Theorem norec2ov 28352

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, 𝐵⟩𝐺(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.

Step	Hyp	Ref	Expression
1		df-ov 7427	. . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2		opelxp 5687	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ (No × No) ↔ (𝐴 ∈ No ∧ 𝐵 ∈ No))
3		eqid 2760	. . . . . . 7 ⊢ {⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} = {⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))}
4		eqid 2760	. . . . . . 7 ⊢ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ (No × No) ∧ 𝑏 ∈ (No × No) ∧ (((1st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))} = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ (No × No) ∧ 𝑏 ∈ (No × No) ∧ (((1st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))}
5	3, 4	noxpordfr 28346	. . . . . 6 ⊢ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ (No × No) ∧ 𝑏 ∈ (No × No) ∧ (((1st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))} Fr (No × No)
6	3, 4	noxpordpo 28345	. . . . . 6 ⊢ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ (No × No) ∧ 𝑏 ∈ (No × No) ∧ (((1st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))} Po (No × No)
7	3, 4	noxpordse 28347	. . . . . 6 ⊢ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ (No × No) ∧ 𝑏 ∈ (No × No) ∧ (((1st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))} Se (No × No)
8	5, 6, 7	3pm3.2i 1358	. . . . 5 ⊢ ({⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ (No × No) ∧ 𝑏 ∈ (No × No) ∧ (((1st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))} Fr (No × No) ∧ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ (No × No) ∧ 𝑏 ∈ (No × No) ∧ (((1st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))} Po (No × No) ∧ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ (No × No) ∧ 𝑏 ∈ (No × No) ∧ (((1st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))} Se (No × No))
9		norec2.1	. . . . . . 7 ⊢ 𝐹 = norec2 (𝐺)
10		df-norec2 28344	. . . . . . 7 ⊢ norec2 (𝐺) = frecs({⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ (No × No) ∧ 𝑏 ∈ (No × No) ∧ (((1st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))}, (No × No), 𝐺)
11	9, 10	eqtri 2783	. . . . . 6 ⊢ 𝐹 = frecs({⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ (No × No) ∧ 𝑏 ∈ (No × No) ∧ (((1st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))}, (No × No), 𝐺)
12	11	fpr2 8330	. . . . 5 ⊢ ((({⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ (No × No) ∧ 𝑏 ∈ (No × No) ∧ (((1st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))} Fr (No × No) ∧ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ (No × No) ∧ 𝑏 ∈ (No × No) ∧ (((1st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))} Po (No × No) ∧ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ (No × No) ∧ 𝑏 ∈ (No × No) ∧ (((1st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))} Se (No × No)) ∧ ⟨𝐴, 𝐵⟩ ∈ (No × No)) → (𝐹‘⟨𝐴, 𝐵⟩) = (⟨𝐴, 𝐵⟩𝐺(𝐹 ↾ Pred({⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ (No × No) ∧ 𝑏 ∈ (No × No) ∧ (((1st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))}, (No × No), ⟨𝐴, 𝐵⟩))))
13	8, 12	mpan 703	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ (No × No) → (𝐹‘⟨𝐴, 𝐵⟩) = (⟨𝐴, 𝐵⟩𝐺(𝐹 ↾ Pred({⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ (No × No) ∧ 𝑏 ∈ (No × No) ∧ (((1st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))}, (No × No), ⟨𝐴, 𝐵⟩))))
14	2, 13	sylbir 238	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (𝐹‘⟨𝐴, 𝐵⟩) = (⟨𝐴, 𝐵⟩𝐺(𝐹 ↾ Pred({⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ (No × No) ∧ 𝑏 ∈ (No × No) ∧ (((1st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))}, (No × No), ⟨𝐴, 𝐵⟩))))
15	1, 14	eqtrid 2807	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (𝐴𝐹𝐵) = (⟨𝐴, 𝐵⟩𝐺(𝐹 ↾ Pred({⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ (No × No) ∧ 𝑏 ∈ (No × No) ∧ (((1st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))}, (No × No), ⟨𝐴, 𝐵⟩))))
16	3, 4	noxpordpred 28348	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → Pred({⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ (No × No) ∧ 𝑏 ∈ (No × No) ∧ (((1st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))}, (No × No), ⟨𝐴, 𝐵⟩) = (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))
17	16	reseq2d 5971	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (𝐹 ↾ Pred({⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ (No × No) ∧ 𝑏 ∈ (No × No) ∧ (((1st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))}, (No × No), ⟨𝐴, 𝐵⟩)) = (𝐹 ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩})))
18	17	oveq2d 7440	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (⟨𝐴, 𝐵⟩𝐺(𝐹 ↾ Pred({⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ (No × No) ∧ 𝑏 ∈ (No × No) ∧ (((1st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))}, (No × No), ⟨𝐴, 𝐵⟩))) = (⟨𝐴, 𝐵⟩𝐺(𝐹 ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))))
19	15, 18	eqtrd 2795	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (𝐴𝐹𝐵) = (⟨𝐴, 𝐵⟩𝐺(𝐹 ↾ (((((L‘𝐴) ∪ (R‘𝐴)) ∪ {𝐴}) × (((L‘𝐵) ∪ (R‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ≠ wne 2955   ∖ cdif 4044   ∪ cun 4148  {csn 4584  ⟨cop 4834   class class class wbr 5102  {copab 5166   Po wpo 5557   Fr wfr 5601   Se wse 5602   × cxp 5649   ↾ cres 5653  Predcpred 6306  ‘cfv 6543  (class class class)co 7424  1^st c1st 8003  2^nd c2nd 8004  frecscfrecs 8306  Nocsur 28006  Lcleft 28220  Rcright 28221   norec2 cnorec2 28343

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8005  df-2nd 8006  df-frecs 8307  df-wrecs 8338  df-recs 8387  df-1o 8484  df-2o 8485  df-no 28009  df-lts 28010  df-bday 28011  df-slts 28153  df-cuts 28155  df-made 28222  df-old 28223  df-left 28225  df-right 28226  df-norec2 28344

This theorem is used by:  addsval  28357  mulsval  28504