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 12381 | . . . . 5 ⊢ 1 ∈ ℝ+ | |
3 | 2 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → 1 ∈ ℝ+) |
4 | simpr 488 | . . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → 𝑣 ∈ 𝐴) | |
5 | vex 3444 | . . . . . . . . . . . . . . . 16 ⊢ 𝑣 ∈ V | |
6 | nfcv 2955 | . . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥𝐵 | |
7 | csbtt 3845 | . . . . . . . . . . . . . . . 16 ⊢ ((𝑣 ∈ V ∧ Ⅎ𝑥𝐵) → ⦋𝑣 / 𝑥⦌𝐵 = 𝐵) | |
8 | 5, 6, 7 | mp2an 691 | . . . . . . . . . . . . . . 15 ⊢ ⦋𝑣 / 𝑥⦌𝐵 = 𝐵 |
9 | 8, 1 | eqeltrid 2894 | . . . . . . . . . . . . . 14 ⊢ (𝜑 → ⦋𝑣 / 𝑥⦌𝐵 ∈ ℂ) |
10 | 9 | adantr 484 | . . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → ⦋𝑣 / 𝑥⦌𝐵 ∈ ℂ) |
11 | constlimc.f | . . . . . . . . . . . . . 14 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
12 | 11 | fvmpts 6748 | . . . . . . . . . . . . 13 ⊢ ((𝑣 ∈ 𝐴 ∧ ⦋𝑣 / 𝑥⦌𝐵 ∈ ℂ) → (𝐹‘𝑣) = ⦋𝑣 / 𝑥⦌𝐵) |
13 | 4, 10, 12 | syl2anc 587 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝐹‘𝑣) = ⦋𝑣 / 𝑥⦌𝐵) |
14 | 13 | oveq1d 7150 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → ((𝐹‘𝑣) − 𝐵) = (⦋𝑣 / 𝑥⦌𝐵 − 𝐵)) |
15 | 8 | oveq1i 7145 | . . . . . . . . . . 11 ⊢ (⦋𝑣 / 𝑥⦌𝐵 − 𝐵) = (𝐵 − 𝐵) |
16 | 14, 15 | eqtrdi 2849 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → ((𝐹‘𝑣) − 𝐵) = (𝐵 − 𝐵)) |
17 | 16 | fveq2d 6649 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (abs‘((𝐹‘𝑣) − 𝐵)) = (abs‘(𝐵 − 𝐵))) |
18 | 1 | subidd 10974 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 − 𝐵) = 0) |
19 | 18 | fveq2d 6649 | . . . . . . . . . 10 ⊢ (𝜑 → (abs‘(𝐵 − 𝐵)) = (abs‘0)) |
20 | 19 | adantr 484 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (abs‘(𝐵 − 𝐵)) = (abs‘0)) |
21 | abs0 14637 | . . . . . . . . . 10 ⊢ (abs‘0) = 0 | |
22 | 21 | a1i 11 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (abs‘0) = 0) |
23 | 17, 20, 22 | 3eqtrd 2837 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (abs‘((𝐹‘𝑣) − 𝐵)) = 0) |
24 | 23 | adantlr 714 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ+) ∧ 𝑣 ∈ 𝐴) → (abs‘((𝐹‘𝑣) − 𝐵)) = 0) |
25 | rpgt0 12389 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ+ → 0 < 𝑦) | |
26 | 25 | ad2antlr 726 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ+) ∧ 𝑣 ∈ 𝐴) → 0 < 𝑦) |
27 | 24, 26 | eqbrtrd 5052 | . . . . . 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 3149 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐶 ∧ (abs‘(𝑣 − 𝐶)) < 1) → (abs‘((𝐹‘𝑣) − 𝐵)) < 𝑦)) |
30 | brimralrspcev 5091 | . . . 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 587 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → ∃𝑤 ∈ ℝ+ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐶 ∧ (abs‘(𝑣 − 𝐶)) < 𝑤) → (abs‘((𝐹‘𝑣) − 𝐵)) < 𝑦)) |
32 | 31 | ralrimiva 3149 | . 2 ⊢ (𝜑 → ∀𝑦 ∈ ℝ+ ∃𝑤 ∈ ℝ+ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐶 ∧ (abs‘(𝑣 − 𝐶)) < 𝑤) → (abs‘((𝐹‘𝑣) − 𝐵)) < 𝑦)) |
33 | 1 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) |
34 | 33, 11 | fmptd 6855 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) |
35 | constlimc.a | . . 3 ⊢ (𝜑 → 𝐴 ⊆ ℂ) | |
36 | constlimc.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
37 | 34, 35, 36 | ellimc3 24482 | . 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 712 | 1 ⊢ (𝜑 → 𝐵 ∈ (𝐹 limℂ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Ⅎwnfc 2936 ≠ wne 2987 ∀wral 3106 ∃wrex 3107 Vcvv 3441 ⦋csb 3828 ⊆ wss 3881 class class class wbr 5030 ↦ cmpt 5110 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 0cc0 10526 1c1 10527 < clt 10664 − cmin 10859 ℝ+crp 12377 abscabs 14585 limℂ climc 24465 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fi 8859 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-fz 12886 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-plusg 16570 df-mulr 16571 df-starv 16572 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-rest 16688 df-topn 16689 df-topgen 16709 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cnp 21833 df-xms 22927 df-ms 22928 df-limc 24469 |
This theorem is referenced by: reclimc 42295 fourierdlem53 42801 fourierdlem60 42808 fourierdlem61 42809 fourierdlem73 42821 fourierdlem74 42822 fourierdlem75 42823 fourierdlem76 42824 fouriersw 42873 |
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