Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > climaddc1 | 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) |
climaddc1.h | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = ((𝐹‘𝑘) + 𝐶)) |
Ref | Expression |
---|---|
climaddc1 | ⊢ (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 8256 | . . 3 ⊢ 0 ∈ ℤ | |
7 | uzssz 8492 | . . . 4 ⊢ (ℤ≥‘0) ⊆ ℤ | |
8 | zex 8254 | . . . 4 ⊢ ℤ ∈ V | |
9 | 7, 8 | climconst2 9812 | . . 3 ⊢ ((𝐶 ∈ ℂ ∧ 0 ∈ ℤ) → (ℤ × {𝐶}) ⇝ 𝐶) |
10 | 5, 6, 9 | sylancl 392 | . 2 ⊢ (𝜑 → (ℤ × {𝐶}) ⇝ 𝐶) |
11 | climaddc1.7 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) | |
12 | eluzelz 8482 | . . . . 5 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → 𝑘 ∈ ℤ) | |
13 | 12, 1 | eleq2s 2132 | . . . 4 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℤ) |
14 | fvconst2g 5375 | . . . 4 ⊢ ((𝐶 ∈ ℂ ∧ 𝑘 ∈ ℤ) → ((ℤ × {𝐶})‘𝑘) = 𝐶) | |
15 | 5, 13, 14 | syl2an 273 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((ℤ × {𝐶})‘𝑘) = 𝐶) |
16 | 5 | adantr 261 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐶 ∈ ℂ) |
17 | 15, 16 | eqeltrd 2114 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((ℤ × {𝐶})‘𝑘) ∈ ℂ) |
18 | climaddc1.h | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = ((𝐹‘𝑘) + 𝐶)) | |
19 | 15 | oveq2d 5528 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) + ((ℤ × {𝐶})‘𝑘)) = ((𝐹‘𝑘) + 𝐶)) |
20 | 18, 19 | eqtr4d 2075 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = ((𝐹‘𝑘) + ((ℤ × {𝐶})‘𝑘))) |
21 | 1, 2, 3, 4, 10, 11, 17, 20 | climadd 9846 | 1 ⊢ (𝜑 → 𝐺 ⇝ (𝐴 + 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∈ wcel 1393 {csn 3375 class class class wbr 3764 × cxp 4343 ‘cfv 4902 (class class class)co 5512 ℂcc 6887 0cc0 6889 + caddc 6892 ℤcz 8245 ℤ≥cuz 8473 ⇝ cli 9799 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-1re 6978 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-mulrcl 6983 ax-addcom 6984 ax-mulcom 6985 ax-addass 6986 ax-mulass 6987 ax-distr 6988 ax-i2m1 6989 ax-1rid 6991 ax-0id 6992 ax-rnegex 6993 ax-precex 6994 ax-cnre 6995 ax-pre-ltirr 6996 ax-pre-ltwlin 6997 ax-pre-lttrn 6998 ax-pre-apti 6999 ax-pre-ltadd 7000 ax-pre-mulgt0 7001 ax-pre-mulext 7002 ax-arch 7003 ax-caucvg 7004 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-reu 2313 df-rmo 2314 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-if 3332 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-frec 5978 df-1o 6001 df-2o 6002 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-i1p 6565 df-iplp 6566 df-iltp 6568 df-enr 6811 df-nr 6812 df-ltr 6815 df-0r 6816 df-1r 6817 df-0 6896 df-1 6897 df-r 6899 df-lt 6902 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 df-le 7066 df-sub 7184 df-neg 7185 df-reap 7566 df-ap 7573 df-div 7652 df-inn 7915 df-2 7973 df-3 7974 df-4 7975 df-n0 8182 df-z 8246 df-uz 8474 df-rp 8584 df-iseq 9212 df-iexp 9255 df-cj 9442 df-re 9443 df-im 9444 df-rsqrt 9596 df-abs 9597 df-clim 9800 |
This theorem is referenced by: climaddc2 9850 clim2iser2 9858 |
Copyright terms: Public domain | W3C validator |