Theorem constr01

Description: 0 and 1 are in all steps of the construction of constructible points. (Contributed by Thierry Arnoux, 25-Jun-2025)

	Ref	Expression
Hypotheses	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 } )
	constrsscn.1	⊢ ( 𝜑 → 𝑁 ∈ On )
Assertion	constr01	⊢ ( 𝜑 → { 0 , 1 } ⊆ ( 𝐶 ‘ 𝑁 ) )

Proof

Step	Hyp	Ref	Expression
1		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 } )
2		constrsscn.1	⊢ ( 𝜑 → 𝑁 ∈ On )
3		fveq2	⊢ ( 𝑚 = ∅ → ( 𝐶 ‘ 𝑚 ) = ( 𝐶 ‘ ∅ ) )
4	3	sseq2d	⊢ ( 𝑚 = ∅ → ( { 0 , 1 } ⊆ ( 𝐶 ‘ 𝑚 ) ↔ { 0 , 1 } ⊆ ( 𝐶 ‘ ∅ ) ) )
5		fveq2	⊢ ( 𝑚 = 𝑛 → ( 𝐶 ‘ 𝑚 ) = ( 𝐶 ‘ 𝑛 ) )
6	5	sseq2d	⊢ ( 𝑚 = 𝑛 → ( { 0 , 1 } ⊆ ( 𝐶 ‘ 𝑚 ) ↔ { 0 , 1 } ⊆ ( 𝐶 ‘ 𝑛 ) ) )
7		fveq2	⊢ ( 𝑚 = suc 𝑛 → ( 𝐶 ‘ 𝑚 ) = ( 𝐶 ‘ suc 𝑛 ) )
8	7	sseq2d	⊢ ( 𝑚 = suc 𝑛 → ( { 0 , 1 } ⊆ ( 𝐶 ‘ 𝑚 ) ↔ { 0 , 1 } ⊆ ( 𝐶 ‘ suc 𝑛 ) ) )
9		fveq2	⊢ ( 𝑚 = 𝑁 → ( 𝐶 ‘ 𝑚 ) = ( 𝐶 ‘ 𝑁 ) )
10	9	sseq2d	⊢ ( 𝑚 = 𝑁 → ( { 0 , 1 } ⊆ ( 𝐶 ‘ 𝑚 ) ↔ { 0 , 1 } ⊆ ( 𝐶 ‘ 𝑁 ) ) )
11	1	constr0	⊢ ( 𝐶 ‘ ∅ ) = { 0 , 1 }
12	11	eqimss2i	⊢ { 0 , 1 } ⊆ ( 𝐶 ‘ ∅ )
13		simpr	⊢ ( ( 𝑛 ∈ On ∧ { 0 , 1 } ⊆ ( 𝐶 ‘ 𝑛 ) ) → { 0 , 1 } ⊆ ( 𝐶 ‘ 𝑛 ) )
14		simpl	⊢ ( ( 𝑛 ∈ On ∧ { 0 , 1 } ⊆ ( 𝐶 ‘ 𝑛 ) ) → 𝑛 ∈ On )
15		0elpr01	⊢ 0 ∈ { 0 , 1 }
16	15	a1i	⊢ ( ( 𝑛 ∈ On ∧ { 0 , 1 } ⊆ ( 𝐶 ‘ 𝑛 ) ) → 0 ∈ { 0 , 1 } )
17	13 16	sseldd	⊢ ( ( 𝑛 ∈ On ∧ { 0 , 1 } ⊆ ( 𝐶 ‘ 𝑛 ) ) → 0 ∈ ( 𝐶 ‘ 𝑛 ) )
18	1 14 17	constrsslem	⊢ ( ( 𝑛 ∈ On ∧ { 0 , 1 } ⊆ ( 𝐶 ‘ 𝑛 ) ) → ( 𝐶 ‘ 𝑛 ) ⊆ ( 𝐶 ‘ suc 𝑛 ) )
19	13 18	sstrd	⊢ ( ( 𝑛 ∈ On ∧ { 0 , 1 } ⊆ ( 𝐶 ‘ 𝑛 ) ) → { 0 , 1 } ⊆ ( 𝐶 ‘ suc 𝑛 ) )
20	19	ex	⊢ ( 𝑛 ∈ On → ( { 0 , 1 } ⊆ ( 𝐶 ‘ 𝑛 ) → { 0 , 1 } ⊆ ( 𝐶 ‘ suc 𝑛 ) ) )
21		0ellim	⊢ ( Lim 𝑚 → ∅ ∈ 𝑚 )
22		fveq2	⊢ ( 𝑜 = ∅ → ( 𝐶 ‘ 𝑜 ) = ( 𝐶 ‘ ∅ ) )
23	22 11	eqtrdi	⊢ ( 𝑜 = ∅ → ( 𝐶 ‘ 𝑜 ) = { 0 , 1 } )
24	23	ssiun2s	⊢ ( ∅ ∈ 𝑚 → { 0 , 1 } ⊆ ∪ 𝑜 ∈ 𝑚 ( 𝐶 ‘ 𝑜 ) )
25	21 24	syl	⊢ ( Lim 𝑚 → { 0 , 1 } ⊆ ∪ 𝑜 ∈ 𝑚 ( 𝐶 ‘ 𝑜 ) )
26		vex	⊢ 𝑚 ∈ V
27	26	a1i	⊢ ( Lim 𝑚 → 𝑚 ∈ V )
28		id	⊢ ( Lim 𝑚 → Lim 𝑚 )
29	1 27 28	constrlim	⊢ ( Lim 𝑚 → ( 𝐶 ‘ 𝑚 ) = ∪ 𝑜 ∈ 𝑚 ( 𝐶 ‘ 𝑜 ) )
30	25 29	sseqtrrd	⊢ ( Lim 𝑚 → { 0 , 1 } ⊆ ( 𝐶 ‘ 𝑚 ) )
31	30	a1d	⊢ ( Lim 𝑚 → ( ∀ 𝑛 ∈ 𝑚 { 0 , 1 } ⊆ ( 𝐶 ‘ 𝑛 ) → { 0 , 1 } ⊆ ( 𝐶 ‘ 𝑚 ) ) )
32	4 6 8 10 12 20 31	tfinds	⊢ ( 𝑁 ∈ On → { 0 , 1 } ⊆ ( 𝐶 ‘ 𝑁 ) )
33	2 32	syl	⊢ ( 𝜑 → { 0 , 1 } ⊆ ( 𝐶 ‘ 𝑁 ) )