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Mirrors > Home > ILE Home > Th. List > frec2uzsucd | GIF version |
Description: The value of 𝐺 (see frec2uz0d 9185) 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 | frec2uz.1 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℤ) | |
2 | frec2uzzd.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ω) | |
3 | zex 8254 | . . . . . . . 8 ⊢ ℤ ∈ V | |
4 | 3 | mptex 5387 | . . . . . . 7 ⊢ (𝑥 ∈ ℤ ↦ (𝑥 + 1)) ∈ V |
5 | vex 2560 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
6 | 4, 5 | fvex 5195 | . . . . . 6 ⊢ ((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑦) ∈ V |
7 | 6 | ax-gen 1338 | . . . . 5 ⊢ ∀𝑦((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑦) ∈ V |
8 | frecsuc 5991 | . . . . 5 ⊢ ((∀𝑦((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑦) ∈ V ∧ 𝐶 ∈ ℤ ∧ 𝐴 ∈ ω) → (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)‘suc 𝐴) = ((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘(frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)‘𝐴))) | |
9 | 7, 8 | mp3an1 1219 | . . . 4 ⊢ ((𝐶 ∈ ℤ ∧ 𝐴 ∈ ω) → (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)‘suc 𝐴) = ((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘(frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)‘𝐴))) |
10 | 1, 2, 9 | syl2anc 391 | . . 3 ⊢ (𝜑 → (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)‘suc 𝐴) = ((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘(frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)‘𝐴))) |
11 | frec2uz.2 | . . . 4 ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) | |
12 | 11 | fveq1i 5179 | . . 3 ⊢ (𝐺‘suc 𝐴) = (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)‘suc 𝐴) |
13 | 11 | fveq1i 5179 | . . . 4 ⊢ (𝐺‘𝐴) = (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)‘𝐴) |
14 | 13 | fveq2i 5181 | . . 3 ⊢ ((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘(𝐺‘𝐴)) = ((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘(frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)‘𝐴)) |
15 | 10, 12, 14 | 3eqtr4g 2097 | . 2 ⊢ (𝜑 → (𝐺‘suc 𝐴) = ((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘(𝐺‘𝐴))) |
16 | 1, 11, 2 | frec2uzzd 9186 | . . 3 ⊢ (𝜑 → (𝐺‘𝐴) ∈ ℤ) |
17 | oveq1 5519 | . . . 4 ⊢ (𝑧 = (𝐺‘𝐴) → (𝑧 + 1) = ((𝐺‘𝐴) + 1)) | |
18 | oveq1 5519 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 + 1) = (𝑧 + 1)) | |
19 | 18 | cbvmptv 3852 | . . . 4 ⊢ (𝑥 ∈ ℤ ↦ (𝑥 + 1)) = (𝑧 ∈ ℤ ↦ (𝑧 + 1)) |
20 | peano2z 8281 | . . . 4 ⊢ (𝑧 ∈ ℤ → (𝑧 + 1) ∈ ℤ) | |
21 | 17, 19, 20 | fvmpt3 5251 | . . 3 ⊢ ((𝐺‘𝐴) ∈ ℤ → ((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘(𝐺‘𝐴)) = ((𝐺‘𝐴) + 1)) |
22 | 16, 21 | syl 14 | . 2 ⊢ (𝜑 → ((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘(𝐺‘𝐴)) = ((𝐺‘𝐴) + 1)) |
23 | 15, 22 | eqtrd 2072 | 1 ⊢ (𝜑 → (𝐺‘suc 𝐴) = ((𝐺‘𝐴) + 1)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1241 = wceq 1243 ∈ wcel 1393 Vcvv 2557 ↦ cmpt 3818 suc csuc 4102 ωcom 4313 ‘cfv 4902 (class class class)co 5512 freccfrec 5977 1c1 6890 + caddc 6892 ℤcz 8245 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-1re 6978 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-addcom 6984 ax-addass 6986 ax-distr 6988 ax-i2m1 6989 ax-0id 6992 ax-rnegex 6993 ax-cnre 6995 |
This theorem depends on definitions: df-bi 110 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-id 4030 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-recs 5920 df-frec 5978 df-sub 7184 df-neg 7185 df-inn 7915 df-n0 8182 df-z 8246 |
This theorem is referenced by: frec2uzuzd 9188 frec2uzltd 9189 frec2uzrand 9191 frec2uzrdg 9195 frecuzrdgsuc 9201 frecfzennn 9203 |
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