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Theorem norec2ov 27840

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 7372	. . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2		opelxp 5667	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ ( No × No ) ↔ (𝐴 ∈ No ∧ 𝐵 ∈ No ))
3		eqid 2729	. . . . . . 7 ⊢ {⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} = {⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))}
4		eqid 2729	. . . . . . 7 ⊢ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ( No × No ) ∧ 𝑏 ∈ ( No × No ) ∧ (((1^st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1^st ‘𝑏) ∨ (1^st ‘𝑎) = (1^st ‘𝑏)) ∧ ((2^nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2^nd ‘𝑏) ∨ (2^nd ‘𝑎) = (2^nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))} = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ( No × No ) ∧ 𝑏 ∈ ( No × No ) ∧ (((1^st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1^st ‘𝑏) ∨ (1^st ‘𝑎) = (1^st ‘𝑏)) ∧ ((2^nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2^nd ‘𝑏) ∨ (2^nd ‘𝑎) = (2^nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))}
5	3, 4	noxpordfr 27834	. . . . . 6 ⊢ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ( No × No ) ∧ 𝑏 ∈ ( No × No ) ∧ (((1^st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1^st ‘𝑏) ∨ (1^st ‘𝑎) = (1^st ‘𝑏)) ∧ ((2^nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2^nd ‘𝑏) ∨ (2^nd ‘𝑎) = (2^nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))} Fr ( No × No )
6	3, 4	noxpordpo 27833	. . . . . 6 ⊢ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ( No × No ) ∧ 𝑏 ∈ ( No × No ) ∧ (((1^st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1^st ‘𝑏) ∨ (1^st ‘𝑎) = (1^st ‘𝑏)) ∧ ((2^nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2^nd ‘𝑏) ∨ (2^nd ‘𝑎) = (2^nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))} Po ( No × No )
7	3, 4	noxpordse 27835	. . . . . 6 ⊢ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ( No × No ) ∧ 𝑏 ∈ ( No × No ) ∧ (((1^st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1^st ‘𝑏) ∨ (1^st ‘𝑎) = (1^st ‘𝑏)) ∧ ((2^nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2^nd ‘𝑏) ∨ (2^nd ‘𝑎) = (2^nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))} Se ( No × No )
8	5, 6, 7	3pm3.2i 1340	. . . . 5 ⊢ ({⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ( No × No ) ∧ 𝑏 ∈ ( No × No ) ∧ (((1^st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1^st ‘𝑏) ∨ (1^st ‘𝑎) = (1^st ‘𝑏)) ∧ ((2^nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2^nd ‘𝑏) ∨ (2^nd ‘𝑎) = (2^nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))} Fr ( No × No ) ∧ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ( No × No ) ∧ 𝑏 ∈ ( No × No ) ∧ (((1^st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1^st ‘𝑏) ∨ (1^st ‘𝑎) = (1^st ‘𝑏)) ∧ ((2^nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2^nd ‘𝑏) ∨ (2^nd ‘𝑎) = (2^nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))} Po ( No × No ) ∧ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ( No × No ) ∧ 𝑏 ∈ ( No × No ) ∧ (((1^st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1^st ‘𝑏) ∨ (1^st ‘𝑎) = (1^st ‘𝑏)) ∧ ((2^nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2^nd ‘𝑏) ∨ (2^nd ‘𝑎) = (2^nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))} Se ( No × No ))
9		norec2.1	. . . . . . 7 ⊢ 𝐹 = norec2 (𝐺)
10		df-norec2 27832	. . . . . . 7 ⊢ norec2 (𝐺) = frecs({⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ( No × No ) ∧ 𝑏 ∈ ( No × No ) ∧ (((1^st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1^st ‘𝑏) ∨ (1^st ‘𝑎) = (1^st ‘𝑏)) ∧ ((2^nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2^nd ‘𝑏) ∨ (2^nd ‘𝑎) = (2^nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))}, ( No × No ), 𝐺)
11	9, 10	eqtri 2752	. . . . . 6 ⊢ 𝐹 = frecs({⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ( No × No ) ∧ 𝑏 ∈ ( No × No ) ∧ (((1^st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1^st ‘𝑏) ∨ (1^st ‘𝑎) = (1^st ‘𝑏)) ∧ ((2^nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2^nd ‘𝑏) ∨ (2^nd ‘𝑎) = (2^nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))}, ( No × No ), 𝐺)
12	11	fpr2 8260	. . . . 5 ⊢ ((({⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ( No × No ) ∧ 𝑏 ∈ ( No × No ) ∧ (((1^st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1^st ‘𝑏) ∨ (1^st ‘𝑎) = (1^st ‘𝑏)) ∧ ((2^nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2^nd ‘𝑏) ∨ (2^nd ‘𝑎) = (2^nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))} Fr ( No × No ) ∧ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ( No × No ) ∧ 𝑏 ∈ ( No × No ) ∧ (((1^st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1^st ‘𝑏) ∨ (1^st ‘𝑎) = (1^st ‘𝑏)) ∧ ((2^nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2^nd ‘𝑏) ∨ (2^nd ‘𝑎) = (2^nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))} Po ( No × No ) ∧ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ( No × No ) ∧ 𝑏 ∈ ( No × No ) ∧ (((1^st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1^st ‘𝑏) ∨ (1^st ‘𝑎) = (1^st ‘𝑏)) ∧ ((2^nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2^nd ‘𝑏) ∨ (2^nd ‘𝑎) = (2^nd ‘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 ) ∧ (((1^st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1^st ‘𝑏) ∨ (1^st ‘𝑎) = (1^st ‘𝑏)) ∧ ((2^nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2^nd ‘𝑏) ∨ (2^nd ‘𝑎) = (2^nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))}, ( No × No ), ⟨𝐴, 𝐵⟩))))
13	8, 12	mpan 690	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ ( No × No ) → (𝐹‘⟨𝐴, 𝐵⟩) = (⟨𝐴, 𝐵⟩𝐺(𝐹 ↾ Pred({⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ( No × No ) ∧ 𝑏 ∈ ( No × No ) ∧ (((1^st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1^st ‘𝑏) ∨ (1^st ‘𝑎) = (1^st ‘𝑏)) ∧ ((2^nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2^nd ‘𝑏) ∨ (2^nd ‘𝑎) = (2^nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))}, ( No × No ), ⟨𝐴, 𝐵⟩))))
14	2, 13	sylbir 235	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐹‘⟨𝐴, 𝐵⟩) = (⟨𝐴, 𝐵⟩𝐺(𝐹 ↾ Pred({⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ( No × No ) ∧ 𝑏 ∈ ( No × No ) ∧ (((1^st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1^st ‘𝑏) ∨ (1^st ‘𝑎) = (1^st ‘𝑏)) ∧ ((2^nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2^nd ‘𝑏) ∨ (2^nd ‘𝑎) = (2^nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))}, ( No × No ), ⟨𝐴, 𝐵⟩))))
15	1, 14	eqtrid 2776	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴𝐹𝐵) = (⟨𝐴, 𝐵⟩𝐺(𝐹 ↾ Pred({⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ( No × No ) ∧ 𝑏 ∈ ( No × No ) ∧ (((1^st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1^st ‘𝑏) ∨ (1^st ‘𝑎) = (1^st ‘𝑏)) ∧ ((2^nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2^nd ‘𝑏) ∨ (2^nd ‘𝑎) = (2^nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))}, ( No × No ), ⟨𝐴, 𝐵⟩))))
16	3, 4	noxpordpred 27836	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → Pred({⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ( No × No ) ∧ 𝑏 ∈ ( No × No ) ∧ (((1^st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1^st ‘𝑏) ∨ (1^st ‘𝑎) = (1^st ‘𝑏)) ∧ ((2^nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2^nd ‘𝑏) ∨ (2^nd ‘𝑎) = (2^nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))}, ( No × No ), ⟨𝐴, 𝐵⟩) = ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))
17	16	reseq2d 5939	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐹 ↾ Pred({⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ( No × No ) ∧ 𝑏 ∈ ( No × No ) ∧ (((1^st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1^st ‘𝑏) ∨ (1^st ‘𝑎) = (1^st ‘𝑏)) ∧ ((2^nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2^nd ‘𝑏) ∨ (2^nd ‘𝑎) = (2^nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))}, ( No × No ), ⟨𝐴, 𝐵⟩)) = (𝐹 ↾ ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩})))
18	17	oveq2d 7385	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (⟨𝐴, 𝐵⟩𝐺(𝐹 ↾ Pred({⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ( No × No ) ∧ 𝑏 ∈ ( No × No ) ∧ (((1^st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1^st ‘𝑏) ∨ (1^st ‘𝑎) = (1^st ‘𝑏)) ∧ ((2^nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2^nd ‘𝑏) ∨ (2^nd ‘𝑎) = (2^nd ‘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 2764	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴𝐹𝐵) = (⟨𝐴, 𝐵⟩𝐺(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 1540 ∈ wcel 2109 ≠ wne 2925 ∖ cdif 3908 ∪ cun 3909 {csn 4585 ⟨cop 4591 class class class wbr 5102 {copab 5164 Po wpo 5537 Fr wfr 5581 Se wse 5582 × cxp 5629 ↾ cres 5633 Predcpred 6261 ‘cfv 6499 (class class class)co 7369 1^st c1st 7945 2^nd c2nd 7946 frecscfrecs 8236 No csur 27527 L cleft 27729 R cright 27730 norec2 cnorec2 27831

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-1st 7947 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-1o 8411 df-2o 8412 df-no 27530 df-slt 27531 df-bday 27532 df-sslt 27669 df-scut 27671 df-made 27731 df-old 27732 df-left 27734 df-right 27735 df-norec2 27832

This theorem is referenced by: addsval 27845 mulsval 27988

W3C validator