Theorem constrlccllem 34432

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 7895	. . . . . 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 6886	. . . . . 6 ⊢ (𝑚 = suc 𝑛 → (𝐶‘𝑚) = (𝐶‘suc 𝑛))
5	4	eleq2d 2846	. . . . 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 7429	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝐴 → (𝑏 − 𝑎) = (𝑏 − 𝐴))
11	10	oveq2d 7437	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐴 → (𝑡 · (𝑏 − 𝑎)) = (𝑡 · (𝑏 − 𝐴)))
12	9, 11	oveq12d 7439	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (𝑎 + (𝑡 · (𝑏 − 𝑎))) = (𝐴 + (𝑡 · (𝑏 − 𝐴))))
13	12	eqeq2d 2771	. . . . . . . . . 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 3186	. . . . . . . 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 3227	. . . . . . 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 7428	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝐵 → (𝑏 − 𝐴) = (𝐵 − 𝐴))
18	17	oveq2d 7437	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝐵 → (𝑡 · (𝑏 − 𝐴)) = (𝑡 · (𝐵 − 𝐴)))
19	18	oveq2d 7437	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐵 → (𝐴 + (𝑡 · (𝑏 − 𝐴))) = (𝐴 + (𝑡 · (𝐵 − 𝐴))))
20	19	eqeq2d 2771	. . . . . . . . . 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 3186	. . . . . . . 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 3227	. . . . . . 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 7429	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝐺 → (𝑋 − 𝑐) = (𝑋 − 𝐺))
25	24	fveq2d 6890	. . . . . . . . . . 11 ⊢ (𝑐 = 𝐺 → (abs‘(𝑋 − 𝑐)) = (abs‘(𝑋 − 𝐺)))
26	25	eqeq1d 2762	. . . . . . . . . 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 3186	. . . . . . . 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 3227	. . . . . . 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 4191	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → {𝐴, 𝐵, 𝐺} ⊆ (𝐶‘𝑛))
34	31, 33	tpssad 33183	. . . . . . 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 33184	. . . . . . 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 33185	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝐺 ∈ (𝐶‘𝑛))
41		oveq1 7428	. . . . . . . . . . 11 ⊢ (𝑒 = 𝐸 → (𝑒 − 𝑓) = (𝐸 − 𝑓))
42	41	fveq2d 6890	. . . . . . . . . 10 ⊢ (𝑒 = 𝐸 → (abs‘(𝑒 − 𝑓)) = (abs‘(𝐸 − 𝑓)))
43	42	eqeq2d 2771	. . . . . . . . 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 7429	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → (𝐸 − 𝑓) = (𝐸 − 𝐹))
46	45	fveq2d 6890	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (abs‘(𝐸 − 𝑓)) = (abs‘(𝐸 − 𝐹)))
47	46	eqeq2d 2771	. . . . . . . . 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 7428	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → (𝑡 · (𝐵 − 𝐴)) = (𝑇 · (𝐵 − 𝐴)))
50	49	oveq2d 7437	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → (𝐴 + (𝑡 · (𝐵 − 𝐴))) = (𝐴 + (𝑇 · (𝐵 − 𝐴))))
51	50	eqeq2d 2771	. . . . . . . . 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 4192	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → {𝐸, 𝐹} ⊆ (𝐶‘𝑛))
56	54, 55	prssad 33173	. . . . . . . 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 33174	. . . . . . . 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 3593	. . . . . . 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 3593	. . . . . 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 7884	. . . . . . 7 ⊢ (𝑛 ∈ ω → 𝑛 ∈ On)
72	71	ad2antlr 740	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝑛 ∈ On)
73		eqid 2760	. . . . . 6 ⊢ (𝐶‘𝑛) = (𝐶‘𝑛)
74	70, 72, 73	constrsuc 34417	. . . . 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 3578	. . 3 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → ∃𝑚 ∈ ω 𝑋 ∈ (𝐶‘𝑚))
77	70	isconstr 34415	. . 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 33182	. . . 4 ⊢ (𝜑 → {𝐴, 𝐵, 𝐺} ⊆ Constr)
80	53, 57	prssd 4777	. . . 4 ⊢ (𝜑 → {𝐸, 𝐹} ⊆ Constr)
81	79, 80	unssd 4190	. . 3 ⊢ (𝜑 → ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ Constr)
82		tpfi 9321	. . . . 5 ⊢ {𝐴, 𝐵, 𝐺} ∈ Fin
83	82	a1i 11	. . . 4 ⊢ (𝜑 → {𝐴, 𝐵, 𝐺} ∈ Fin)
84		prfi 9319	. . . . 5 ⊢ {𝐸, 𝐹} ∈ Fin
85	84	a1i 11	. . . 4 ⊢ (𝜑 → {𝐸, 𝐹} ∈ Fin)
86	83, 85	unfid 9191	. . 3 ⊢ (𝜑 → ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ∈ Fin)
87	70, 81, 86	constrfiss 34430	. 2 ⊢ (𝜑 → ∃𝑛 ∈ ω ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛))
88	78, 87	r19.29a 3170	1 ⊢ (𝜑 → 𝑋 ∈ Constr)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ w3o 1102   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∃wrex 3086  {crab 3412  Vcvv 3450   ⊆ wss 3895   ∪ cun 4149  {cpr 4586  {ctp 4588   ↦ cmpt 5185  Oncon0 6362  suc csuc 6364  ‘cfv 6539  (class class class)co 7421  ωcom 7878  reccrdg 8421  Fincfn 8977  ℂcc 11202  ℝcr 11203  0cc0 11204  1c1 11205   + caddc 11207   · cmul 11209   − cmin 11545  ∗ccj 15269  ℑcim 15271  abscabs 15407  Constrcconstr 34408

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752  ax-cnex 11260  ax-resscn 11261  ax-1cn 11262  ax-icn 11263  ax-addcl 11264  ax-addrcl 11265  ax-mulcl 11266  ax-mulrcl 11267  ax-mulcom 11268  ax-addass 11269  ax-mulass 11270  ax-distr 11271  ax-i2m1 11272  ax-1ne0 11273  ax-1rid 11274  ax-rnegex 11275  ax-rrecex 11276  ax-cnre 11277  ax-pre-lttri 11278  ax-pre-lttrn 11279  ax-pre-ltadd 11280

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7378  df-ov 7424  df-oprab 7425  df-mpo 7426  df-om 7879  df-2nd 8003  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-rdg 8422  df-1o 8480  df-2o 8481  df-er 8721  df-en 8978  df-dom 8979  df-sdom 8980  df-fin 8981  df-pnf 11349  df-mnf 11350  df-ltxr 11352  df-sub 11547  df-constr 34409

This theorem is used by:  constrlccl  34436