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Mirrors > Home > ILE Home > Th. List > climeu | GIF version |
Description: An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 25-Dec-2005.) |
Ref | Expression |
---|---|
climeu | ⊢ (𝐹 ⇝ 𝐴 → ∃!𝑥 𝐹 ⇝ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climcl 10495 | . . . 4 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ) | |
2 | breq2 3815 | . . . . 5 ⊢ (𝑦 = 𝐴 → (𝐹 ⇝ 𝑦 ↔ 𝐹 ⇝ 𝐴)) | |
3 | 2 | spcegv 2697 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝐹 ⇝ 𝐴 → ∃𝑦 𝐹 ⇝ 𝑦)) |
4 | 1, 3 | mpcom 36 | . . 3 ⊢ (𝐹 ⇝ 𝐴 → ∃𝑦 𝐹 ⇝ 𝑦) |
5 | climuni 10506 | . . . 4 ⊢ ((𝐹 ⇝ 𝑦 ∧ 𝐹 ⇝ 𝑥) → 𝑦 = 𝑥) | |
6 | 5 | gen2 1380 | . . 3 ⊢ ∀𝑦∀𝑥((𝐹 ⇝ 𝑦 ∧ 𝐹 ⇝ 𝑥) → 𝑦 = 𝑥) |
7 | 4, 6 | jctir 306 | . 2 ⊢ (𝐹 ⇝ 𝐴 → (∃𝑦 𝐹 ⇝ 𝑦 ∧ ∀𝑦∀𝑥((𝐹 ⇝ 𝑦 ∧ 𝐹 ⇝ 𝑥) → 𝑦 = 𝑥))) |
8 | nfv 1462 | . . . 4 ⊢ Ⅎ𝑦 𝐹 ⇝ 𝑥 | |
9 | nfv 1462 | . . . 4 ⊢ Ⅎ𝑥 𝐹 ⇝ 𝑦 | |
10 | breq2 3815 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐹 ⇝ 𝑥 ↔ 𝐹 ⇝ 𝑦)) | |
11 | 8, 9, 10 | cbveu 1967 | . . 3 ⊢ (∃!𝑥 𝐹 ⇝ 𝑥 ↔ ∃!𝑦 𝐹 ⇝ 𝑦) |
12 | breq2 3815 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝐹 ⇝ 𝑦 ↔ 𝐹 ⇝ 𝑥)) | |
13 | 12 | eu4 2005 | . . 3 ⊢ (∃!𝑦 𝐹 ⇝ 𝑦 ↔ (∃𝑦 𝐹 ⇝ 𝑦 ∧ ∀𝑦∀𝑥((𝐹 ⇝ 𝑦 ∧ 𝐹 ⇝ 𝑥) → 𝑦 = 𝑥))) |
14 | 11, 13 | bitri 182 | . 2 ⊢ (∃!𝑥 𝐹 ⇝ 𝑥 ↔ (∃𝑦 𝐹 ⇝ 𝑦 ∧ ∀𝑦∀𝑥((𝐹 ⇝ 𝑦 ∧ 𝐹 ⇝ 𝑥) → 𝑦 = 𝑥))) |
15 | 7, 14 | sylibr 132 | 1 ⊢ (𝐹 ⇝ 𝐴 → ∃!𝑥 𝐹 ⇝ 𝑥) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∀wal 1283 ∃wex 1422 ∈ wcel 1434 ∃!weu 1943 class class class wbr 3811 ℂcc 7251 ⇝ cli 10491 |
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 3919 ax-sep 3922 ax-nul 3930 ax-pow 3974 ax-pr 4000 ax-un 4224 ax-setind 4316 ax-iinf 4366 ax-cnex 7339 ax-resscn 7340 ax-1cn 7341 ax-1re 7342 ax-icn 7343 ax-addcl 7344 ax-addrcl 7345 ax-mulcl 7346 ax-mulrcl 7347 ax-addcom 7348 ax-mulcom 7349 ax-addass 7350 ax-mulass 7351 ax-distr 7352 ax-i2m1 7353 ax-0lt1 7354 ax-1rid 7355 ax-0id 7356 ax-rnegex 7357 ax-precex 7358 ax-cnre 7359 ax-pre-ltirr 7360 ax-pre-ltwlin 7361 ax-pre-lttrn 7362 ax-pre-apti 7363 ax-pre-ltadd 7364 ax-pre-mulgt0 7365 ax-pre-mulext 7366 ax-arch 7367 ax-caucvg 7368 |
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 2614 df-sbc 2827 df-csb 2920 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-nul 3270 df-if 3374 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-iun 3706 df-br 3812 df-opab 3866 df-mpt 3867 df-tr 3902 df-id 4084 df-po 4087 df-iso 4088 df-iord 4157 df-on 4159 df-ilim 4160 df-suc 4162 df-iom 4369 df-xp 4407 df-rel 4408 df-cnv 4409 df-co 4410 df-dm 4411 df-rn 4412 df-res 4413 df-ima 4414 df-iota 4934 df-fun 4971 df-fn 4972 df-f 4973 df-f1 4974 df-fo 4975 df-f1o 4976 df-fv 4977 df-riota 5547 df-ov 5594 df-oprab 5595 df-mpt2 5596 df-1st 5846 df-2nd 5847 df-recs 6002 df-frec 6088 df-pnf 7427 df-mnf 7428 df-xr 7429 df-ltxr 7430 df-le 7431 df-sub 7558 df-neg 7559 df-reap 7952 df-ap 7959 df-div 8038 df-inn 8317 df-2 8375 df-3 8376 df-4 8377 df-n0 8566 df-z 8647 df-uz 8915 df-rp 9030 df-iseq 9741 df-iexp 9792 df-cj 10103 df-re 10104 df-im 10105 df-rsqrt 10258 df-abs 10259 df-clim 10492 |
This theorem is referenced by: climreu 10510 climmo 10511 |
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