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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > constlimc | Structured version Visualization version GIF version |
Description: Limit of constant function. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
constlimc.f | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
constlimc.a | ⊢ (𝜑 → 𝐴 ⊆ ℂ) |
constlimc.b | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
constlimc.c | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
Ref | Expression |
---|---|
constlimc | ⊢ (𝜑 → 𝐵 ∈ (𝐹 limℂ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | constlimc.b | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
2 | 1rp 13013 | . . . . 5 ⊢ 1 ∈ ℝ+ | |
3 | 2 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → 1 ∈ ℝ+) |
4 | simpr 483 | . . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → 𝑣 ∈ 𝐴) | |
5 | vex 3465 | . . . . . . . . . . . . . . . 16 ⊢ 𝑣 ∈ V | |
6 | nfcv 2891 | . . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥𝐵 | |
7 | csbtt 3906 | . . . . . . . . . . . . . . . 16 ⊢ ((𝑣 ∈ V ∧ Ⅎ𝑥𝐵) → ⦋𝑣 / 𝑥⦌𝐵 = 𝐵) | |
8 | 5, 6, 7 | mp2an 690 | . . . . . . . . . . . . . . 15 ⊢ ⦋𝑣 / 𝑥⦌𝐵 = 𝐵 |
9 | 8, 1 | eqeltrid 2829 | . . . . . . . . . . . . . 14 ⊢ (𝜑 → ⦋𝑣 / 𝑥⦌𝐵 ∈ ℂ) |
10 | 9 | adantr 479 | . . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → ⦋𝑣 / 𝑥⦌𝐵 ∈ ℂ) |
11 | constlimc.f | . . . . . . . . . . . . . 14 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
12 | 11 | fvmpts 7007 | . . . . . . . . . . . . 13 ⊢ ((𝑣 ∈ 𝐴 ∧ ⦋𝑣 / 𝑥⦌𝐵 ∈ ℂ) → (𝐹‘𝑣) = ⦋𝑣 / 𝑥⦌𝐵) |
13 | 4, 10, 12 | syl2anc 582 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝐹‘𝑣) = ⦋𝑣 / 𝑥⦌𝐵) |
14 | 13 | oveq1d 7434 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → ((𝐹‘𝑣) − 𝐵) = (⦋𝑣 / 𝑥⦌𝐵 − 𝐵)) |
15 | 8 | oveq1i 7429 | . . . . . . . . . . 11 ⊢ (⦋𝑣 / 𝑥⦌𝐵 − 𝐵) = (𝐵 − 𝐵) |
16 | 14, 15 | eqtrdi 2781 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → ((𝐹‘𝑣) − 𝐵) = (𝐵 − 𝐵)) |
17 | 16 | fveq2d 6900 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (abs‘((𝐹‘𝑣) − 𝐵)) = (abs‘(𝐵 − 𝐵))) |
18 | 1 | subidd 11591 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 − 𝐵) = 0) |
19 | 18 | fveq2d 6900 | . . . . . . . . . 10 ⊢ (𝜑 → (abs‘(𝐵 − 𝐵)) = (abs‘0)) |
20 | 19 | adantr 479 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (abs‘(𝐵 − 𝐵)) = (abs‘0)) |
21 | abs0 15268 | . . . . . . . . . 10 ⊢ (abs‘0) = 0 | |
22 | 21 | a1i 11 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (abs‘0) = 0) |
23 | 17, 20, 22 | 3eqtrd 2769 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (abs‘((𝐹‘𝑣) − 𝐵)) = 0) |
24 | 23 | adantlr 713 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ+) ∧ 𝑣 ∈ 𝐴) → (abs‘((𝐹‘𝑣) − 𝐵)) = 0) |
25 | rpgt0 13021 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ+ → 0 < 𝑦) | |
26 | 25 | ad2antlr 725 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ+) ∧ 𝑣 ∈ 𝐴) → 0 < 𝑦) |
27 | 24, 26 | eqbrtrd 5171 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ+) ∧ 𝑣 ∈ 𝐴) → (abs‘((𝐹‘𝑣) − 𝐵)) < 𝑦) |
28 | 27 | a1d 25 | . . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ+) ∧ 𝑣 ∈ 𝐴) → ((𝑣 ≠ 𝐶 ∧ (abs‘(𝑣 − 𝐶)) < 1) → (abs‘((𝐹‘𝑣) − 𝐵)) < 𝑦)) |
29 | 28 | ralrimiva 3135 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐶 ∧ (abs‘(𝑣 − 𝐶)) < 1) → (abs‘((𝐹‘𝑣) − 𝐵)) < 𝑦)) |
30 | brimralrspcev 5210 | . . . 4 ⊢ ((1 ∈ ℝ+ ∧ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐶 ∧ (abs‘(𝑣 − 𝐶)) < 1) → (abs‘((𝐹‘𝑣) − 𝐵)) < 𝑦)) → ∃𝑤 ∈ ℝ+ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐶 ∧ (abs‘(𝑣 − 𝐶)) < 𝑤) → (abs‘((𝐹‘𝑣) − 𝐵)) < 𝑦)) | |
31 | 3, 29, 30 | syl2anc 582 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → ∃𝑤 ∈ ℝ+ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐶 ∧ (abs‘(𝑣 − 𝐶)) < 𝑤) → (abs‘((𝐹‘𝑣) − 𝐵)) < 𝑦)) |
32 | 31 | ralrimiva 3135 | . 2 ⊢ (𝜑 → ∀𝑦 ∈ ℝ+ ∃𝑤 ∈ ℝ+ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐶 ∧ (abs‘(𝑣 − 𝐶)) < 𝑤) → (abs‘((𝐹‘𝑣) − 𝐵)) < 𝑦)) |
33 | 1 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) |
34 | 33, 11 | fmptd 7123 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) |
35 | constlimc.a | . . 3 ⊢ (𝜑 → 𝐴 ⊆ ℂ) | |
36 | constlimc.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
37 | 34, 35, 36 | ellimc3 25852 | . 2 ⊢ (𝜑 → (𝐵 ∈ (𝐹 limℂ 𝐶) ↔ (𝐵 ∈ ℂ ∧ ∀𝑦 ∈ ℝ+ ∃𝑤 ∈ ℝ+ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐶 ∧ (abs‘(𝑣 − 𝐶)) < 𝑤) → (abs‘((𝐹‘𝑣) − 𝐵)) < 𝑦)))) |
38 | 1, 32, 37 | mpbir2and 711 | 1 ⊢ (𝜑 → 𝐵 ∈ (𝐹 limℂ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Ⅎwnfc 2875 ≠ wne 2929 ∀wral 3050 ∃wrex 3059 Vcvv 3461 ⦋csb 3889 ⊆ wss 3944 class class class wbr 5149 ↦ cmpt 5232 ‘cfv 6549 (class class class)co 7419 ℂcc 11138 0cc0 11140 1c1 11141 < clt 11280 − cmin 11476 ℝ+crp 13009 abscabs 15217 limℂ climc 25835 |
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-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 |
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 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fi 9436 df-sup 9467 df-inf 9468 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-q 12966 df-rp 13010 df-xneg 13127 df-xadd 13128 df-xmul 13129 df-fz 13520 df-seq 14003 df-exp 14063 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-struct 17119 df-slot 17154 df-ndx 17166 df-base 17184 df-plusg 17249 df-mulr 17250 df-starv 17251 df-tset 17255 df-ple 17256 df-ds 17258 df-unif 17259 df-rest 17407 df-topn 17408 df-topgen 17428 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-cnfld 21297 df-top 22840 df-topon 22857 df-topsp 22879 df-bases 22893 df-cnp 23176 df-xms 24270 df-ms 24271 df-limc 25839 |
This theorem is referenced by: reclimc 45179 fourierdlem53 45685 fourierdlem60 45692 fourierdlem61 45693 fourierdlem73 45705 fourierdlem74 45706 fourierdlem75 45707 fourierdlem76 45708 fouriersw 45757 |
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