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Theorem constrsscn 33745
Description: Closure of the constructible points in the complex numbers. (Contributed by Thierry Arnoux, 25-Jun-2025.)
Hypotheses
Ref Expression
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
Ref Expression
constrsscn (𝜑 → (𝐶𝑁) ⊆ ℂ)
Distinct variable groups:   𝐶,𝑎,𝑠,𝑥   𝐶,𝑏,𝑠,𝑥   𝐶,𝑐,𝑠,𝑥   𝐶,𝑑,𝑠,𝑥   𝐶,𝑒,𝑠,𝑥   𝐶,𝑓,𝑠,𝑥   𝑠,𝑟,𝑥   𝑡,𝑠,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑡,𝑒,𝑓,𝑠,𝑟,𝑎,𝑏,𝑐,𝑑)   𝐶(𝑡,𝑟)   𝑁(𝑥,𝑡,𝑒,𝑓,𝑠,𝑟,𝑎,𝑏,𝑐,𝑑)

Proof of Theorem constrsscn
Dummy variables 𝑛 𝑜 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 constrsscn.1 . 2 (𝜑𝑁 ∈ On)
2 fveq2 6907 . . . 4 (𝑚 = ∅ → (𝐶𝑚) = (𝐶‘∅))
32sseq1d 4027 . . 3 (𝑚 = ∅ → ((𝐶𝑚) ⊆ ℂ ↔ (𝐶‘∅) ⊆ ℂ))
4 fveq2 6907 . . . 4 (𝑚 = 𝑛 → (𝐶𝑚) = (𝐶𝑛))
54sseq1d 4027 . . 3 (𝑚 = 𝑛 → ((𝐶𝑚) ⊆ ℂ ↔ (𝐶𝑛) ⊆ ℂ))
6 fveq2 6907 . . . 4 (𝑚 = suc 𝑛 → (𝐶𝑚) = (𝐶‘suc 𝑛))
76sseq1d 4027 . . 3 (𝑚 = suc 𝑛 → ((𝐶𝑚) ⊆ ℂ ↔ (𝐶‘suc 𝑛) ⊆ ℂ))
8 fveq2 6907 . . . 4 (𝑚 = 𝑁 → (𝐶𝑚) = (𝐶𝑁))
98sseq1d 4027 . . 3 (𝑚 = 𝑁 → ((𝐶𝑚) ⊆ ℂ ↔ (𝐶𝑁) ⊆ ℂ))
10 constr0.1 . . . . 5 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})
1110constr0 33742 . . . 4 (𝐶‘∅) = {0, 1}
12 0cn 11251 . . . . 5 0 ∈ ℂ
13 ax-1cn 11211 . . . . 5 1 ∈ ℂ
14 prssi 4826 . . . . 5 ((0 ∈ ℂ ∧ 1 ∈ ℂ) → {0, 1} ⊆ ℂ)
1512, 13, 14mp2an 692 . . . 4 {0, 1} ⊆ ℂ
1611, 15eqsstri 4030 . . 3 (𝐶‘∅) ⊆ ℂ
17 simpl 482 . . . . . . . . 9 ((𝑛 ∈ On ∧ (𝐶𝑛) ⊆ ℂ) → 𝑛 ∈ On)
18 eqid 2735 . . . . . . . . 9 (𝐶𝑛) = (𝐶𝑛)
1910, 17, 18constrsuc 33743 . . . . . . . 8 ((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)))))))
2019biimpa 476 . . . . . . 7 (((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))))))
2120simpld 494 . . . . . 6 (((𝑛 ∈ On ∧ (𝐶𝑛) ⊆ ℂ) ∧ 𝑥 ∈ (𝐶‘suc 𝑛)) → 𝑥 ∈ ℂ)
2221ex 412 . . . . 5 ((𝑛 ∈ On ∧ (𝐶𝑛) ⊆ ℂ) → (𝑥 ∈ (𝐶‘suc 𝑛) → 𝑥 ∈ ℂ))
2322ssrdv 4001 . . . 4 ((𝑛 ∈ On ∧ (𝐶𝑛) ⊆ ℂ) → (𝐶‘suc 𝑛) ⊆ ℂ)
2423ex 412 . . 3 (𝑛 ∈ On → ((𝐶𝑛) ⊆ ℂ → (𝐶‘suc 𝑛) ⊆ ℂ))
25 vex 3482 . . . . . . 7 𝑚 ∈ V
2625a1i 11 . . . . . 6 ((Lim 𝑚 ∧ ∀𝑛𝑚 (𝐶𝑛) ⊆ ℂ) → 𝑚 ∈ V)
27 simpl 482 . . . . . 6 ((Lim 𝑚 ∧ ∀𝑛𝑚 (𝐶𝑛) ⊆ ℂ) → Lim 𝑚)
2810, 26, 27constrlim 33744 . . . . 5 ((Lim 𝑚 ∧ ∀𝑛𝑚 (𝐶𝑛) ⊆ ℂ) → (𝐶𝑚) = 𝑜𝑚 (𝐶𝑜))
29 fveq2 6907 . . . . . . . 8 (𝑛 = 𝑜 → (𝐶𝑛) = (𝐶𝑜))
3029sseq1d 4027 . . . . . . 7 (𝑛 = 𝑜 → ((𝐶𝑛) ⊆ ℂ ↔ (𝐶𝑜) ⊆ ℂ))
31 simplr 769 . . . . . . 7 (((Lim 𝑚 ∧ ∀𝑛𝑚 (𝐶𝑛) ⊆ ℂ) ∧ 𝑜𝑚) → ∀𝑛𝑚 (𝐶𝑛) ⊆ ℂ)
32 simpr 484 . . . . . . 7 (((Lim 𝑚 ∧ ∀𝑛𝑚 (𝐶𝑛) ⊆ ℂ) ∧ 𝑜𝑚) → 𝑜𝑚)
3330, 31, 32rspcdva 3623 . . . . . 6 (((Lim 𝑚 ∧ ∀𝑛𝑚 (𝐶𝑛) ⊆ ℂ) ∧ 𝑜𝑚) → (𝐶𝑜) ⊆ ℂ)
3433iunssd 5055 . . . . 5 ((Lim 𝑚 ∧ ∀𝑛𝑚 (𝐶𝑛) ⊆ ℂ) → 𝑜𝑚 (𝐶𝑜) ⊆ ℂ)
3528, 34eqsstrd 4034 . . . 4 ((Lim 𝑚 ∧ ∀𝑛𝑚 (𝐶𝑛) ⊆ ℂ) → (𝐶𝑚) ⊆ ℂ)
3635ex 412 . . 3 (Lim 𝑚 → (∀𝑛𝑚 (𝐶𝑛) ⊆ ℂ → (𝐶𝑚) ⊆ ℂ))
373, 5, 7, 9, 16, 24, 36tfinds 7881 . 2 (𝑁 ∈ On → (𝐶𝑁) ⊆ ℂ)
381, 37syl 17 1 (𝜑 → (𝐶𝑁) ⊆ ℂ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3o 1085  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  {crab 3433  Vcvv 3478  wss 3963  c0 4339  {cpr 4633   ciun 4996  cmpt 5231  Oncon0 6386  Lim wlim 6387  suc csuc 6388  cfv 6563  (class class class)co 7431  reccrdg 8448  cc 11151  cr 11152  0cc0 11153  1c1 11154   + caddc 11156   · cmul 11158  cmin 11490  ccj 15132  cim 15134  abscabs 15270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-mulcl 11215  ax-i2m1 11221
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449
This theorem is referenced by:  constrsslem  33746  constrconj  33750  constrfin  33751  constrelextdg2  33752
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