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Theorem norec2fn 27823
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 2726 . . 3 {⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} = {⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))}
2 eqid 2726 . . 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 27818 . 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 27817 . 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 27819 . 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 27816 . . . 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 2754 . . 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 8286 . 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 844  w3a 1084   = wceq 1533  wcel 2098  wne 2934  cun 3941   class class class wbr 5141  {copab 5203   Po wpo 5579   Fr wfr 5621   Se wse 5622   × cxp 5667   Fn wfn 6531  cfv 6536  1st c1st 7969  2nd c2nd 7970  frecscfrecs 8263   No csur 27523   L cleft 27722   R cright 27723   norec2 cnorec2 27815
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-1o 8464  df-2o 8465  df-no 27526  df-slt 27527  df-bday 27528  df-sslt 27664  df-scut 27666  df-made 27724  df-old 27725  df-left 27727  df-right 27728  df-norec2 27816
This theorem is referenced by:  addsfn  27828  mulsfn  27958
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