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Mirrors > Home > NFE Home > Th. List > frec0 | GIF version |
Description: Calculate the value of the finite recursive function generator at zero. (Contributed by Scott Fenton, 31-Jul-2019.) |
Ref | Expression |
---|---|
frec0.1 | ⊢ F = FRec (G, I) |
frec0.2 | ⊢ (φ → G ∈ Funs ) |
frec0.3 | ⊢ (φ → I ∈ dom G) |
frec0.4 | ⊢ (φ → ran G ⊆ dom G) |
Ref | Expression |
---|---|
frec0 | ⊢ (φ → (F ‘0c) = I) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano1 4403 | . . . . . 6 ⊢ 0c ∈ Nn | |
2 | frec0.3 | . . . . . 6 ⊢ (φ → I ∈ dom G) | |
3 | opexg 4588 | . . . . . 6 ⊢ ((0c ∈ Nn ∧ I ∈ dom G) → 〈0c, I〉 ∈ V) | |
4 | 1, 2, 3 | sylancr 644 | . . . . 5 ⊢ (φ → 〈0c, I〉 ∈ V) |
5 | snidg 3759 | . . . . 5 ⊢ (〈0c, I〉 ∈ V → 〈0c, I〉 ∈ {〈0c, I〉}) | |
6 | 4, 5 | syl 15 | . . . 4 ⊢ (φ → 〈0c, I〉 ∈ {〈0c, I〉}) |
7 | 6 | orcd 381 | . . 3 ⊢ (φ → (〈0c, I〉 ∈ {〈0c, I〉} ∨ ∃y ∈ F y PProd ((x ∈ V ↦ (x +c 1c)), G)〈0c, I〉)) |
8 | snex 4112 | . . . 4 ⊢ {〈0c, I〉} ∈ V | |
9 | csucex 6260 | . . . . 5 ⊢ (x ∈ V ↦ (x +c 1c)) ∈ V | |
10 | frec0.2 | . . . . 5 ⊢ (φ → G ∈ Funs ) | |
11 | pprodexg 5838 | . . . . 5 ⊢ (((x ∈ V ↦ (x +c 1c)) ∈ V ∧ G ∈ Funs ) → PProd ((x ∈ V ↦ (x +c 1c)), G) ∈ V) | |
12 | 9, 10, 11 | sylancr 644 | . . . 4 ⊢ (φ → PProd ((x ∈ V ↦ (x +c 1c)), G) ∈ V) |
13 | frec0.1 | . . . . . 6 ⊢ F = FRec (G, I) | |
14 | df-frec 6311 | . . . . . 6 ⊢ FRec (G, I) = Clos1 ({〈0c, I〉}, PProd ((x ∈ V ↦ (x +c 1c)), G)) | |
15 | 13, 14 | eqtri 2373 | . . . . 5 ⊢ F = Clos1 ({〈0c, I〉}, PProd ((x ∈ V ↦ (x +c 1c)), G)) |
16 | 15 | clos1basesucg 5885 | . . . 4 ⊢ (({〈0c, I〉} ∈ V ∧ PProd ((x ∈ V ↦ (x +c 1c)), G) ∈ V) → (〈0c, I〉 ∈ F ↔ (〈0c, I〉 ∈ {〈0c, I〉} ∨ ∃y ∈ F y PProd ((x ∈ V ↦ (x +c 1c)), G)〈0c, I〉))) |
17 | 8, 12, 16 | sylancr 644 | . . 3 ⊢ (φ → (〈0c, I〉 ∈ F ↔ (〈0c, I〉 ∈ {〈0c, I〉} ∨ ∃y ∈ F y PProd ((x ∈ V ↦ (x +c 1c)), G)〈0c, I〉))) |
18 | 7, 17 | mpbird 223 | . 2 ⊢ (φ → 〈0c, I〉 ∈ F) |
19 | frec0.4 | . . . 4 ⊢ (φ → ran G ⊆ dom G) | |
20 | 13, 10, 2, 19 | fnfrec 6321 | . . 3 ⊢ (φ → F Fn Nn ) |
21 | fnopfvb 5360 | . . 3 ⊢ ((F Fn Nn ∧ 0c ∈ Nn ) → ((F ‘0c) = I ↔ 〈0c, I〉 ∈ F)) | |
22 | 20, 1, 21 | sylancl 643 | . 2 ⊢ (φ → ((F ‘0c) = I ↔ 〈0c, I〉 ∈ F)) |
23 | 18, 22 | mpbird 223 | 1 ⊢ (φ → (F ‘0c) = I) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∨ wo 357 = wceq 1642 ∈ wcel 1710 ∃wrex 2616 Vcvv 2860 ⊆ wss 3258 {csn 3738 1cc1c 4135 Nn cnnc 4374 0cc0c 4375 +c cplc 4376 〈cop 4562 class class class wbr 4640 dom cdm 4773 ran crn 4774 Fn wfn 4777 ‘cfv 4782 ↦ cmpt 5652 PProd cpprod 5738 Funs cfuns 5760 Clos1 cclos1 5873 FRec cfrec 6310 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4079 ax-xp 4080 ax-cnv 4081 ax-1c 4082 ax-sset 4083 ax-si 4084 ax-ins2 4085 ax-ins3 4086 ax-typlower 4087 ax-sn 4088 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2479 df-ne 2519 df-ral 2620 df-rex 2621 df-reu 2622 df-rmo 2623 df-rab 2624 df-v 2862 df-sbc 3048 df-csb 3138 df-nin 3212 df-compl 3213 df-in 3214 df-un 3215 df-dif 3216 df-symdif 3217 df-ss 3260 df-pss 3262 df-nul 3552 df-if 3664 df-pw 3725 df-sn 3742 df-pr 3743 df-uni 3893 df-int 3928 df-iun 3972 df-opk 4059 df-1c 4137 df-pw1 4138 df-uni1 4139 df-xpk 4186 df-cnvk 4187 df-ins2k 4188 df-ins3k 4189 df-imak 4190 df-cok 4191 df-p6 4192 df-sik 4193 df-ssetk 4194 df-imagek 4195 df-idk 4196 df-iota 4340 df-0c 4378 df-addc 4379 df-nnc 4380 df-fin 4381 df-lefin 4441 df-ltfin 4442 df-ncfin 4443 df-tfin 4444 df-evenfin 4445 df-oddfin 4446 df-sfin 4447 df-spfin 4448 df-phi 4566 df-op 4567 df-proj1 4568 df-proj2 4569 df-opab 4624 df-br 4641 df-1st 4724 df-swap 4725 df-sset 4726 df-co 4727 df-ima 4728 df-si 4729 df-id 4768 df-xp 4785 df-cnv 4786 df-rn 4787 df-dm 4788 df-res 4789 df-fun 4790 df-fn 4791 df-f 4792 df-fo 4794 df-fv 4796 df-2nd 4798 df-ov 5527 df-oprab 5529 df-mpt 5653 df-mpt2 5655 df-txp 5737 df-pprod 5739 df-fix 5741 df-cup 5743 df-disj 5745 df-addcfn 5747 df-ins2 5751 df-ins3 5753 df-image 5755 df-ins4 5757 df-si3 5759 df-funs 5761 df-clos1 5874 df-frec 6311 |
This theorem is referenced by: (None) |
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