MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  norec2ov Structured version   Visualization version   GIF version

Theorem norec2ov 27829
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 7408 . . 3 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2 opelxp 5705 . . . 4 (⟨𝐴, 𝐵⟩ ∈ ( No × No ) ↔ (𝐴 No 𝐵 No ))
3 eqid 2726 . . . . . . 7 {⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} = {⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))}
4 eqid 2726 . . . . . . 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 27823 . . . . . 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 27822 . . . . . 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 27824 . . . . . 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 1336 . . . . 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 27821 . . . . . . 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 2754 . . . . . 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 8290 . . . . 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 687 . . . 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 234 . . 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 2778 . 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 27825 . . . 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 5975 . . 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 7421 . 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 2766 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 844  w3a 1084   = wceq 1533  wcel 2098  wne 2934  cdif 3940  cun 3941  {csn 4623  cop 4629   class class class wbr 5141  {copab 5203   Po wpo 5579   Fr wfr 5621   Se wse 5622   × cxp 5667  cres 5671  Predcpred 6293  cfv 6537  (class class class)co 7405  1st c1st 7972  2nd c2nd 7973  frecscfrecs 8266   No csur 27528   L cleft 27727   R cright 27728   norec2 cnorec2 27820
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 7722
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 6294  df-ord 6361  df-on 6362  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-1o 8467  df-2o 8468  df-no 27531  df-slt 27532  df-bday 27533  df-sslt 27669  df-scut 27671  df-made 27729  df-old 27730  df-left 27732  df-right 27733  df-norec2 27821
This theorem is referenced by:  addsval  27834  mulsval  27964
  Copyright terms: Public domain W3C validator