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Mirrors > Home > ILE Home > Th. List > climle | GIF version |
Description: Comparison of the limits of two sequences. (Contributed by Paul Chapman, 10-Sep-2007.) (Revised by Mario Carneiro, 1-Feb-2014.) |
Ref | Expression |
---|---|
climadd.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climadd.2 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
climadd.4 | ⊢ (𝜑 → 𝐹 ⇝ 𝐴) |
climle.5 | ⊢ (𝜑 → 𝐺 ⇝ 𝐵) |
climle.6 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) |
climle.7 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℝ) |
climle.8 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ≤ (𝐺‘𝑘)) |
Ref | Expression |
---|---|
climle | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climadd.1 | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | climadd.2 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
3 | climle.5 | . . . 4 ⊢ (𝜑 → 𝐺 ⇝ 𝐵) | |
4 | zex 9238 | . . . . . . . 8 ⊢ ℤ ∈ V | |
5 | uzssz 9523 | . . . . . . . 8 ⊢ (ℤ≥‘𝑀) ⊆ ℤ | |
6 | 4, 5 | ssexi 4138 | . . . . . . 7 ⊢ (ℤ≥‘𝑀) ∈ V |
7 | 1, 6 | eqeltri 2250 | . . . . . 6 ⊢ 𝑍 ∈ V |
8 | 7 | mptex 5737 | . . . . 5 ⊢ (𝑗 ∈ 𝑍 ↦ ((𝐺‘𝑗) − (𝐹‘𝑗))) ∈ V |
9 | 8 | a1i 9 | . . . 4 ⊢ (𝜑 → (𝑗 ∈ 𝑍 ↦ ((𝐺‘𝑗) − (𝐹‘𝑗))) ∈ V) |
10 | climadd.4 | . . . 4 ⊢ (𝜑 → 𝐹 ⇝ 𝐴) | |
11 | climle.7 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℝ) | |
12 | 11 | recnd 7963 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ) |
13 | climle.6 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) | |
14 | 13 | recnd 7963 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
15 | simpr 110 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝑍) | |
16 | 11, 13 | resubcld 8315 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐺‘𝑘) − (𝐹‘𝑘)) ∈ ℝ) |
17 | fveq2 5510 | . . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝐺‘𝑗) = (𝐺‘𝑘)) | |
18 | fveq2 5510 | . . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝐹‘𝑗) = (𝐹‘𝑘)) | |
19 | 17, 18 | oveq12d 5886 | . . . . . 6 ⊢ (𝑗 = 𝑘 → ((𝐺‘𝑗) − (𝐹‘𝑗)) = ((𝐺‘𝑘) − (𝐹‘𝑘))) |
20 | eqid 2177 | . . . . . 6 ⊢ (𝑗 ∈ 𝑍 ↦ ((𝐺‘𝑗) − (𝐹‘𝑗))) = (𝑗 ∈ 𝑍 ↦ ((𝐺‘𝑗) − (𝐹‘𝑗))) | |
21 | 19, 20 | fvmptg 5587 | . . . . 5 ⊢ ((𝑘 ∈ 𝑍 ∧ ((𝐺‘𝑘) − (𝐹‘𝑘)) ∈ ℝ) → ((𝑗 ∈ 𝑍 ↦ ((𝐺‘𝑗) − (𝐹‘𝑗)))‘𝑘) = ((𝐺‘𝑘) − (𝐹‘𝑘))) |
22 | 15, 16, 21 | syl2anc 411 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑗 ∈ 𝑍 ↦ ((𝐺‘𝑗) − (𝐹‘𝑗)))‘𝑘) = ((𝐺‘𝑘) − (𝐹‘𝑘))) |
23 | 1, 2, 3, 9, 10, 12, 14, 22 | climsub 11307 | . . 3 ⊢ (𝜑 → (𝑗 ∈ 𝑍 ↦ ((𝐺‘𝑗) − (𝐹‘𝑗))) ⇝ (𝐵 − 𝐴)) |
24 | 22, 16 | eqeltrd 2254 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑗 ∈ 𝑍 ↦ ((𝐺‘𝑗) − (𝐹‘𝑗)))‘𝑘) ∈ ℝ) |
25 | climle.8 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ≤ (𝐺‘𝑘)) | |
26 | 11, 13 | subge0d 8469 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (0 ≤ ((𝐺‘𝑘) − (𝐹‘𝑘)) ↔ (𝐹‘𝑘) ≤ (𝐺‘𝑘))) |
27 | 25, 26 | mpbird 167 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ ((𝐺‘𝑘) − (𝐹‘𝑘))) |
28 | 27, 22 | breqtrrd 4028 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ ((𝑗 ∈ 𝑍 ↦ ((𝐺‘𝑗) − (𝐹‘𝑗)))‘𝑘)) |
29 | 1, 2, 23, 24, 28 | climge0 11304 | . 2 ⊢ (𝜑 → 0 ≤ (𝐵 − 𝐴)) |
30 | 1, 2, 3, 11 | climrecl 11303 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) |
31 | 1, 2, 10, 13 | climrecl 11303 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
32 | 30, 31 | subge0d 8469 | . 2 ⊢ (𝜑 → (0 ≤ (𝐵 − 𝐴) ↔ 𝐴 ≤ 𝐵)) |
33 | 29, 32 | mpbid 147 | 1 ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 Vcvv 2737 class class class wbr 4000 ↦ cmpt 4061 ‘cfv 5211 (class class class)co 5868 ℝcr 7788 0cc0 7789 ≤ cle 7970 − cmin 8105 ℤcz 9229 ℤ≥cuz 9504 ⇝ cli 11257 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-iinf 4583 ax-cnex 7880 ax-resscn 7881 ax-1cn 7882 ax-1re 7883 ax-icn 7884 ax-addcl 7885 ax-addrcl 7886 ax-mulcl 7887 ax-mulrcl 7888 ax-addcom 7889 ax-mulcom 7890 ax-addass 7891 ax-mulass 7892 ax-distr 7893 ax-i2m1 7894 ax-0lt1 7895 ax-1rid 7896 ax-0id 7897 ax-rnegex 7898 ax-precex 7899 ax-cnre 7900 ax-pre-ltirr 7901 ax-pre-ltwlin 7902 ax-pre-lttrn 7903 ax-pre-apti 7904 ax-pre-ltadd 7905 ax-pre-mulgt0 7906 ax-pre-mulext 7907 ax-arch 7908 ax-caucvg 7909 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4289 df-po 4292 df-iso 4293 df-iord 4362 df-on 4364 df-ilim 4365 df-suc 4367 df-iom 4586 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 df-fv 5219 df-riota 5824 df-ov 5871 df-oprab 5872 df-mpo 5873 df-1st 6134 df-2nd 6135 df-recs 6299 df-frec 6385 df-pnf 7971 df-mnf 7972 df-xr 7973 df-ltxr 7974 df-le 7975 df-sub 8107 df-neg 8108 df-reap 8509 df-ap 8516 df-div 8606 df-inn 8896 df-2 8954 df-3 8955 df-4 8956 df-n0 9153 df-z 9230 df-uz 9505 df-rp 9628 df-seqfrec 10419 df-exp 10493 df-cj 10822 df-re 10823 df-im 10824 df-rsqrt 10978 df-abs 10979 df-clim 11258 |
This theorem is referenced by: climlec2 11320 iserle 11321 iserabs 11454 cvgcmpub 11455 |
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