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Theorem norec2ov 28005
Description: The value of the double-recursion surreal function. (Contributed by Scott Fenton, 20-Aug-2024.)
Hypothesis
Ref Expression
norec2.1 𝐹 = norec2 (𝐺)
Assertion
Ref Expression
norec2ov ((𝐴 No 𝐵 No ) → (𝐴𝐹𝐵) = (⟨𝐴, 𝐵𝐺(𝐹 ↾ ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))))

Proof of Theorem norec2ov
Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ov 7434 . . 3 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2 opelxp 5725 . . . 4 (⟨𝐴, 𝐵⟩ ∈ ( No × No ) ↔ (𝐴 No 𝐵 No ))
3 eqid 2735 . . . . . . 7 {⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} = {⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))}
4 eqid 2735 . . . . . . 7 {⟨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))}
53, 4noxpordfr 27999 . . . . . 6 {⟨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 )
63, 4noxpordpo 27998 . . . . . 6 {⟨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 )
73, 4noxpordse 28000 . . . . . 6 {⟨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 )
85, 6, 73pm3.2i 1338 . . . . 5 ({⟨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 ))
9 norec2.1 . . . . . . 7 𝐹 = norec2 (𝐺)
10 df-norec2 27997 . . . . . . 7 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)
119, 10eqtri 2763 . . . . . 6 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)
1211fpr2 8328 . . . . 5 ((({⟨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 )) ∧ ⟨𝐴, 𝐵⟩ ∈ ( No × No )) → (𝐹‘⟨𝐴, 𝐵⟩) = (⟨𝐴, 𝐵𝐺(𝐹 ↾ Pred({⟨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 ), ⟨𝐴, 𝐵⟩))))
138, 12mpan 690 . . . 4 (⟨𝐴, 𝐵⟩ ∈ ( No × No ) → (𝐹‘⟨𝐴, 𝐵⟩) = (⟨𝐴, 𝐵𝐺(𝐹 ↾ Pred({⟨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 ), ⟨𝐴, 𝐵⟩))))
142, 13sylbir 235 . . 3 ((𝐴 No 𝐵 No ) → (𝐹‘⟨𝐴, 𝐵⟩) = (⟨𝐴, 𝐵𝐺(𝐹 ↾ Pred({⟨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 ), ⟨𝐴, 𝐵⟩))))
151, 14eqtrid 2787 . 2 ((𝐴 No 𝐵 No ) → (𝐴𝐹𝐵) = (⟨𝐴, 𝐵𝐺(𝐹 ↾ Pred({⟨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 ), ⟨𝐴, 𝐵⟩))))
163, 4noxpordpred 28001 . . . 4 ((𝐴 No 𝐵 No ) → Pred({⟨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 ), ⟨𝐴, 𝐵⟩) = ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))
1716reseq2d 6000 . . 3 ((𝐴 No 𝐵 No ) → (𝐹 ↾ Pred({⟨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 ), ⟨𝐴, 𝐵⟩)) = (𝐹 ↾ ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩})))
1817oveq2d 7447 . 2 ((𝐴 No 𝐵 No ) → (⟨𝐴, 𝐵𝐺(𝐹 ↾ Pred({⟨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 ), ⟨𝐴, 𝐵⟩))) = (⟨𝐴, 𝐵𝐺(𝐹 ↾ ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))))
1915, 18eqtrd 2775 1 ((𝐴 No 𝐵 No ) → (𝐴𝐹𝐵) = (⟨𝐴, 𝐵𝐺(𝐹 ↾ ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 847  w3a 1086   = wceq 1537  wcel 2106  wne 2938  cdif 3960  cun 3961  {csn 4631  cop 4637   class class class wbr 5148  {copab 5210   Po wpo 5595   Fr wfr 5638   Se wse 5639   × cxp 5687  cres 5691  Predcpred 6322  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  frecscfrecs 8304   No csur 27699   L cleft 27899   R cright 27900   norec2 cnorec2 27996
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-pow 5371  ax-pr 5438  ax-un 7754
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-rmo 3378  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-tp 4636  df-op 4638  df-uni 4913  df-int 4952  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-se 5642  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-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-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-1o 8505  df-2o 8506  df-no 27702  df-slt 27703  df-bday 27704  df-sslt 27841  df-scut 27843  df-made 27901  df-old 27902  df-left 27904  df-right 27905  df-norec2 27997
This theorem is referenced by:  addsval  28010  mulsval  28150
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