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Mirrors > Home > ILE Home > Th. List > frec2uzzd | GIF version |
Description: The value of 𝐺 (see frec2uz0d 9955) is an integer. (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 |
---|---|
frec2uzzd | ⊢ (𝜑 → (𝐺‘𝐴) ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frec2uz.2 | . . 3 ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) | |
2 | 1 | fveq1i 5341 | . 2 ⊢ (𝐺‘𝐴) = (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)‘𝐴) |
3 | frec2uz.1 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℤ) | |
4 | simpr 109 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → 𝑘 ∈ ℤ) | |
5 | 4 | peano2zd 8970 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (𝑘 + 1) ∈ ℤ) |
6 | oveq1 5697 | . . . . . 6 ⊢ (𝑥 = 𝑘 → (𝑥 + 1) = (𝑘 + 1)) | |
7 | eqid 2095 | . . . . . 6 ⊢ (𝑥 ∈ ℤ ↦ (𝑥 + 1)) = (𝑥 ∈ ℤ ↦ (𝑥 + 1)) | |
8 | 6, 7 | fvmptg 5415 | . . . . 5 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 + 1) ∈ ℤ) → ((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑘) = (𝑘 + 1)) |
9 | 4, 5, 8 | syl2anc 404 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑘) = (𝑘 + 1)) |
10 | 9, 5 | eqeltrd 2171 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → ((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑘) ∈ ℤ) |
11 | frec2uzzd.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ω) | |
12 | 3, 10, 11 | freccl 6206 | . 2 ⊢ (𝜑 → (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)‘𝐴) ∈ ℤ) |
13 | 2, 12 | syl5eqel 2181 | 1 ⊢ (𝜑 → (𝐺‘𝐴) ∈ ℤ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1296 ∈ wcel 1445 ↦ cmpt 3921 ωcom 4433 ‘cfv 5049 (class class class)co 5690 freccfrec 6193 1c1 7448 + caddc 7450 ℤcz 8848 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-coll 3975 ax-sep 3978 ax-nul 3986 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-iinf 4431 ax-cnex 7533 ax-resscn 7534 ax-1cn 7535 ax-1re 7536 ax-icn 7537 ax-addcl 7538 ax-addrcl 7539 ax-mulcl 7540 ax-addcom 7542 ax-addass 7544 ax-distr 7546 ax-i2m1 7547 ax-0id 7550 ax-rnegex 7551 ax-cnre 7553 |
This theorem depends on definitions: df-bi 116 df-3or 928 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-ral 2375 df-rex 2376 df-reu 2377 df-rab 2379 df-v 2635 df-sbc 2855 df-csb 2948 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-nul 3303 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-int 3711 df-iun 3754 df-br 3868 df-opab 3922 df-mpt 3923 df-tr 3959 df-id 4144 df-iord 4217 df-on 4219 df-ilim 4220 df-suc 4222 df-iom 4434 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-f1 5054 df-fo 5055 df-f1o 5056 df-fv 5057 df-riota 5646 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-recs 6108 df-frec 6194 df-sub 7752 df-neg 7753 df-inn 8521 df-n0 8772 df-z 8849 |
This theorem is referenced by: frec2uzsucd 9957 frec2uzltd 9959 frec2uzlt2d 9960 frec2uzf1od 9962 frec2uzrdg 9965 frec2uzled 9985 |
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