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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 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐶 ∈ ℂ) |
8 | climaddc2.h | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐶 + (𝐹‘𝑘))) | |
9 | 7, 6, 8 | comraddd 11458 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = ((𝐹‘𝑘) + 𝐶)) |
10 | 1, 2, 3, 4, 5, 6, 9 | climaddc1 15611 | . 2 ⊢ (𝜑 → 𝐺 ⇝ (𝐴 + 𝐶)) |
11 | climcl 15475 | . . . 4 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ) | |
12 | 3, 11 | syl 17 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
13 | 12, 4 | addcomd 11446 | . 2 ⊢ (𝜑 → (𝐴 + 𝐶) = (𝐶 + 𝐴)) |
14 | 10, 13 | breqtrd 5169 | 1 ⊢ (𝜑 → 𝐺 ⇝ (𝐶 + 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 class class class wbr 5143 ‘cfv 6543 (class class class)co 7416 ℂcc 11136 + caddc 11141 ℤcz 12588 ℤ≥cuz 12852 ⇝ cli 15460 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-sup 9465 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-n0 12503 df-z 12589 df-uz 12853 df-rp 13007 df-seq 13999 df-exp 14059 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-clim 15464 |
This theorem is referenced by: isumsplit 15818 divcnvlin 35384 |
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