Theorem norec2fn 28342

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.

Step	Hyp	Ref	Expression
1		eqid 2760	. . 3 ⊢ {⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))} = {⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ ((L‘𝑑) ∪ (R‘𝑑))}
2		eqid 2760	. . 3 ⊢ {⟨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))}
3	1, 2	noxpordfr 28337	. 2 ⊢ {⟨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)
4	1, 2	noxpordpo 28336	. 2 ⊢ {⟨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)
5	1, 2	noxpordse 28338	. 2 ⊢ {⟨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)
6		norec2.1	. . . 4 ⊢ 𝐹 = norec2 (𝐺)
7		df-norec2 28335	. . . 4 ⊢ 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)
8	6, 7	eqtri 2783	. . 3 ⊢ 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)
9	8	fpr1 8321	. 2 ⊢ (({⟨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)) → 𝐹 Fn (No × No))
10	3, 4, 5, 9	mp3an 1490	1 ⊢ 𝐹 Fn (No × No)

This proof depends on syntax axioms:   ∨ wo 861   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ≠ wne 2955   ∪ cun 4149   class class class wbr 5102  {copab 5166   Po wpo 5556   Fr wfr 5600   Se wse 5601   × cxp 5648   Fn wfn 6533  ‘cfv 6538  1^st c1st 7999  2^nd c2nd 8000  frecscfrecs 8298  Nocsur 27997  Lcleft 28211  Rcright 28212   norec2 cnorec2 28334

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 5326  ax-pr 5390  ax-un 7751

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 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-1o 8476  df-2o 8477  df-no 28000  df-lts 28001  df-bday 28002  df-slts 28144  df-cuts 28146  df-made 28213  df-old 28214  df-left 28216  df-right 28217  df-norec2 28335

This theorem is used by:  addsfn  28347  mulsfn  28494