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Theorem norec2fn 27893
Description: The double-recursion operator on surreals yields a function on pairs of surreals. (Contributed by Scott Fenton, 20-Aug-2024.)
Hypothesis
Ref Expression
norec2.1 𝐹 = norec2 (𝐺)
Assertion
Ref Expression
norec2fn 𝐹 Fn ( No × No )

Proof of Theorem norec2fn
Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2728 . . 3 {⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} = {⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))}
2 eqid 2728 . . 3 {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ( No × No ) ∧ 𝑏 ∈ ( No × No ) ∧ (((1st𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1st𝑏) ∨ (1st𝑎) = (1st𝑏)) ∧ ((2nd𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2nd𝑏) ∨ (2nd𝑎) = (2nd𝑏)) ∧ 𝑎𝑏))} = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ( No × No ) ∧ 𝑏 ∈ ( No × No ) ∧ (((1st𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1st𝑏) ∨ (1st𝑎) = (1st𝑏)) ∧ ((2nd𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2nd𝑏) ∨ (2nd𝑎) = (2nd𝑏)) ∧ 𝑎𝑏))}
31, 2noxpordfr 27888 . 2 {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ( No × No ) ∧ 𝑏 ∈ ( No × No ) ∧ (((1st𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1st𝑏) ∨ (1st𝑎) = (1st𝑏)) ∧ ((2nd𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2nd𝑏) ∨ (2nd𝑎) = (2nd𝑏)) ∧ 𝑎𝑏))} Fr ( No × No )
41, 2noxpordpo 27887 . 2 {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ( No × No ) ∧ 𝑏 ∈ ( No × No ) ∧ (((1st𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1st𝑏) ∨ (1st𝑎) = (1st𝑏)) ∧ ((2nd𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2nd𝑏) ∨ (2nd𝑎) = (2nd𝑏)) ∧ 𝑎𝑏))} Po ( No × No )
51, 2noxpordse 27889 . 2 {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ( No × No ) ∧ 𝑏 ∈ ( No × No ) ∧ (((1st𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1st𝑏) ∨ (1st𝑎) = (1st𝑏)) ∧ ((2nd𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2nd𝑏) ∨ (2nd𝑎) = (2nd𝑏)) ∧ 𝑎𝑏))} Se ( No × No )
6 norec2.1 . . . 4 𝐹 = norec2 (𝐺)
7 df-norec2 27886 . . . 4 norec2 (𝐺) = frecs({⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ( No × No ) ∧ 𝑏 ∈ ( No × No ) ∧ (((1st𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1st𝑏) ∨ (1st𝑎) = (1st𝑏)) ∧ ((2nd𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2nd𝑏) ∨ (2nd𝑎) = (2nd𝑏)) ∧ 𝑎𝑏))}, ( No × No ), 𝐺)
86, 7eqtri 2756 . . 3 𝐹 = frecs({⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ( No × No ) ∧ 𝑏 ∈ ( No × No ) ∧ (((1st𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1st𝑏) ∨ (1st𝑎) = (1st𝑏)) ∧ ((2nd𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2nd𝑏) ∨ (2nd𝑎) = (2nd𝑏)) ∧ 𝑎𝑏))}, ( No × No ), 𝐺)
98fpr1 8315 . 2 (({⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ( No × No ) ∧ 𝑏 ∈ ( No × No ) ∧ (((1st𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1st𝑏) ∨ (1st𝑎) = (1st𝑏)) ∧ ((2nd𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2nd𝑏) ∨ (2nd𝑎) = (2nd𝑏)) ∧ 𝑎𝑏))} Fr ( No × No ) ∧ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ( No × No ) ∧ 𝑏 ∈ ( No × No ) ∧ (((1st𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1st𝑏) ∨ (1st𝑎) = (1st𝑏)) ∧ ((2nd𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2nd𝑏) ∨ (2nd𝑎) = (2nd𝑏)) ∧ 𝑎𝑏))} Po ( No × No ) ∧ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ( No × No ) ∧ 𝑏 ∈ ( No × No ) ∧ (((1st𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1st𝑏) ∨ (1st𝑎) = (1st𝑏)) ∧ ((2nd𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2nd𝑏) ∨ (2nd𝑎) = (2nd𝑏)) ∧ 𝑎𝑏))} Se ( No × No )) → 𝐹 Fn ( No × No ))
103, 4, 5, 9mp3an 1457 1 𝐹 Fn ( No × No )
Colors of variables: wff setvar class
Syntax hints:  wo 845  w3a 1084   = wceq 1533  wcel 2098  wne 2937  cun 3947   class class class wbr 5152  {copab 5214   Po wpo 5592   Fr wfr 5634   Se wse 5635   × cxp 5680   Fn wfn 6548  cfv 6553  1st c1st 7997  2nd c2nd 7998  frecscfrecs 8292   No csur 27593   L cleft 27792   R cright 27793   norec2 cnorec2 27885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-tp 4637  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-1o 8493  df-2o 8494  df-no 27596  df-slt 27597  df-bday 27598  df-sslt 27734  df-scut 27736  df-made 27794  df-old 27795  df-left 27797  df-right 27798  df-norec2 27886
This theorem is referenced by:  addsfn  27898  mulsfn  28028
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