Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  noetainflem1 Structured version   Visualization version   GIF version

Theorem noetainflem1 33573
Description: Lemma for noeta 33579. Establish that this particular construction gives a surreal. (Contributed by Scott Fenton, 9-Aug-2024.)
Hypotheses
Ref Expression
noetainflem.1 𝑇 = if(∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥, ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
noetainflem.2 𝑊 = (𝑇 ∪ ((suc ( bday 𝐴) ∖ dom 𝑇) × {2o}))
Assertion
Ref Expression
noetainflem1 ((𝐴 ∈ V ∧ 𝐵 No 𝐵 ∈ V) → 𝑊 No )
Distinct variable group:   𝐵,𝑔,𝑢,𝑣,𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑣,𝑢,𝑔)   𝑇(𝑥,𝑦,𝑣,𝑢,𝑔)   𝑊(𝑥,𝑦,𝑣,𝑢,𝑔)

Proof of Theorem noetainflem1
StepHypRef Expression
1 noetainflem.2 . 2 𝑊 = (𝑇 ∪ ((suc ( bday 𝐴) ∖ dom 𝑇) × {2o}))
2 noetainflem.1 . . . . 5 𝑇 = if(∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥, ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
32noinfno 33554 . . . 4 ((𝐵 No 𝐵 ∈ V) → 𝑇 No )
433adant1 1131 . . 3 ((𝐴 ∈ V ∧ 𝐵 No 𝐵 ∈ V) → 𝑇 No )
5 bdayimaon 33529 . . . 4 (𝐴 ∈ V → suc ( bday 𝐴) ∈ On)
653ad2ant1 1134 . . 3 ((𝐴 ∈ V ∧ 𝐵 No 𝐵 ∈ V) → suc ( bday 𝐴) ∈ On)
7 2oex 8141 . . . . 5 2o ∈ V
87prid2 4651 . . . 4 2o ∈ {1o, 2o}
98noextendseq 33503 . . 3 ((𝑇 No ∧ suc ( bday 𝐴) ∈ On) → (𝑇 ∪ ((suc ( bday 𝐴) ∖ dom 𝑇) × {2o})) ∈ No )
104, 6, 9syl2anc 587 . 2 ((𝐴 ∈ V ∧ 𝐵 No 𝐵 ∈ V) → (𝑇 ∪ ((suc ( bday 𝐴) ∖ dom 𝑇) × {2o})) ∈ No )
111, 10eqeltrid 2837 1 ((𝐴 ∈ V ∧ 𝐵 No 𝐵 ∈ V) → 𝑊 No )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  w3a 1088   = wceq 1542  wcel 2113  {cab 2716  wral 3053  wrex 3054  Vcvv 3397  cdif 3838  cun 3839  wss 3841  ifcif 4411  {csn 4513  cop 4519   cuni 4793   class class class wbr 5027  cmpt 5107   × cxp 5517  dom cdm 5519  cres 5521  cima 5522  Oncon0 6166  suc csuc 6168  cio 6289  cfv 6333  crio 7120  1oc1o 8117  2oc2o 8118   No csur 33476   <s cslt 33477   bday cbday 33478
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 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pr 5293  ax-un 7473
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-pss 3860  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-tp 4518  df-op 4520  df-uni 4794  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-tr 5134  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6169  df-on 6170  df-suc 6172  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-riota 7121  df-1o 8124  df-2o 8125  df-no 33479  df-slt 33480  df-bday 33481
This theorem is referenced by:  noetainflem3  33575  noetainflem4  33576  noetalem1  33577
  Copyright terms: Public domain W3C validator