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Mirrors > Home > MPE Home > Th. List > norecov | Structured version Visualization version GIF version |
Description: Calculate the value of the surreal recursion operation. (Contributed by Scott Fenton, 19-Aug-2024.) |
Ref | Expression |
---|---|
norec.1 | ⊢ 𝐹 = norec (𝐺) |
Ref | Expression |
---|---|
norecov | ⊢ (𝐴 ∈ No → (𝐹‘𝐴) = (𝐴𝐺(𝐹 ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2725 | . . . . 5 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))} = {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))} | |
2 | 1 | lrrecfr 27949 | . . . 4 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))} Fr No |
3 | 1 | lrrecpo 27947 | . . . 4 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))} Po No |
4 | 1 | lrrecse 27948 | . . . 4 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))} Se No |
5 | 2, 3, 4 | 3pm3.2i 1336 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))} Fr No ∧ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))} Po No ∧ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))} Se No ) |
6 | norec.1 | . . . . 5 ⊢ 𝐹 = norec (𝐺) | |
7 | df-norec 27944 | . . . . 5 ⊢ norec (𝐺) = frecs({〈𝑥, 𝑦〉 ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝐺) | |
8 | 6, 7 | eqtri 2753 | . . . 4 ⊢ 𝐹 = frecs({〈𝑥, 𝑦〉 ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝐺) |
9 | 8 | fpr2 8318 | . . 3 ⊢ ((({〈𝑥, 𝑦〉 ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))} Fr No ∧ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))} Po No ∧ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))} Se No ) ∧ 𝐴 ∈ No ) → (𝐹‘𝐴) = (𝐴𝐺(𝐹 ↾ Pred({〈𝑥, 𝑦〉 ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝐴)))) |
10 | 5, 9 | mpan 688 | . 2 ⊢ (𝐴 ∈ No → (𝐹‘𝐴) = (𝐴𝐺(𝐹 ↾ Pred({〈𝑥, 𝑦〉 ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝐴)))) |
11 | 1 | lrrecpred 27950 | . . . 4 ⊢ (𝐴 ∈ No → Pred({〈𝑥, 𝑦〉 ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝐴) = (( L ‘𝐴) ∪ ( R ‘𝐴))) |
12 | 11 | reseq2d 5988 | . . 3 ⊢ (𝐴 ∈ No → (𝐹 ↾ Pred({〈𝑥, 𝑦〉 ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝐴)) = (𝐹 ↾ (( L ‘𝐴) ∪ ( R ‘𝐴)))) |
13 | 12 | oveq2d 7439 | . 2 ⊢ (𝐴 ∈ No → (𝐴𝐺(𝐹 ↾ Pred({〈𝑥, 𝑦〉 ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝐴))) = (𝐴𝐺(𝐹 ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))))) |
14 | 10, 13 | eqtrd 2765 | 1 ⊢ (𝐴 ∈ No → (𝐹‘𝐴) = (𝐴𝐺(𝐹 ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∪ cun 3944 {copab 5214 Po wpo 5591 Fr wfr 5633 Se wse 5634 ↾ cres 5683 Predcpred 6310 ‘cfv 6553 (class class class)co 7423 frecscfrecs 8294 No csur 27661 L cleft 27861 R cright 27862 norec cnorec 27943 |
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 2166 ax-ext 2696 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-se 5637 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-suc 6381 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-2nd 8003 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-1o 8495 df-2o 8496 df-no 27664 df-slt 27665 df-bday 27666 df-sslt 27803 df-scut 27805 df-made 27863 df-old 27864 df-left 27866 df-right 27867 df-norec 27944 |
This theorem is referenced by: negsval 28027 |
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