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Mirrors > Home > MPE Home > Th. List > climaddc2 | Structured version Visualization version GIF version |
Description: Limit of a constant 𝐶 added to each term of a sequence. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 3-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) |
climaddc2.h | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐶 + (𝐹‘𝑘))) |
Ref | Expression |
---|---|
climaddc2 | ⊢ (𝜑 → 𝐺 ⇝ (𝐶 + 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climadd.1 | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | climadd.2 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
3 | climadd.4 | . . 3 ⊢ (𝜑 → 𝐹 ⇝ 𝐴) | |
4 | climaddc1.5 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
5 | climaddc1.6 | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑊) | |
6 | climaddc1.7 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) | |
7 | 4 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐶 ∈ ℂ) |
8 | climaddc2.h | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐶 + (𝐹‘𝑘))) | |
9 | 7, 6, 8 | comraddd 10843 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = ((𝐹‘𝑘) + 𝐶)) |
10 | 1, 2, 3, 4, 5, 6, 9 | climaddc1 14983 | . 2 ⊢ (𝜑 → 𝐺 ⇝ (𝐴 + 𝐶)) |
11 | climcl 14848 | . . . 4 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ) | |
12 | 3, 11 | syl 17 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
13 | 12, 4 | addcomd 10831 | . 2 ⊢ (𝜑 → (𝐴 + 𝐶) = (𝐶 + 𝐴)) |
14 | 10, 13 | breqtrd 5056 | 1 ⊢ (𝜑 → 𝐺 ⇝ (𝐶 + 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 + caddc 10529 ℤcz 11969 ℤ≥cuz 12231 ⇝ cli 14833 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 |
This theorem is referenced by: isumsplit 15187 divcnvlin 33077 |
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