Theorem constrlccllem 33937

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 7823	. . . . . 6 ⊢ (𝑛 ∈ ω ↔ suc 𝑛 ∈ ω)
2	1	biimpi 217	. . . . 5 ⊢ (𝑛 ∈ ω → suc 𝑛 ∈ ω)
3	2	ad2antlr 733	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → suc 𝑛 ∈ ω)
4		fveq2 6827	. . . . . 6 ⊢ (𝑚 = suc 𝑛 → (𝐶‘𝑚) = (𝐶‘suc 𝑛))
5	4	eleq2d 2825	. . . . 5 ⊢ (𝑚 = suc 𝑛 → (𝑋 ∈ (𝐶‘𝑚) ↔ 𝑋 ∈ (𝐶‘suc 𝑛)))
6	5	adantl 482	. . . 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 732	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝑋 ∈ ℂ)
9		id 22	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐴 → 𝑎 = 𝐴)
10		oveq2 7364	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝐴 → (𝑏 − 𝑎) = (𝑏 − 𝐴))
11	10	oveq2d 7372	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐴 → (𝑡 · (𝑏 − 𝑎)) = (𝑡 · (𝑏 − 𝐴)))
12	9, 11	oveq12d 7374	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (𝑎 + (𝑡 · (𝑏 − 𝑎))) = (𝐴 + (𝑡 · (𝑏 − 𝐴))))
13	12	eqeq2d 2750	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → (𝑋 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ↔ 𝑋 = (𝐴 + (𝑡 · (𝑏 − 𝐴)))))
14	13	anbi1d 637	. . . . . . . . 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 3163	. . . . . . . 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 3204	. . . . . . 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 7363	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝐵 → (𝑏 − 𝐴) = (𝐵 − 𝐴))
18	17	oveq2d 7372	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝐵 → (𝑡 · (𝑏 − 𝐴)) = (𝑡 · (𝐵 − 𝐴)))
19	18	oveq2d 7372	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐵 → (𝐴 + (𝑡 · (𝑏 − 𝐴))) = (𝐴 + (𝑡 · (𝐵 − 𝐴))))
20	19	eqeq2d 2750	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → (𝑋 = (𝐴 + (𝑡 · (𝑏 − 𝐴))) ↔ 𝑋 = (𝐴 + (𝑡 · (𝐵 − 𝐴)))))
21	20	anbi1d 637	. . . . . . . . 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 3163	. . . . . . . 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 3204	. . . . . . 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 7364	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝐺 → (𝑋 − 𝑐) = (𝑋 − 𝐺))
25	24	fveq2d 6831	. . . . . . . . . . 11 ⊢ (𝑐 = 𝐺 → (abs‘(𝑋 − 𝑐)) = (abs‘(𝑋 − 𝐺)))
26	25	eqeq1d 2741	. . . . . . . . . 10 ⊢ (𝑐 = 𝐺 → ((abs‘(𝑋 − 𝑐)) = (abs‘(𝑒 − 𝑓)) ↔ (abs‘(𝑋 − 𝐺)) = (abs‘(𝑒 − 𝑓))))
27	26	anbi2d 636	. . . . . . . . 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 3163	. . . . . . . 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 3204	. . . . . . 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 732	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝐴 ∈ Constr)
32		simpr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛))
33	32	unssad 4122	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → {𝐴, 𝐵, 𝐺} ⊆ (𝐶‘𝑛))
34	31, 33	tpssad 32627	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝐴 ∈ (𝐶‘𝑛))
35		constrlccllem.b	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ Constr)
36	35	ad2antrr 732	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝐵 ∈ Constr)
37	36, 33	tpssbd 32628	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝐵 ∈ (𝐶‘𝑛))
38		constrlccllem.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ Constr)
39	38	ad2antrr 732	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝐺 ∈ Constr)
40	39, 33	tpsscd 32629	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝐺 ∈ (𝐶‘𝑛))
41		oveq1 7363	. . . . . . . . . . 11 ⊢ (𝑒 = 𝐸 → (𝑒 − 𝑓) = (𝐸 − 𝑓))
42	41	fveq2d 6831	. . . . . . . . . 10 ⊢ (𝑒 = 𝐸 → (abs‘(𝑒 − 𝑓)) = (abs‘(𝐸 − 𝑓)))
43	42	eqeq2d 2750	. . . . . . . . 9 ⊢ (𝑒 = 𝐸 → ((abs‘(𝑋 − 𝐺)) = (abs‘(𝑒 − 𝑓)) ↔ (abs‘(𝑋 − 𝐺)) = (abs‘(𝐸 − 𝑓))))
44	43	anbi2d 636	. . . . . . . 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 7364	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → (𝐸 − 𝑓) = (𝐸 − 𝐹))
46	45	fveq2d 6831	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (abs‘(𝐸 − 𝑓)) = (abs‘(𝐸 − 𝐹)))
47	46	eqeq2d 2750	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → ((abs‘(𝑋 − 𝐺)) = (abs‘(𝐸 − 𝑓)) ↔ (abs‘(𝑋 − 𝐺)) = (abs‘(𝐸 − 𝐹))))
48	47	anbi2d 636	. . . . . . . 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 7363	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → (𝑡 · (𝐵 − 𝐴)) = (𝑇 · (𝐵 − 𝐴)))
50	49	oveq2d 7372	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → (𝐴 + (𝑡 · (𝐵 − 𝐴))) = (𝐴 + (𝑇 · (𝐵 − 𝐴))))
51	50	eqeq2d 2750	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (𝑋 = (𝐴 + (𝑡 · (𝐵 − 𝐴))) ↔ 𝑋 = (𝐴 + (𝑇 · (𝐵 − 𝐴)))))
52	51	anbi1d 637	. . . . . . . 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 732	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝐸 ∈ Constr)
55	32	unssbd 4123	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → {𝐸, 𝐹} ⊆ (𝐶‘𝑛))
56	54, 55	prssad 32617	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝐸 ∈ (𝐶‘𝑛))
57		constrlccllem.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ Constr)
58	57	ad2antrr 732	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝐹 ∈ Constr)
59	58, 55	prssbd 32618	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝐹 ∈ (𝐶‘𝑛))
60		constrlccllem.t	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ ℝ)
61	60	ad2antrr 732	. . . . . . . 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 732	. . . . . . . . 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 732	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → (abs‘(𝑋 − 𝐺)) = (abs‘(𝐸 − 𝐹)))
66	63, 65	jca 516	. . . . . . . 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 3578	. . . . . . 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 3578	. . . . . 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 1344	. . . . 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 7812	. . . . . . 7 ⊢ (𝑛 ∈ ω → 𝑛 ∈ On)
72	71	ad2antlr 733	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝑛 ∈ On)
73		eqid 2739	. . . . . 6 ⊢ (𝐶‘𝑛) = (𝐶‘𝑛)
74	70, 72, 73	constrsuc 33922	. . . . 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 719	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝑋 ∈ (𝐶‘suc 𝑛))
76	3, 6, 75	rspcedvd 3562	. . 3 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → ∃𝑚 ∈ ω 𝑋 ∈ (𝐶‘𝑚))
77	70	isconstr 33920	. . 3 ⊢ (𝑋 ∈ Constr ↔ ∃𝑚 ∈ ω 𝑋 ∈ (𝐶‘𝑚))
78	76, 77	sylibr 235	. 2 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝑋 ∈ Constr)
79	30, 35, 38	tpssd 32626	. . . 4 ⊢ (𝜑 → {𝐴, 𝐵, 𝐺} ⊆ Constr)
80	53, 57	prssd 4753	. . . 4 ⊢ (𝜑 → {𝐸, 𝐹} ⊆ Constr)
81	79, 80	unssd 4121	. . 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 33935	. 2 ⊢ (𝜑 → ∃𝑛 ∈ ω ({𝐴, 𝐵, 𝐺} ∪ {𝐸, 𝐹}) ⊆ (𝐶‘𝑛))
88	78, 87	r19.29a 3147	1 ⊢ (𝜑 → 𝑋 ∈ Constr)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∨ w3o 1091   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  ∃wrex 3063  {crab 3391  Vcvv 3431   ∪ cun 3881   ⊆ wss 3883  {cpr 4557  {ctp 4559   ↦ cmpt 5153  Oncon0 6310  suc csuc 6312  ‘cfv 6485  (class class class)co 7356  ωcom 7806  reccrdg 8338  Fincfn 8883  ℂcc 11027  ℝcr 11028  0cc0 11029  1c1 11030   + caddc 11032   · cmul 11034   − cmin 11368  ∗ccj 15049  ℑcim 15051  abscabs 15187  Constrcconstr 33913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-tp 4560  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-er 8633  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11172  df-mnf 11173  df-ltxr 11175  df-sub 11370  df-constr 33914

This theorem is referenced by:  constrlccl  33941