Metamath Proof Explorer


Theorem constrsscn

Description: Closure of the constructible points in the complex numbers. (Contributed by Thierry Arnoux, 25-Jun-2025)

Ref Expression
Hypotheses constr0.1 𝐶 = rec ( ( 𝑠 ∈ V ↦ { 𝑥 ∈ ℂ ∣ ( ∃ 𝑎𝑠𝑏𝑠𝑐𝑠𝑑𝑠𝑡 ∈ ℝ ∃ 𝑟 ∈ ℝ ( 𝑥 = ( 𝑎 + ( 𝑡 · ( 𝑏𝑎 ) ) ) ∧ 𝑥 = ( 𝑐 + ( 𝑟 · ( 𝑑𝑐 ) ) ) ∧ ( ℑ ‘ ( ( ∗ ‘ ( 𝑏𝑎 ) ) · ( 𝑑𝑐 ) ) ) ≠ 0 ) ∨ ∃ 𝑎𝑠𝑏𝑠𝑐𝑠𝑒𝑠𝑓𝑠𝑡 ∈ ℝ ( 𝑥 = ( 𝑎 + ( 𝑡 · ( 𝑏𝑎 ) ) ) ∧ ( abs ‘ ( 𝑥𝑐 ) ) = ( abs ‘ ( 𝑒𝑓 ) ) ) ∨ ∃ 𝑎𝑠𝑏𝑠𝑐𝑠𝑑𝑠𝑒𝑠𝑓𝑠 ( 𝑎𝑑 ∧ ( abs ‘ ( 𝑥𝑎 ) ) = ( abs ‘ ( 𝑏𝑐 ) ) ∧ ( abs ‘ ( 𝑥𝑑 ) ) = ( abs ‘ ( 𝑒𝑓 ) ) ) ) } ) , { 0 , 1 } )
constrsscn.1 ( 𝜑𝑁 ∈ On )
Assertion constrsscn ( 𝜑 → ( 𝐶𝑁 ) ⊆ ℂ )

Proof

Step Hyp Ref Expression
1 constr0.1 𝐶 = rec ( ( 𝑠 ∈ V ↦ { 𝑥 ∈ ℂ ∣ ( ∃ 𝑎𝑠𝑏𝑠𝑐𝑠𝑑𝑠𝑡 ∈ ℝ ∃ 𝑟 ∈ ℝ ( 𝑥 = ( 𝑎 + ( 𝑡 · ( 𝑏𝑎 ) ) ) ∧ 𝑥 = ( 𝑐 + ( 𝑟 · ( 𝑑𝑐 ) ) ) ∧ ( ℑ ‘ ( ( ∗ ‘ ( 𝑏𝑎 ) ) · ( 𝑑𝑐 ) ) ) ≠ 0 ) ∨ ∃ 𝑎𝑠𝑏𝑠𝑐𝑠𝑒𝑠𝑓𝑠𝑡 ∈ ℝ ( 𝑥 = ( 𝑎 + ( 𝑡 · ( 𝑏𝑎 ) ) ) ∧ ( abs ‘ ( 𝑥𝑐 ) ) = ( abs ‘ ( 𝑒𝑓 ) ) ) ∨ ∃ 𝑎𝑠𝑏𝑠𝑐𝑠𝑑𝑠𝑒𝑠𝑓𝑠 ( 𝑎𝑑 ∧ ( abs ‘ ( 𝑥𝑎 ) ) = ( abs ‘ ( 𝑏𝑐 ) ) ∧ ( abs ‘ ( 𝑥𝑑 ) ) = ( abs ‘ ( 𝑒𝑓 ) ) ) ) } ) , { 0 , 1 } )
2 constrsscn.1 ( 𝜑𝑁 ∈ On )
3 fveq2 ( 𝑚 = ∅ → ( 𝐶𝑚 ) = ( 𝐶 ‘ ∅ ) )
4 3 sseq1d ( 𝑚 = ∅ → ( ( 𝐶𝑚 ) ⊆ ℂ ↔ ( 𝐶 ‘ ∅ ) ⊆ ℂ ) )
5 fveq2 ( 𝑚 = 𝑛 → ( 𝐶𝑚 ) = ( 𝐶𝑛 ) )
6 5 sseq1d ( 𝑚 = 𝑛 → ( ( 𝐶𝑚 ) ⊆ ℂ ↔ ( 𝐶𝑛 ) ⊆ ℂ ) )
7 fveq2 ( 𝑚 = suc 𝑛 → ( 𝐶𝑚 ) = ( 𝐶 ‘ suc 𝑛 ) )
8 7 sseq1d ( 𝑚 = suc 𝑛 → ( ( 𝐶𝑚 ) ⊆ ℂ ↔ ( 𝐶 ‘ suc 𝑛 ) ⊆ ℂ ) )
9 fveq2 ( 𝑚 = 𝑁 → ( 𝐶𝑚 ) = ( 𝐶𝑁 ) )
10 9 sseq1d ( 𝑚 = 𝑁 → ( ( 𝐶𝑚 ) ⊆ ℂ ↔ ( 𝐶𝑁 ) ⊆ ℂ ) )
11 1 constr0 ( 𝐶 ‘ ∅ ) = { 0 , 1 }
12 0cn 0 ∈ ℂ
13 ax-1cn 1 ∈ ℂ
14 prssi ( ( 0 ∈ ℂ ∧ 1 ∈ ℂ ) → { 0 , 1 } ⊆ ℂ )
15 12 13 14 mp2an { 0 , 1 } ⊆ ℂ
16 11 15 eqsstri ( 𝐶 ‘ ∅ ) ⊆ ℂ
17 simpl ( ( 𝑛 ∈ On ∧ ( 𝐶𝑛 ) ⊆ ℂ ) → 𝑛 ∈ On )
18 eqid ( 𝐶𝑛 ) = ( 𝐶𝑛 )
19 1 17 18 constrsuc ( ( 𝑛 ∈ On ∧ ( 𝐶𝑛 ) ⊆ ℂ ) → ( 𝑥 ∈ ( 𝐶 ‘ suc 𝑛 ) ↔ ( 𝑥 ∈ ℂ ∧ ( ∃ 𝑎 ∈ ( 𝐶𝑛 ) ∃ 𝑏 ∈ ( 𝐶𝑛 ) ∃ 𝑐 ∈ ( 𝐶𝑛 ) ∃ 𝑑 ∈ ( 𝐶𝑛 ) ∃ 𝑡 ∈ ℝ ∃ 𝑟 ∈ ℝ ( 𝑥 = ( 𝑎 + ( 𝑡 · ( 𝑏𝑎 ) ) ) ∧ 𝑥 = ( 𝑐 + ( 𝑟 · ( 𝑑𝑐 ) ) ) ∧ ( ℑ ‘ ( ( ∗ ‘ ( 𝑏𝑎 ) ) · ( 𝑑𝑐 ) ) ) ≠ 0 ) ∨ ∃ 𝑎 ∈ ( 𝐶𝑛 ) ∃ 𝑏 ∈ ( 𝐶𝑛 ) ∃ 𝑐 ∈ ( 𝐶𝑛 ) ∃ 𝑒 ∈ ( 𝐶𝑛 ) ∃ 𝑓 ∈ ( 𝐶𝑛 ) ∃ 𝑡 ∈ ℝ ( 𝑥 = ( 𝑎 + ( 𝑡 · ( 𝑏𝑎 ) ) ) ∧ ( abs ‘ ( 𝑥𝑐 ) ) = ( abs ‘ ( 𝑒𝑓 ) ) ) ∨ ∃ 𝑎 ∈ ( 𝐶𝑛 ) ∃ 𝑏 ∈ ( 𝐶𝑛 ) ∃ 𝑐 ∈ ( 𝐶𝑛 ) ∃ 𝑑 ∈ ( 𝐶𝑛 ) ∃ 𝑒 ∈ ( 𝐶𝑛 ) ∃ 𝑓 ∈ ( 𝐶𝑛 ) ( 𝑎𝑑 ∧ ( abs ‘ ( 𝑥𝑎 ) ) = ( abs ‘ ( 𝑏𝑐 ) ) ∧ ( abs ‘ ( 𝑥𝑑 ) ) = ( abs ‘ ( 𝑒𝑓 ) ) ) ) ) ) )
20 19 biimpa ( ( ( 𝑛 ∈ On ∧ ( 𝐶𝑛 ) ⊆ ℂ ) ∧ 𝑥 ∈ ( 𝐶 ‘ suc 𝑛 ) ) → ( 𝑥 ∈ ℂ ∧ ( ∃ 𝑎 ∈ ( 𝐶𝑛 ) ∃ 𝑏 ∈ ( 𝐶𝑛 ) ∃ 𝑐 ∈ ( 𝐶𝑛 ) ∃ 𝑑 ∈ ( 𝐶𝑛 ) ∃ 𝑡 ∈ ℝ ∃ 𝑟 ∈ ℝ ( 𝑥 = ( 𝑎 + ( 𝑡 · ( 𝑏𝑎 ) ) ) ∧ 𝑥 = ( 𝑐 + ( 𝑟 · ( 𝑑𝑐 ) ) ) ∧ ( ℑ ‘ ( ( ∗ ‘ ( 𝑏𝑎 ) ) · ( 𝑑𝑐 ) ) ) ≠ 0 ) ∨ ∃ 𝑎 ∈ ( 𝐶𝑛 ) ∃ 𝑏 ∈ ( 𝐶𝑛 ) ∃ 𝑐 ∈ ( 𝐶𝑛 ) ∃ 𝑒 ∈ ( 𝐶𝑛 ) ∃ 𝑓 ∈ ( 𝐶𝑛 ) ∃ 𝑡 ∈ ℝ ( 𝑥 = ( 𝑎 + ( 𝑡 · ( 𝑏𝑎 ) ) ) ∧ ( abs ‘ ( 𝑥𝑐 ) ) = ( abs ‘ ( 𝑒𝑓 ) ) ) ∨ ∃ 𝑎 ∈ ( 𝐶𝑛 ) ∃ 𝑏 ∈ ( 𝐶𝑛 ) ∃ 𝑐 ∈ ( 𝐶𝑛 ) ∃ 𝑑 ∈ ( 𝐶𝑛 ) ∃ 𝑒 ∈ ( 𝐶𝑛 ) ∃ 𝑓 ∈ ( 𝐶𝑛 ) ( 𝑎𝑑 ∧ ( abs ‘ ( 𝑥𝑎 ) ) = ( abs ‘ ( 𝑏𝑐 ) ) ∧ ( abs ‘ ( 𝑥𝑑 ) ) = ( abs ‘ ( 𝑒𝑓 ) ) ) ) ) )
21 20 simpld ( ( ( 𝑛 ∈ On ∧ ( 𝐶𝑛 ) ⊆ ℂ ) ∧ 𝑥 ∈ ( 𝐶 ‘ suc 𝑛 ) ) → 𝑥 ∈ ℂ )
22 21 ex ( ( 𝑛 ∈ On ∧ ( 𝐶𝑛 ) ⊆ ℂ ) → ( 𝑥 ∈ ( 𝐶 ‘ suc 𝑛 ) → 𝑥 ∈ ℂ ) )
23 22 ssrdv ( ( 𝑛 ∈ On ∧ ( 𝐶𝑛 ) ⊆ ℂ ) → ( 𝐶 ‘ suc 𝑛 ) ⊆ ℂ )
24 23 ex ( 𝑛 ∈ On → ( ( 𝐶𝑛 ) ⊆ ℂ → ( 𝐶 ‘ suc 𝑛 ) ⊆ ℂ ) )
25 vex 𝑚 ∈ V
26 25 a1i ( ( Lim 𝑚 ∧ ∀ 𝑛𝑚 ( 𝐶𝑛 ) ⊆ ℂ ) → 𝑚 ∈ V )
27 simpl ( ( Lim 𝑚 ∧ ∀ 𝑛𝑚 ( 𝐶𝑛 ) ⊆ ℂ ) → Lim 𝑚 )
28 1 26 27 constrlim ( ( Lim 𝑚 ∧ ∀ 𝑛𝑚 ( 𝐶𝑛 ) ⊆ ℂ ) → ( 𝐶𝑚 ) = 𝑜𝑚 ( 𝐶𝑜 ) )
29 fveq2 ( 𝑛 = 𝑜 → ( 𝐶𝑛 ) = ( 𝐶𝑜 ) )
30 29 sseq1d ( 𝑛 = 𝑜 → ( ( 𝐶𝑛 ) ⊆ ℂ ↔ ( 𝐶𝑜 ) ⊆ ℂ ) )
31 simplr ( ( ( Lim 𝑚 ∧ ∀ 𝑛𝑚 ( 𝐶𝑛 ) ⊆ ℂ ) ∧ 𝑜𝑚 ) → ∀ 𝑛𝑚 ( 𝐶𝑛 ) ⊆ ℂ )
32 simpr ( ( ( Lim 𝑚 ∧ ∀ 𝑛𝑚 ( 𝐶𝑛 ) ⊆ ℂ ) ∧ 𝑜𝑚 ) → 𝑜𝑚 )
33 30 31 32 rspcdva ( ( ( Lim 𝑚 ∧ ∀ 𝑛𝑚 ( 𝐶𝑛 ) ⊆ ℂ ) ∧ 𝑜𝑚 ) → ( 𝐶𝑜 ) ⊆ ℂ )
34 33 iunssd ( ( Lim 𝑚 ∧ ∀ 𝑛𝑚 ( 𝐶𝑛 ) ⊆ ℂ ) → 𝑜𝑚 ( 𝐶𝑜 ) ⊆ ℂ )
35 28 34 eqsstrd ( ( Lim 𝑚 ∧ ∀ 𝑛𝑚 ( 𝐶𝑛 ) ⊆ ℂ ) → ( 𝐶𝑚 ) ⊆ ℂ )
36 35 ex ( Lim 𝑚 → ( ∀ 𝑛𝑚 ( 𝐶𝑛 ) ⊆ ℂ → ( 𝐶𝑚 ) ⊆ ℂ ) )
37 4 6 8 10 16 24 36 tfinds ( 𝑁 ∈ On → ( 𝐶𝑁 ) ⊆ ℂ )
38 2 37 syl ( 𝜑 → ( 𝐶𝑁 ) ⊆ ℂ )