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Theorem constrsscn 33781
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 6906 . . . 4 (𝑚 = ∅ → (𝐶𝑚) = (𝐶‘∅))
32sseq1d 4015 . . 3 (𝑚 = ∅ → ((𝐶𝑚) ⊆ ℂ ↔ (𝐶‘∅) ⊆ ℂ))
4 fveq2 6906 . . . 4 (𝑚 = 𝑛 → (𝐶𝑚) = (𝐶𝑛))
54sseq1d 4015 . . 3 (𝑚 = 𝑛 → ((𝐶𝑚) ⊆ ℂ ↔ (𝐶𝑛) ⊆ ℂ))
6 fveq2 6906 . . . 4 (𝑚 = suc 𝑛 → (𝐶𝑚) = (𝐶‘suc 𝑛))
76sseq1d 4015 . . 3 (𝑚 = suc 𝑛 → ((𝐶𝑚) ⊆ ℂ ↔ (𝐶‘suc 𝑛) ⊆ ℂ))
8 fveq2 6906 . . . 4 (𝑚 = 𝑁 → (𝐶𝑚) = (𝐶𝑁))
98sseq1d 4015 . . 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 33778 . . . 4 (𝐶‘∅) = {0, 1}
12 0cn 11253 . . . . 5 0 ∈ ℂ
13 ax-1cn 11213 . . . . 5 1 ∈ ℂ
14 prssi 4821 . . . . 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 2737 . . . . . . . . 9 (𝐶𝑛) = (𝐶𝑛)
1910, 17, 18constrsuc 33779 . . . . . . . 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 3989 . . . 4 ((𝑛 ∈ On ∧ (𝐶𝑛) ⊆ ℂ) → (𝐶‘suc 𝑛) ⊆ ℂ)
2423ex 412 . . 3 (𝑛 ∈ On → ((𝐶𝑛) ⊆ ℂ → (𝐶‘suc 𝑛) ⊆ ℂ))
25 vex 3484 . . . . . . 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 33780 . . . . 5 ((Lim 𝑚 ∧ ∀𝑛𝑚 (𝐶𝑛) ⊆ ℂ) → (𝐶𝑚) = 𝑜𝑚 (𝐶𝑜))
29 fveq2 6906 . . . . . . . 8 (𝑛 = 𝑜 → (𝐶𝑛) = (𝐶𝑜))
3029sseq1d 4015 . . . . . . 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 5050 . . . . 5 ((Lim 𝑚 ∧ ∀𝑛𝑚 (𝐶𝑛) ⊆ ℂ) → 𝑜𝑚 (𝐶𝑜) ⊆ ℂ)
3528, 34eqsstrd 4018 . . . 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 1086  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  {crab 3436  Vcvv 3480  wss 3951  c0 4333  {cpr 4628   ciun 4991  cmpt 5225  Oncon0 6384  Lim wlim 6385  suc csuc 6386  cfv 6561  (class class class)co 7431  reccrdg 8449  cc 11153  cr 11154  0cc0 11155  1c1 11156   + caddc 11158   · cmul 11160  cmin 11492  ccj 15135  cim 15137  abscabs 15273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-mulcl 11217  ax-i2m1 11223
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450
This theorem is referenced by:  constrsslem  33782  constrconj  33786  constrfin  33787  constrelextdg2  33788  constrextdg2lem  33789
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