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Theorem norec2fn 27936

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 2735	. . 3 ⊢ {⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} = {⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))}
2		eqid 2735	. . 3 ⊢ {⟨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))}
3	1, 2	noxpordfr 27931	. 2 ⊢ {⟨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 )
4	1, 2	noxpordpo 27930	. 2 ⊢ {⟨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 )
5	1, 2	noxpordse 27932	. 2 ⊢ {⟨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 )
6		norec2.1	. . . 4 ⊢ 𝐹 = norec2 (𝐺)
7		df-norec2 27929	. . . 4 ⊢ 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)
8	6, 7	eqtri 2758	. . 3 ⊢ 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)
9	8	fpr1 8242	. 2 ⊢ (({⟨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 )) → 𝐹 Fn ( No × No ))
10	3, 4, 5, 9	mp3an 1464	1 ⊢ 𝐹 Fn ( No × No )

Colors of variables: wff setvar class

Syntax hints: ∨ wo 848 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2930 ∪ cun 3883 class class class wbr 5074 {copab 5136 Po wpo 5526 Fr wfr 5570 Se wse 5571 × cxp 5618 Fn wfn 6482 ‘cfv 6487 1^st c1st 7929 2^nd c2nd 7930 frecscfrecs 8219 No csur 27591 L cleft 27805 R cright 27806 norec2 cnorec2 27928

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2184 ax-ext 2707 ax-rep 5201 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7678

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2931 df-ral 3050 df-rex 3060 df-rmo 3340 df-reu 3341 df-rab 3388 df-v 3429 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4841 df-int 4880 df-iun 4925 df-br 5075 df-opab 5137 df-mpt 5156 df-tr 5182 df-id 5515 df-eprel 5520 df-po 5528 df-so 5529 df-fr 5573 df-se 5574 df-we 5575 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6254 df-ord 6315 df-on 6316 df-suc 6318 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-1st 7931 df-2nd 7932 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-1o 8394 df-2o 8395 df-no 27594 df-lts 27595 df-bday 27596 df-slts 27738 df-cuts 27740 df-made 27807 df-old 27808 df-left 27810 df-right 27811 df-norec2 27929

This theorem is referenced by: addsfn 27941 mulsfn 28088

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