Theorem constrcccllem 33945

Description: Constructible numbers are closed under circle-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})
constrcccllem.a	⊢ (𝜑 → 𝐴 ∈ Constr)
constrcccllem.b	⊢ (𝜑 → 𝐵 ∈ Constr)
constrcccllem.c	⊢ (𝜑 → 𝐺 ∈ Constr)
constrcccllem.d	⊢ (𝜑 → 𝐷 ∈ Constr)
constrcccllem.e	⊢ (𝜑 → 𝐸 ∈ Constr)
constrcccllem.f	⊢ (𝜑 → 𝐹 ∈ Constr)
constrcccllem.x	⊢ (𝜑 → 𝑋 ∈ ℂ)
constrcccllem.1	⊢ (𝜑 → 𝐴 ≠ 𝐷)
constrcccllem.2	⊢ (𝜑 → (abs‘(𝑋 − 𝐴)) = (abs‘(𝐵 − 𝐺)))
constrcccllem.3	⊢ (𝜑 → (abs‘(𝑋 − 𝐷)) = (abs‘(𝐸 − 𝐹)))

Assertion

Ref	Expression
constrcccllem	⊢ (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   𝐷,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   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)   𝐷(𝑟)   𝐸(𝑟,𝑑)   𝐹(𝑟,𝑑)   𝐺(𝑟)   𝑋(𝑥,𝑠)

Proof of Theorem constrcccllem

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

Step	Hyp	Ref	Expression
1		peano2b 7830	. . . . . 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} ∪ {𝐷, 𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → suc 𝑛 ∈ ω)
4		fveq2 6834	. . . . . 6 ⊢ (𝑚 = suc 𝑛 → (𝐶‘𝑚) = (𝐶‘suc 𝑛))
5	4	eleq2d 2826	. . . . 5 ⊢ (𝑚 = suc 𝑛 → (𝑋 ∈ (𝐶‘𝑚) ↔ 𝑋 ∈ (𝐶‘suc 𝑛)))
6	5	adantl 482	. . . 4 ⊢ ((((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐷, 𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) ∧ 𝑚 = suc 𝑛) → (𝑋 ∈ (𝐶‘𝑚) ↔ 𝑋 ∈ (𝐶‘suc 𝑛)))
7		constrcccllem.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ ℂ)
8	7	ad2antrr 732	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐷, 𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝑋 ∈ ℂ)
9		neeq1 2997	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → (𝑎 ≠ 𝑑 ↔ 𝐴 ≠ 𝑑))
10		oveq2 7371	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐴 → (𝑋 − 𝑎) = (𝑋 − 𝐴))
11	10	fveq2d 6838	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (abs‘(𝑋 − 𝑎)) = (abs‘(𝑋 − 𝐴)))
12	11	eqeq1d 2742	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → ((abs‘(𝑋 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ↔ (abs‘(𝑋 − 𝐴)) = (abs‘(𝑏 − 𝑐))))
13	9, 12	3anbi12d 1445	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → ((𝑎 ≠ 𝑑 ∧ (abs‘(𝑋 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑋 − 𝑑)) = (abs‘(𝑒 − 𝑓))) ↔ (𝐴 ≠ 𝑑 ∧ (abs‘(𝑋 − 𝐴)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑋 − 𝑑)) = (abs‘(𝑒 − 𝑓)))))
14	13	rexbidv 3164	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (∃𝑓 ∈ (𝐶‘𝑛)(𝑎 ≠ 𝑑 ∧ (abs‘(𝑋 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑋 − 𝑑)) = (abs‘(𝑒 − 𝑓))) ↔ ∃𝑓 ∈ (𝐶‘𝑛)(𝐴 ≠ 𝑑 ∧ (abs‘(𝑋 − 𝐴)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑋 − 𝑑)) = (abs‘(𝑒 − 𝑓)))))
15	14	2rexbidv 3205	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (∃𝑑 ∈ (𝐶‘𝑛)∃𝑒 ∈ (𝐶‘𝑛)∃𝑓 ∈ (𝐶‘𝑛)(𝑎 ≠ 𝑑 ∧ (abs‘(𝑋 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑋 − 𝑑)) = (abs‘(𝑒 − 𝑓))) ↔ ∃𝑑 ∈ (𝐶‘𝑛)∃𝑒 ∈ (𝐶‘𝑛)∃𝑓 ∈ (𝐶‘𝑛)(𝐴 ≠ 𝑑 ∧ (abs‘(𝑋 − 𝐴)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑋 − 𝑑)) = (abs‘(𝑒 − 𝑓)))))
16		oveq1 7370	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝐵 → (𝑏 − 𝑐) = (𝐵 − 𝑐))
17	16	fveq2d 6838	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐵 → (abs‘(𝑏 − 𝑐)) = (abs‘(𝐵 − 𝑐)))
18	17	eqeq2d 2751	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → ((abs‘(𝑋 − 𝐴)) = (abs‘(𝑏 − 𝑐)) ↔ (abs‘(𝑋 − 𝐴)) = (abs‘(𝐵 − 𝑐))))
19	18	3anbi2d 1449	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → ((𝐴 ≠ 𝑑 ∧ (abs‘(𝑋 − 𝐴)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑋 − 𝑑)) = (abs‘(𝑒 − 𝑓))) ↔ (𝐴 ≠ 𝑑 ∧ (abs‘(𝑋 − 𝐴)) = (abs‘(𝐵 − 𝑐)) ∧ (abs‘(𝑋 − 𝑑)) = (abs‘(𝑒 − 𝑓)))))
20	19	rexbidv 3164	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → (∃𝑓 ∈ (𝐶‘𝑛)(𝐴 ≠ 𝑑 ∧ (abs‘(𝑋 − 𝐴)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑋 − 𝑑)) = (abs‘(𝑒 − 𝑓))) ↔ ∃𝑓 ∈ (𝐶‘𝑛)(𝐴 ≠ 𝑑 ∧ (abs‘(𝑋 − 𝐴)) = (abs‘(𝐵 − 𝑐)) ∧ (abs‘(𝑋 − 𝑑)) = (abs‘(𝑒 − 𝑓)))))
21	20	2rexbidv 3205	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (∃𝑑 ∈ (𝐶‘𝑛)∃𝑒 ∈ (𝐶‘𝑛)∃𝑓 ∈ (𝐶‘𝑛)(𝐴 ≠ 𝑑 ∧ (abs‘(𝑋 − 𝐴)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑋 − 𝑑)) = (abs‘(𝑒 − 𝑓))) ↔ ∃𝑑 ∈ (𝐶‘𝑛)∃𝑒 ∈ (𝐶‘𝑛)∃𝑓 ∈ (𝐶‘𝑛)(𝐴 ≠ 𝑑 ∧ (abs‘(𝑋 − 𝐴)) = (abs‘(𝐵 − 𝑐)) ∧ (abs‘(𝑋 − 𝑑)) = (abs‘(𝑒 − 𝑓)))))
22		oveq2 7371	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝐺 → (𝐵 − 𝑐) = (𝐵 − 𝐺))
23	22	fveq2d 6838	. . . . . . . . . . 11 ⊢ (𝑐 = 𝐺 → (abs‘(𝐵 − 𝑐)) = (abs‘(𝐵 − 𝐺)))
24	23	eqeq2d 2751	. . . . . . . . . 10 ⊢ (𝑐 = 𝐺 → ((abs‘(𝑋 − 𝐴)) = (abs‘(𝐵 − 𝑐)) ↔ (abs‘(𝑋 − 𝐴)) = (abs‘(𝐵 − 𝐺))))
25	24	3anbi2d 1449	. . . . . . . . 9 ⊢ (𝑐 = 𝐺 → ((𝐴 ≠ 𝑑 ∧ (abs‘(𝑋 − 𝐴)) = (abs‘(𝐵 − 𝑐)) ∧ (abs‘(𝑋 − 𝑑)) = (abs‘(𝑒 − 𝑓))) ↔ (𝐴 ≠ 𝑑 ∧ (abs‘(𝑋 − 𝐴)) = (abs‘(𝐵 − 𝐺)) ∧ (abs‘(𝑋 − 𝑑)) = (abs‘(𝑒 − 𝑓)))))
26	25	rexbidv 3164	. . . . . . . 8 ⊢ (𝑐 = 𝐺 → (∃𝑓 ∈ (𝐶‘𝑛)(𝐴 ≠ 𝑑 ∧ (abs‘(𝑋 − 𝐴)) = (abs‘(𝐵 − 𝑐)) ∧ (abs‘(𝑋 − 𝑑)) = (abs‘(𝑒 − 𝑓))) ↔ ∃𝑓 ∈ (𝐶‘𝑛)(𝐴 ≠ 𝑑 ∧ (abs‘(𝑋 − 𝐴)) = (abs‘(𝐵 − 𝐺)) ∧ (abs‘(𝑋 − 𝑑)) = (abs‘(𝑒 − 𝑓)))))
27	26	2rexbidv 3205	. . . . . . 7 ⊢ (𝑐 = 𝐺 → (∃𝑑 ∈ (𝐶‘𝑛)∃𝑒 ∈ (𝐶‘𝑛)∃𝑓 ∈ (𝐶‘𝑛)(𝐴 ≠ 𝑑 ∧ (abs‘(𝑋 − 𝐴)) = (abs‘(𝐵 − 𝑐)) ∧ (abs‘(𝑋 − 𝑑)) = (abs‘(𝑒 − 𝑓))) ↔ ∃𝑑 ∈ (𝐶‘𝑛)∃𝑒 ∈ (𝐶‘𝑛)∃𝑓 ∈ (𝐶‘𝑛)(𝐴 ≠ 𝑑 ∧ (abs‘(𝑋 − 𝐴)) = (abs‘(𝐵 − 𝐺)) ∧ (abs‘(𝑋 − 𝑑)) = (abs‘(𝑒 − 𝑓)))))
28		constrcccllem.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ Constr)
29	28	ad2antrr 732	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐷, 𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝐴 ∈ Constr)
30		simpr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐷, 𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → ({𝐴, 𝐵, 𝐺} ∪ {𝐷, 𝐸, 𝐹}) ⊆ (𝐶‘𝑛))
31	30	unssad 4129	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐷, 𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → {𝐴, 𝐵, 𝐺} ⊆ (𝐶‘𝑛))
32	29, 31	tpssad 32634	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐷, 𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝐴 ∈ (𝐶‘𝑛))
33		constrcccllem.b	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ Constr)
34	33	ad2antrr 732	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐷, 𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝐵 ∈ Constr)
35	34, 31	tpssbd 32635	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐷, 𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝐵 ∈ (𝐶‘𝑛))
36		constrcccllem.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ Constr)
37	36	ad2antrr 732	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐷, 𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝐺 ∈ Constr)
38	37, 31	tpsscd 32636	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐷, 𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝐺 ∈ (𝐶‘𝑛))
39		neeq2 2998	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → (𝐴 ≠ 𝑑 ↔ 𝐴 ≠ 𝐷))
40		oveq2 7371	. . . . . . . . . . 11 ⊢ (𝑑 = 𝐷 → (𝑋 − 𝑑) = (𝑋 − 𝐷))
41	40	fveq2d 6838	. . . . . . . . . 10 ⊢ (𝑑 = 𝐷 → (abs‘(𝑋 − 𝑑)) = (abs‘(𝑋 − 𝐷)))
42	41	eqeq1d 2742	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → ((abs‘(𝑋 − 𝑑)) = (abs‘(𝑒 − 𝑓)) ↔ (abs‘(𝑋 − 𝐷)) = (abs‘(𝑒 − 𝑓))))
43	39, 42	3anbi13d 1446	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → ((𝐴 ≠ 𝑑 ∧ (abs‘(𝑋 − 𝐴)) = (abs‘(𝐵 − 𝐺)) ∧ (abs‘(𝑋 − 𝑑)) = (abs‘(𝑒 − 𝑓))) ↔ (𝐴 ≠ 𝐷 ∧ (abs‘(𝑋 − 𝐴)) = (abs‘(𝐵 − 𝐺)) ∧ (abs‘(𝑋 − 𝐷)) = (abs‘(𝑒 − 𝑓)))))
44		oveq1 7370	. . . . . . . . . . 11 ⊢ (𝑒 = 𝐸 → (𝑒 − 𝑓) = (𝐸 − 𝑓))
45	44	fveq2d 6838	. . . . . . . . . 10 ⊢ (𝑒 = 𝐸 → (abs‘(𝑒 − 𝑓)) = (abs‘(𝐸 − 𝑓)))
46	45	eqeq2d 2751	. . . . . . . . 9 ⊢ (𝑒 = 𝐸 → ((abs‘(𝑋 − 𝐷)) = (abs‘(𝑒 − 𝑓)) ↔ (abs‘(𝑋 − 𝐷)) = (abs‘(𝐸 − 𝑓))))
47	46	3anbi3d 1450	. . . . . . . 8 ⊢ (𝑒 = 𝐸 → ((𝐴 ≠ 𝐷 ∧ (abs‘(𝑋 − 𝐴)) = (abs‘(𝐵 − 𝐺)) ∧ (abs‘(𝑋 − 𝐷)) = (abs‘(𝑒 − 𝑓))) ↔ (𝐴 ≠ 𝐷 ∧ (abs‘(𝑋 − 𝐴)) = (abs‘(𝐵 − 𝐺)) ∧ (abs‘(𝑋 − 𝐷)) = (abs‘(𝐸 − 𝑓)))))
48		oveq2 7371	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → (𝐸 − 𝑓) = (𝐸 − 𝐹))
49	48	fveq2d 6838	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (abs‘(𝐸 − 𝑓)) = (abs‘(𝐸 − 𝐹)))
50	49	eqeq2d 2751	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → ((abs‘(𝑋 − 𝐷)) = (abs‘(𝐸 − 𝑓)) ↔ (abs‘(𝑋 − 𝐷)) = (abs‘(𝐸 − 𝐹))))
51	50	3anbi3d 1450	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((𝐴 ≠ 𝐷 ∧ (abs‘(𝑋 − 𝐴)) = (abs‘(𝐵 − 𝐺)) ∧ (abs‘(𝑋 − 𝐷)) = (abs‘(𝐸 − 𝑓))) ↔ (𝐴 ≠ 𝐷 ∧ (abs‘(𝑋 − 𝐴)) = (abs‘(𝐵 − 𝐺)) ∧ (abs‘(𝑋 − 𝐷)) = (abs‘(𝐸 − 𝐹)))))
52		constrcccllem.d	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ Constr)
53	52	ad2antrr 732	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐷, 𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝐷 ∈ Constr)
54	30	unssbd 4130	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐷, 𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → {𝐷, 𝐸, 𝐹} ⊆ (𝐶‘𝑛))
55	53, 54	tpssad 32634	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐷, 𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝐷 ∈ (𝐶‘𝑛))
56		constrcccllem.e	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ Constr)
57	56	ad2antrr 732	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐷, 𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝐸 ∈ Constr)
58	57, 54	tpssbd 32635	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐷, 𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝐸 ∈ (𝐶‘𝑛))
59		constrcccllem.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ Constr)
60	59	ad2antrr 732	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐷, 𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝐹 ∈ Constr)
61	60, 54	tpsscd 32636	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐷, 𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝐹 ∈ (𝐶‘𝑛))
62		constrcccllem.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ≠ 𝐷)
63	62	ad2antrr 732	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐷, 𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝐴 ≠ 𝐷)
64		constrcccllem.2	. . . . . . . . . 10 ⊢ (𝜑 → (abs‘(𝑋 − 𝐴)) = (abs‘(𝐵 − 𝐺)))
65	64	ad2antrr 732	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐷, 𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → (abs‘(𝑋 − 𝐴)) = (abs‘(𝐵 − 𝐺)))
66		constrcccllem.3	. . . . . . . . . 10 ⊢ (𝜑 → (abs‘(𝑋 − 𝐷)) = (abs‘(𝐸 − 𝐹)))
67	66	ad2antrr 732	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐷, 𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → (abs‘(𝑋 − 𝐷)) = (abs‘(𝐸 − 𝐹)))
68	63, 65, 67	3jca 1134	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐷, 𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → (𝐴 ≠ 𝐷 ∧ (abs‘(𝑋 − 𝐴)) = (abs‘(𝐵 − 𝐺)) ∧ (abs‘(𝑋 − 𝐷)) = (abs‘(𝐸 − 𝐹))))
69	43, 47, 51, 55, 58, 61, 68	3rspcedvdw 3585	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐷, 𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → ∃𝑑 ∈ (𝐶‘𝑛)∃𝑒 ∈ (𝐶‘𝑛)∃𝑓 ∈ (𝐶‘𝑛)(𝐴 ≠ 𝑑 ∧ (abs‘(𝑋 − 𝐴)) = (abs‘(𝐵 − 𝐺)) ∧ (abs‘(𝑋 − 𝑑)) = (abs‘(𝑒 − 𝑓))))
70	15, 21, 27, 32, 35, 38, 69	3rspcedvdw 3585	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐷, 𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → ∃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))))
71	70	3mix3d 1345	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐷, 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)))))
72		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})
73		nnon 7819	. . . . . . 7 ⊢ (𝑛 ∈ ω → 𝑛 ∈ On)
74	73	ad2antlr 733	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐷, 𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝑛 ∈ On)
75		eqid 2740	. . . . . 6 ⊢ (𝐶‘𝑛) = (𝐶‘𝑛)
76	72, 74, 75	constrsuc 33929	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐷, 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)))))))
77	8, 71, 76	mpbir2and 719	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐷, 𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝑋 ∈ (𝐶‘suc 𝑛))
78	3, 6, 77	rspcedvd 3569	. . 3 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐷, 𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → ∃𝑚 ∈ ω 𝑋 ∈ (𝐶‘𝑚))
79	72	isconstr 33927	. . 3 ⊢ (𝑋 ∈ Constr ↔ ∃𝑚 ∈ ω 𝑋 ∈ (𝐶‘𝑚))
80	78, 79	sylibr 235	. 2 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ({𝐴, 𝐵, 𝐺} ∪ {𝐷, 𝐸, 𝐹}) ⊆ (𝐶‘𝑛)) → 𝑋 ∈ Constr)
81	28, 33, 36	tpssd 32633	. . . 4 ⊢ (𝜑 → {𝐴, 𝐵, 𝐺} ⊆ Constr)
82	52, 56, 59	tpssd 32633	. . . 4 ⊢ (𝜑 → {𝐷, 𝐸, 𝐹} ⊆ Constr)
83	81, 82	unssd 4128	. . 3 ⊢ (𝜑 → ({𝐴, 𝐵, 𝐺} ∪ {𝐷, 𝐸, 𝐹}) ⊆ Constr)
84		tpfi 9233	. . . . 5 ⊢ {𝐴, 𝐵, 𝐺} ∈ Fin
85	84	a1i 11	. . . 4 ⊢ (𝜑 → {𝐴, 𝐵, 𝐺} ∈ Fin)
86		tpfi 9233	. . . . 5 ⊢ {𝐷, 𝐸, 𝐹} ∈ Fin
87	86	a1i 11	. . . 4 ⊢ (𝜑 → {𝐷, 𝐸, 𝐹} ∈ Fin)
88	85, 87	unfid 9103	. . 3 ⊢ (𝜑 → ({𝐴, 𝐵, 𝐺} ∪ {𝐷, 𝐸, 𝐹}) ∈ Fin)
89	72, 83, 88	constrfiss 33942	. 2 ⊢ (𝜑 → ∃𝑛 ∈ ω ({𝐴, 𝐵, 𝐺} ∪ {𝐷, 𝐸, 𝐹}) ⊆ (𝐶‘𝑛))
90	80, 89	r19.29a 3148	1 ⊢ (𝜑 → 𝑋 ∈ Constr)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∨ w3o 1091   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ≠ wne 2935  ∃wrex 3064  {crab 3392  Vcvv 3432   ∪ cun 3888   ⊆ wss 3890  {cpr 4564  {ctp 4566   ↦ cmpt 5160  Oncon0 6317  suc csuc 6319  ‘cfv 6492  (class class class)co 7363  ωcom 7813  reccrdg 8345  Fincfn 8890  ℂcc 11034  ℝcr 11035  0cc0 11036  1c1 11037   + caddc 11039   · cmul 11041   − cmin 11375  ∗ccj 15056  ℑcim 15058  abscabs 15194  Constrcconstr 33920

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 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112

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 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  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 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-2o 8403  df-er 8640  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-pnf 11179  df-mnf 11180  df-ltxr 11182  df-sub 11377  df-constr 33921

This theorem is referenced by:  constrcccl  33949