Theorem constrlccllem 33910

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 7825	. . . . . 6 ⊢ (𝑛 ∈ ω ↔ suc 𝑛 ∈ ω)
2	1	biimpi 216	. . . . 5 ⊢ (𝑛 ∈ ω → suc 𝑛 ∈ ω)
3	2	ad2antlr 727	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → suc 𝑛 ∈ ω)
4		fveq2 6834	. . . . . 6 ⊢ (𝑚 = suc 𝑛 → (𝐶‘𝑚) = (𝐶‘suc 𝑛))
5	4	eleq2d 2822	. . . . 5 ⊢ (𝑚 = suc 𝑛 → (𝑋 ∈ (𝐶‘𝑚) ↔ 𝑋 ∈ (𝐶‘suc 𝑛)))
6	5	adantl 481	. . . 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 726	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝑋 ∈ ℂ)
9		id 22	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐴 → 𝑎 = 𝐴)
10		oveq2 7366	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝐴 → (𝑏 − 𝑎) = (𝑏 − 𝐴))
11	10	oveq2d 7374	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐴 → (𝑡 · (𝑏 − 𝑎)) = (𝑡 · (𝑏 − 𝐴)))
12	9, 11	oveq12d 7376	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (𝑎 + (𝑡 · (𝑏 − 𝑎))) = (𝐴 + (𝑡 · (𝑏 − 𝐴))))
13	12	eqeq2d 2747	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → (𝑋 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ↔ 𝑋 = (𝐴 + (𝑡 · (𝑏 − 𝐴)))))
14	13	anbi1d 631	. . . . . . . . 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 3160	. . . . . . . 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 3201	. . . . . . 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 7365	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝐵 → (𝑏 − 𝐴) = (𝐵 − 𝐴))
18	17	oveq2d 7374	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝐵 → (𝑡 · (𝑏 − 𝐴)) = (𝑡 · (𝐵 − 𝐴)))
19	18	oveq2d 7374	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐵 → (𝐴 + (𝑡 · (𝑏 − 𝐴))) = (𝐴 + (𝑡 · (𝐵 − 𝐴))))
20	19	eqeq2d 2747	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → (𝑋 = (𝐴 + (𝑡 · (𝑏 − 𝐴))) ↔ 𝑋 = (𝐴 + (𝑡 · (𝐵 − 𝐴)))))
21	20	anbi1d 631	. . . . . . . . 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 3160	. . . . . . . 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 3201	. . . . . . 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 7366	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝐺 → (𝑋 − 𝑐) = (𝑋 − 𝐺))
25	24	fveq2d 6838	. . . . . . . . . . 11 ⊢ (𝑐 = 𝐺 → (abs‘(𝑋 − 𝑐)) = (abs‘(𝑋 − 𝐺)))
26	25	eqeq1d 2738	. . . . . . . . . 10 ⊢ (𝑐 = 𝐺 → ((abs‘(𝑋 − 𝑐)) = (abs‘(𝑒 − 𝑓)) ↔ (abs‘(𝑋 − 𝐺)) = (abs‘(𝑒 − 𝑓))))
27	26	anbi2d 630	. . . . . . . . 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 3160	. . . . . . . 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 3201	. . . . . . 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 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝐴 ∈ Constr)
32		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛))
33	32	unssad 4145	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → {𝐴, 𝐵, 𝐺} ⊆ (𝐶‘𝑛))
34	31, 33	tpssad 32614	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝐴 ∈ (𝐶‘𝑛))
35		constrlccllem.b	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ Constr)
36	35	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝐵 ∈ Constr)
37	36, 33	tpssbd 32615	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝐵 ∈ (𝐶‘𝑛))
38		constrlccllem.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ Constr)
39	38	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝐺 ∈ Constr)
40	39, 33	tpsscd 32616	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝐺 ∈ (𝐶‘𝑛))
41		oveq1 7365	. . . . . . . . . . 11 ⊢ (𝑒 = 𝐸 → (𝑒 − 𝑓) = (𝐸 − 𝑓))
42	41	fveq2d 6838	. . . . . . . . . 10 ⊢ (𝑒 = 𝐸 → (abs‘(𝑒 − 𝑓)) = (abs‘(𝐸 − 𝑓)))
43	42	eqeq2d 2747	. . . . . . . . 9 ⊢ (𝑒 = 𝐸 → ((abs‘(𝑋 − 𝐺)) = (abs‘(𝑒 − 𝑓)) ↔ (abs‘(𝑋 − 𝐺)) = (abs‘(𝐸 − 𝑓))))
44	43	anbi2d 630	. . . . . . . 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 7366	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → (𝐸 − 𝑓) = (𝐸 − 𝐹))
46	45	fveq2d 6838	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (abs‘(𝐸 − 𝑓)) = (abs‘(𝐸 − 𝐹)))
47	46	eqeq2d 2747	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → ((abs‘(𝑋 − 𝐺)) = (abs‘(𝐸 − 𝑓)) ↔ (abs‘(𝑋 − 𝐺)) = (abs‘(𝐸 − 𝐹))))
48	47	anbi2d 630	. . . . . . . 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 7365	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → (𝑡 · (𝐵 − 𝐴)) = (𝑇 · (𝐵 − 𝐴)))
50	49	oveq2d 7374	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → (𝐴 + (𝑡 · (𝐵 − 𝐴))) = (𝐴 + (𝑇 · (𝐵 − 𝐴))))
51	50	eqeq2d 2747	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (𝑋 = (𝐴 + (𝑡 · (𝐵 − 𝐴))) ↔ 𝑋 = (𝐴 + (𝑇 · (𝐵 − 𝐴)))))
52	51	anbi1d 631	. . . . . . . 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 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝐸 ∈ Constr)
55	32	unssbd 4146	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → {𝐸, 𝐹} ⊆ (𝐶‘𝑛))
56	54, 55	prssad 32604	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝐸 ∈ (𝐶‘𝑛))
57		constrlccllem.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ Constr)
58	57	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝐹 ∈ Constr)
59	58, 55	prssbd 32605	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝐹 ∈ (𝐶‘𝑛))
60		constrlccllem.t	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ ℝ)
61	60	ad2antrr 726	. . . . . . . 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 726	. . . . . . . . 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 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → (abs‘(𝑋 − 𝐺)) = (abs‘(𝐸 − 𝐹)))
66	63, 65	jca 511	. . . . . . . 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 1338	. . . . 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 7814	. . . . . . 7 ⊢ (𝑛 ∈ ω → 𝑛 ∈ On)
72	71	ad2antlr 727	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝑛 ∈ On)
73		eqid 2736	. . . . . 6 ⊢ (𝐶‘𝑛) = (𝐶‘𝑛)
74	70, 72, 73	constrsuc 33895	. . . . 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 713	. . . 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 33893	. . 3 ⊢ (𝑋 ∈ Constr ↔ ∃𝑚 ∈ ω 𝑋 ∈ (𝐶‘𝑚))
78	76, 77	sylibr 234	. 2 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝑋 ∈ Constr)
79	30, 35, 38	tpssd 32613	. . . 4 ⊢ (𝜑 → {𝐴, 𝐵, 𝐺} ⊆ Constr)
80	53, 57	prssd 4778	. . . 4 ⊢ (𝜑 → {𝐸, 𝐹} ⊆ Constr)
81	79, 80	unssd 4144	. . 3 ⊢ (𝜑 → ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ Constr)
82		tpfi 9226	. . . . 5 ⊢ {𝐴, 𝐵, 𝐺} ∈ Fin
83	82	a1i 11	. . . 4 ⊢ (𝜑 → {𝐴, 𝐵, 𝐺} ∈ Fin)
84		prfi 9224	. . . . 5 ⊢ {𝐸, 𝐹} ∈ Fin
85	84	a1i 11	. . . 4 ⊢ (𝜑 → {𝐸, 𝐹} ∈ Fin)
86	83, 85	unfid 9096	. . 3 ⊢ (𝜑 → ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ∈ Fin)
87	70, 81, 86	constrfiss 33908	. 2 ⊢ (𝜑 → ∃𝑛 ∈ ω ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛))
88	78, 87	r19.29a 3144	1 ⊢ (𝜑 → 𝑋 ∈ Constr)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ w3o 1085   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2932  ∃wrex 3060  {crab 3399  Vcvv 3440   ∪ cun 3899   ⊆ wss 3901  {cpr 4582  {ctp 4584   ↦ cmpt 5179  Oncon0 6317  suc csuc 6319  ‘cfv 6492  (class class class)co 7358  ωcom 7808  reccrdg 8340  Fincfn 8883  ℂcc 11024  ℝcr 11025  0cc0 11026  1c1 11027   + caddc 11029   · cmul 11031   − cmin 11364  ∗ccj 15019  ℑcim 15021  abscabs 15157  Constrcconstr 33886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-er 8635  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11168  df-mnf 11169  df-ltxr 11171  df-sub 11366  df-constr 33887

This theorem is referenced by:  constrlccl  33914