. . . 4 ⊢ (𝜑 → {𝐸, 𝐹} ∈ Fin)
86	83, 85	unfid 9180	. . 3 ⊢ (𝜑 → ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ∈ Fin)
87	70, 81, 86	constrfiss 34376	. 2 ⊢ (𝜑 → ∃𝑛 ∈ ω ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛))
88	78, 87	r19.29a 3171	1 ⊢ (𝜑 → 𝑋 ∈ Constr)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ w3o 1102   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∃wrex 3087  {crab 3413  Vcvv 3451   ∪ cun 3897   ⊆ wss 3899  {cpr 4586  {ctp 4588   ↦ cmpt 5186  Oncon0 6361  suc csuc 6363  ‘cfv 6537  (class class class)co 7418  ωcom 7875  reccrdg 8410  Fincfn 8966  ℂcc 11191  ℝcr 11192  0cc0 11193  1c1 11194   + caddc 11196   · cmul 11198   − cmin 11534  ∗ccj 15256  ℑcim 15258  abscabs 15394  Constrcconstr 34354

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-2o 8470  df-er 8710  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-pnf 11338  df-mnf 11339  df-ltxr 11341  df-sub 11536  df-constr 34355

This theorem is used by:  constrlccl  34382