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Mirrors > Home > ILE Home > Th. List > climconst2 | GIF version |
Description: A constant sequence converges to its value. (Contributed by NM, 6-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.) |
Ref | Expression |
---|---|
climconst2.1 | ⊢ (ℤ≥‘𝑀) ⊆ 𝑍 |
climconst2.2 | ⊢ 𝑍 ∈ V |
Ref | Expression |
---|---|
climconst2 | ⊢ ((𝐴 ∈ ℂ ∧ 𝑀 ∈ ℤ) → (𝑍 × {𝐴}) ⇝ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2083 | . 2 ⊢ (ℤ≥‘𝑀) = (ℤ≥‘𝑀) | |
2 | simpr 108 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ ℤ) | |
3 | climconst2.2 | . . 3 ⊢ 𝑍 ∈ V | |
4 | snexg 3977 | . . . 4 ⊢ (𝐴 ∈ ℂ → {𝐴} ∈ V) | |
5 | 4 | adantr 270 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑀 ∈ ℤ) → {𝐴} ∈ V) |
6 | xpexg 4501 | . . 3 ⊢ ((𝑍 ∈ V ∧ {𝐴} ∈ V) → (𝑍 × {𝐴}) ∈ V) | |
7 | 3, 5, 6 | sylancr 405 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑀 ∈ ℤ) → (𝑍 × {𝐴}) ∈ V) |
8 | simpl 107 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑀 ∈ ℤ) → 𝐴 ∈ ℂ) | |
9 | climconst2.1 | . . . 4 ⊢ (ℤ≥‘𝑀) ⊆ 𝑍 | |
10 | 9 | sseli 3005 | . . 3 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → 𝑘 ∈ 𝑍) |
11 | fvconst2g 5428 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ 𝑍) → ((𝑍 × {𝐴})‘𝑘) = 𝐴) | |
12 | 8, 10, 11 | syl2an 283 | . 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → ((𝑍 × {𝐴})‘𝑘) = 𝐴) |
13 | 1, 2, 7, 8, 12 | climconst 10314 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑀 ∈ ℤ) → (𝑍 × {𝐴}) ⇝ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1285 ∈ wcel 1434 Vcvv 2610 ⊆ wss 2983 {csn 3417 class class class wbr 3806 × cxp 4390 ‘cfv 4953 ℂcc 7077 ℤcz 8468 ℤ≥cuz 8736 ⇝ cli 10302 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-coll 3914 ax-sep 3917 ax-nul 3925 ax-pow 3969 ax-pr 3993 ax-un 4217 ax-setind 4309 ax-iinf 4358 ax-cnex 7165 ax-resscn 7166 ax-1cn 7167 ax-1re 7168 ax-icn 7169 ax-addcl 7170 ax-addrcl 7171 ax-mulcl 7172 ax-mulrcl 7173 ax-addcom 7174 ax-mulcom 7175 ax-addass 7176 ax-mulass 7177 ax-distr 7178 ax-i2m1 7179 ax-0lt1 7180 ax-1rid 7181 ax-0id 7182 ax-rnegex 7183 ax-precex 7184 ax-cnre 7185 ax-pre-ltirr 7186 ax-pre-ltwlin 7187 ax-pre-lttrn 7188 ax-pre-apti 7189 ax-pre-ltadd 7190 ax-pre-mulgt0 7191 ax-pre-mulext 7192 |
This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rmo 2361 df-rab 2362 df-v 2612 df-sbc 2826 df-csb 2919 df-dif 2985 df-un 2987 df-in 2989 df-ss 2996 df-nul 3269 df-if 3370 df-pw 3403 df-sn 3423 df-pr 3424 df-op 3426 df-uni 3623 df-int 3658 df-iun 3701 df-br 3807 df-opab 3861 df-mpt 3862 df-tr 3897 df-id 4077 df-po 4080 df-iso 4081 df-iord 4150 df-on 4152 df-ilim 4153 df-suc 4155 df-iom 4361 df-xp 4398 df-rel 4399 df-cnv 4400 df-co 4401 df-dm 4402 df-rn 4403 df-res 4404 df-ima 4405 df-iota 4918 df-fun 4955 df-fn 4956 df-f 4957 df-f1 4958 df-fo 4959 df-f1o 4960 df-fv 4961 df-riota 5520 df-ov 5567 df-oprab 5568 df-mpt2 5569 df-1st 5819 df-2nd 5820 df-recs 5975 df-frec 6061 df-pnf 7253 df-mnf 7254 df-xr 7255 df-ltxr 7256 df-le 7257 df-sub 7384 df-neg 7385 df-reap 7778 df-ap 7785 df-div 7864 df-inn 8143 df-2 8201 df-n0 8392 df-z 8469 df-uz 8737 df-rp 8852 df-iseq 9558 df-iexp 9609 df-cj 9914 df-rsqrt 10069 df-abs 10070 df-clim 10303 |
This theorem is referenced by: climz 10316 iserclim0 10329 climaddc1 10352 climmulc2 10354 climsubc1 10355 climsubc2 10356 climlec2 10364 |
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