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Mirrors > Home > MPE Home > Th. List > climlec2 | Structured version Visualization version GIF version |
Description: Comparison of a constant to the limit of a sequence. (Contributed by NM, 28-Feb-2008.) (Revised by Mario Carneiro, 1-Feb-2014.) |
Ref | Expression |
---|---|
clim2ser.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climlec2.2 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
climlec2.3 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
climlec2.4 | ⊢ (𝜑 → 𝐹 ⇝ 𝐵) |
climlec2.5 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) |
climlec2.6 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ≤ (𝐹‘𝑘)) |
Ref | Expression |
---|---|
climlec2 | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clim2ser.1 | . 2 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | climlec2.2 | . 2 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
3 | climlec2.3 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
4 | 3 | recnd 11188 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
5 | 0z 12515 | . . 3 ⊢ 0 ∈ ℤ | |
6 | uzssz 12789 | . . . 4 ⊢ (ℤ≥‘0) ⊆ ℤ | |
7 | zex 12513 | . . . 4 ⊢ ℤ ∈ V | |
8 | 6, 7 | climconst2 15436 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℤ) → (ℤ × {𝐴}) ⇝ 𝐴) |
9 | 4, 5, 8 | sylancl 587 | . 2 ⊢ (𝜑 → (ℤ × {𝐴}) ⇝ 𝐴) |
10 | climlec2.4 | . 2 ⊢ (𝜑 → 𝐹 ⇝ 𝐵) | |
11 | eluzelz 12778 | . . . . 5 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → 𝑘 ∈ ℤ) | |
12 | 11, 1 | eleq2s 2852 | . . . 4 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℤ) |
13 | fvconst2g 7152 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℤ) → ((ℤ × {𝐴})‘𝑘) = 𝐴) | |
14 | 3, 12, 13 | syl2an 597 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((ℤ × {𝐴})‘𝑘) = 𝐴) |
15 | 3 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℝ) |
16 | 14, 15 | eqeltrd 2834 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((ℤ × {𝐴})‘𝑘) ∈ ℝ) |
17 | climlec2.5 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) | |
18 | climlec2.6 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ≤ (𝐹‘𝑘)) | |
19 | 14, 18 | eqbrtrd 5128 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((ℤ × {𝐴})‘𝑘) ≤ (𝐹‘𝑘)) |
20 | 1, 2, 9, 10, 16, 17, 19 | climle 15528 | 1 ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {csn 4587 class class class wbr 5106 × cxp 5632 ‘cfv 6497 ℂcc 11054 ℝcr 11055 0cc0 11056 ≤ cle 11195 ℤcz 12504 ℤ≥cuz 12768 ⇝ cli 15372 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-pm 8771 df-en 8887 df-dom 8888 df-sdom 8889 df-sup 9383 df-inf 9384 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-n0 12419 df-z 12505 df-uz 12769 df-rp 12921 df-fl 13703 df-seq 13913 df-exp 13974 df-cj 14990 df-re 14991 df-im 14992 df-sqrt 15126 df-abs 15127 df-clim 15376 df-rlim 15377 |
This theorem is referenced by: climub 15552 climlec3 34362 dvgrat 42680 |
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