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Mirrors > Home > MPE Home > Th. List > climsubc1 | Structured version Visualization version GIF version |
Description: Limit of a constant 𝐶 subtracted from each term of a sequence. (Contributed by Mario Carneiro, 9-Feb-2014.) |
Ref | Expression |
---|---|
climadd.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climadd.2 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
climadd.4 | ⊢ (𝜑 → 𝐹 ⇝ 𝐴) |
climaddc1.5 | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
climaddc1.6 | ⊢ (𝜑 → 𝐺 ∈ 𝑊) |
climaddc1.7 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
climsubc1.h | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = ((𝐹‘𝑘) − 𝐶)) |
Ref | Expression |
---|---|
climsubc1 | ⊢ (𝜑 → 𝐺 ⇝ (𝐴 − 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climadd.1 | . 2 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | climadd.2 | . 2 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
3 | climadd.4 | . 2 ⊢ (𝜑 → 𝐹 ⇝ 𝐴) | |
4 | climaddc1.6 | . 2 ⊢ (𝜑 → 𝐺 ∈ 𝑊) | |
5 | climaddc1.5 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
6 | 0z 12621 | . . 3 ⊢ 0 ∈ ℤ | |
7 | uzssz 12895 | . . . 4 ⊢ (ℤ≥‘0) ⊆ ℤ | |
8 | zex 12619 | . . . 4 ⊢ ℤ ∈ V | |
9 | 7, 8 | climconst2 15550 | . . 3 ⊢ ((𝐶 ∈ ℂ ∧ 0 ∈ ℤ) → (ℤ × {𝐶}) ⇝ 𝐶) |
10 | 5, 6, 9 | sylancl 584 | . 2 ⊢ (𝜑 → (ℤ × {𝐶}) ⇝ 𝐶) |
11 | climaddc1.7 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) | |
12 | eluzelz 12884 | . . . . 5 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → 𝑘 ∈ ℤ) | |
13 | 12, 1 | eleq2s 2844 | . . . 4 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℤ) |
14 | fvconst2g 7219 | . . . 4 ⊢ ((𝐶 ∈ ℂ ∧ 𝑘 ∈ ℤ) → ((ℤ × {𝐶})‘𝑘) = 𝐶) | |
15 | 5, 13, 14 | syl2an 594 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((ℤ × {𝐶})‘𝑘) = 𝐶) |
16 | 5 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐶 ∈ ℂ) |
17 | 15, 16 | eqeltrd 2826 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((ℤ × {𝐶})‘𝑘) ∈ ℂ) |
18 | climsubc1.h | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = ((𝐹‘𝑘) − 𝐶)) | |
19 | 15 | oveq2d 7440 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) − ((ℤ × {𝐶})‘𝑘)) = ((𝐹‘𝑘) − 𝐶)) |
20 | 18, 19 | eqtr4d 2769 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = ((𝐹‘𝑘) − ((ℤ × {𝐶})‘𝑘))) |
21 | 1, 2, 3, 4, 10, 11, 17, 20 | climsub 15636 | 1 ⊢ (𝜑 → 𝐺 ⇝ (𝐴 − 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 {csn 4633 class class class wbr 5153 × cxp 5680 ‘cfv 6554 (class class class)co 7424 ℂcc 11156 0cc0 11158 − cmin 11494 ℤcz 12610 ℤ≥cuz 12874 ⇝ cli 15486 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-sup 9485 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12611 df-uz 12875 df-rp 13029 df-seq 14022 df-exp 14082 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-clim 15490 |
This theorem is referenced by: clim2ser 15659 ulmdvlem1 26429 fourierdlem112 45839 |
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