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Mirrors > Home > ILE Home > Th. List > frec2uzsucd | GIF version |
Description: The value of 𝐺 (see frec2uz0d 10324) at a successor. (Contributed by Jim Kingdon, 16-May-2020.) |
Ref | Expression |
---|---|
frec2uz.1 | ⊢ (𝜑 → 𝐶 ∈ ℤ) |
frec2uz.2 | ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) |
frec2uzzd.a | ⊢ (𝜑 → 𝐴 ∈ ω) |
Ref | Expression |
---|---|
frec2uzsucd | ⊢ (𝜑 → (𝐺‘suc 𝐴) = ((𝐺‘𝐴) + 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano2z 9218 | . . . . . . 7 ⊢ (𝑧 ∈ ℤ → (𝑧 + 1) ∈ ℤ) | |
2 | oveq1 5843 | . . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑥 + 1) = (𝑧 + 1)) | |
3 | eqid 2164 | . . . . . . . 8 ⊢ (𝑥 ∈ ℤ ↦ (𝑥 + 1)) = (𝑥 ∈ ℤ ↦ (𝑥 + 1)) | |
4 | 2, 3 | fvmptg 5556 | . . . . . . 7 ⊢ ((𝑧 ∈ ℤ ∧ (𝑧 + 1) ∈ ℤ) → ((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) = (𝑧 + 1)) |
5 | 1, 4 | mpdan 418 | . . . . . 6 ⊢ (𝑧 ∈ ℤ → ((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) = (𝑧 + 1)) |
6 | 5, 1 | eqeltrd 2241 | . . . . 5 ⊢ (𝑧 ∈ ℤ → ((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) ∈ ℤ) |
7 | 6 | rgen 2517 | . . . 4 ⊢ ∀𝑧 ∈ ℤ ((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) ∈ ℤ |
8 | frec2uz.1 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℤ) | |
9 | frec2uzzd.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ω) | |
10 | frecsuc 6366 | . . . 4 ⊢ ((∀𝑧 ∈ ℤ ((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) ∈ ℤ ∧ 𝐶 ∈ ℤ ∧ 𝐴 ∈ ω) → (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)‘suc 𝐴) = ((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘(frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)‘𝐴))) | |
11 | 7, 8, 9, 10 | mp3an2i 1331 | . . 3 ⊢ (𝜑 → (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)‘suc 𝐴) = ((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘(frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)‘𝐴))) |
12 | frec2uz.2 | . . . 4 ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) | |
13 | 12 | fveq1i 5481 | . . 3 ⊢ (𝐺‘suc 𝐴) = (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)‘suc 𝐴) |
14 | 12 | fveq1i 5481 | . . . 4 ⊢ (𝐺‘𝐴) = (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)‘𝐴) |
15 | 14 | fveq2i 5483 | . . 3 ⊢ ((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘(𝐺‘𝐴)) = ((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘(frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)‘𝐴)) |
16 | 11, 13, 15 | 3eqtr4g 2222 | . 2 ⊢ (𝜑 → (𝐺‘suc 𝐴) = ((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘(𝐺‘𝐴))) |
17 | 8, 12, 9 | frec2uzzd 10325 | . . 3 ⊢ (𝜑 → (𝐺‘𝐴) ∈ ℤ) |
18 | oveq1 5843 | . . . 4 ⊢ (𝑧 = (𝐺‘𝐴) → (𝑧 + 1) = ((𝐺‘𝐴) + 1)) | |
19 | 2 | cbvmptv 4072 | . . . 4 ⊢ (𝑥 ∈ ℤ ↦ (𝑥 + 1)) = (𝑧 ∈ ℤ ↦ (𝑧 + 1)) |
20 | 18, 19, 1 | fvmpt3 5559 | . . 3 ⊢ ((𝐺‘𝐴) ∈ ℤ → ((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘(𝐺‘𝐴)) = ((𝐺‘𝐴) + 1)) |
21 | 17, 20 | syl 14 | . 2 ⊢ (𝜑 → ((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘(𝐺‘𝐴)) = ((𝐺‘𝐴) + 1)) |
22 | 16, 21 | eqtrd 2197 | 1 ⊢ (𝜑 → (𝐺‘suc 𝐴) = ((𝐺‘𝐴) + 1)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ∈ wcel 2135 ∀wral 2442 ↦ cmpt 4037 suc csuc 4337 ωcom 4561 ‘cfv 5182 (class class class)co 5836 freccfrec 6349 1c1 7745 + caddc 7747 ℤcz 9182 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-recs 6264 df-frec 6350 df-sub 8062 df-neg 8063 df-inn 8849 df-n0 9106 df-z 9183 |
This theorem is referenced by: frec2uzuzd 10327 frec2uzltd 10328 frec2uzrand 10330 frec2uzrdg 10334 frecuzrdgsuc 10339 frecuzrdgg 10341 frecfzennn 10351 1tonninf 10365 omgadd 10704 ennnfonelemkh 12288 ennnfonelemhf1o 12289 ennnfonelemnn0 12298 012of 13716 2o01f 13717 isomninnlem 13750 iswomninnlem 13769 ismkvnnlem 13772 |
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