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Mirrors > Home > ILE Home > Th. List > climmpt | GIF version |
Description: Exhibit a function 𝐺 with the same convergence properties as the not-quite-function 𝐹. (Contributed by Mario Carneiro, 31-Jan-2014.) |
Ref | Expression |
---|---|
2clim.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climmpt.2 | ⊢ 𝐺 = (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) |
Ref | Expression |
---|---|
climmpt | ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2clim.1 | . 2 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | simpr 109 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → 𝐹 ∈ 𝑉) | |
3 | climmpt.2 | . . . 4 ⊢ 𝐺 = (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) | |
4 | uzf 9490 | . . . . . . . 8 ⊢ ℤ≥:ℤ⟶𝒫 ℤ | |
5 | 4 | ffvelrni 5630 | . . . . . . 7 ⊢ (𝑀 ∈ ℤ → (ℤ≥‘𝑀) ∈ 𝒫 ℤ) |
6 | elex 2741 | . . . . . . 7 ⊢ ((ℤ≥‘𝑀) ∈ 𝒫 ℤ → (ℤ≥‘𝑀) ∈ V) | |
7 | 5, 6 | syl 14 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → (ℤ≥‘𝑀) ∈ V) |
8 | 1, 7 | eqeltrid 2257 | . . . . 5 ⊢ (𝑀 ∈ ℤ → 𝑍 ∈ V) |
9 | mptexg 5721 | . . . . 5 ⊢ (𝑍 ∈ V → (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) ∈ V) | |
10 | 8, 9 | syl 14 | . . . 4 ⊢ (𝑀 ∈ ℤ → (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) ∈ V) |
11 | 3, 10 | eqeltrid 2257 | . . 3 ⊢ (𝑀 ∈ ℤ → 𝐺 ∈ V) |
12 | 11 | adantr 274 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → 𝐺 ∈ V) |
13 | simpl 108 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → 𝑀 ∈ ℤ) | |
14 | simpr 109 | . . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) ∧ 𝑚 ∈ 𝑍) → 𝑚 ∈ 𝑍) | |
15 | fvexg 5515 | . . . . 5 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) ∈ V) | |
16 | 15 | adantll 473 | . . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) ∈ V) |
17 | fveq2 5496 | . . . . 5 ⊢ (𝑘 = 𝑚 → (𝐹‘𝑘) = (𝐹‘𝑚)) | |
18 | 17, 3 | fvmptg 5572 | . . . 4 ⊢ ((𝑚 ∈ 𝑍 ∧ (𝐹‘𝑚) ∈ V) → (𝐺‘𝑚) = (𝐹‘𝑚)) |
19 | 14, 16, 18 | syl2anc 409 | . . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) ∧ 𝑚 ∈ 𝑍) → (𝐺‘𝑚) = (𝐹‘𝑚)) |
20 | 19 | eqcomd 2176 | . 2 ⊢ (((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) = (𝐺‘𝑚)) |
21 | 1, 2, 12, 13, 20 | climeq 11262 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1348 ∈ wcel 2141 Vcvv 2730 𝒫 cpw 3566 class class class wbr 3989 ↦ cmpt 4050 ‘cfv 5198 ℤcz 9212 ℤ≥cuz 9487 ⇝ cli 11241 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-addass 7876 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-inn 8879 df-n0 9136 df-z 9213 df-uz 9488 df-clim 11242 |
This theorem is referenced by: (None) |
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