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 11712 | . . . . 5 ⊢ 1 ∈ ℝ+ | |
3 | 2 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → 1 ∈ ℝ+) |
4 | simpr 476 | . . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → 𝑣 ∈ 𝐴) | |
5 | vex 3176 | . . . . . . . . . . . . . . . 16 ⊢ 𝑣 ∈ V | |
6 | nfcv 2751 | . . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥𝐵 | |
7 | csbtt 3510 | . . . . . . . . . . . . . . . 16 ⊢ ((𝑣 ∈ V ∧ Ⅎ𝑥𝐵) → ⦋𝑣 / 𝑥⦌𝐵 = 𝐵) | |
8 | 5, 6, 7 | mp2an 704 | . . . . . . . . . . . . . . 15 ⊢ ⦋𝑣 / 𝑥⦌𝐵 = 𝐵 |
9 | 8, 1 | syl5eqel 2692 | . . . . . . . . . . . . . 14 ⊢ (𝜑 → ⦋𝑣 / 𝑥⦌𝐵 ∈ ℂ) |
10 | 9 | adantr 480 | . . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → ⦋𝑣 / 𝑥⦌𝐵 ∈ ℂ) |
11 | constlimc.f | . . . . . . . . . . . . . 14 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
12 | 11 | fvmpts 6194 | . . . . . . . . . . . . 13 ⊢ ((𝑣 ∈ 𝐴 ∧ ⦋𝑣 / 𝑥⦌𝐵 ∈ ℂ) → (𝐹‘𝑣) = ⦋𝑣 / 𝑥⦌𝐵) |
13 | 4, 10, 12 | syl2anc 691 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝐹‘𝑣) = ⦋𝑣 / 𝑥⦌𝐵) |
14 | 13 | oveq1d 6564 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → ((𝐹‘𝑣) − 𝐵) = (⦋𝑣 / 𝑥⦌𝐵 − 𝐵)) |
15 | 8 | oveq1i 6559 | . . . . . . . . . . 11 ⊢ (⦋𝑣 / 𝑥⦌𝐵 − 𝐵) = (𝐵 − 𝐵) |
16 | 14, 15 | syl6eq 2660 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → ((𝐹‘𝑣) − 𝐵) = (𝐵 − 𝐵)) |
17 | 16 | fveq2d 6107 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (abs‘((𝐹‘𝑣) − 𝐵)) = (abs‘(𝐵 − 𝐵))) |
18 | 1 | subidd 10259 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 − 𝐵) = 0) |
19 | 18 | fveq2d 6107 | . . . . . . . . . 10 ⊢ (𝜑 → (abs‘(𝐵 − 𝐵)) = (abs‘0)) |
20 | 19 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (abs‘(𝐵 − 𝐵)) = (abs‘0)) |
21 | abs0 13873 | . . . . . . . . . 10 ⊢ (abs‘0) = 0 | |
22 | 21 | a1i 11 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (abs‘0) = 0) |
23 | 17, 20, 22 | 3eqtrd 2648 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (abs‘((𝐹‘𝑣) − 𝐵)) = 0) |
24 | 23 | adantlr 747 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ+) ∧ 𝑣 ∈ 𝐴) → (abs‘((𝐹‘𝑣) − 𝐵)) = 0) |
25 | rpgt0 11720 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ+ → 0 < 𝑦) | |
26 | 25 | ad2antlr 759 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ+) ∧ 𝑣 ∈ 𝐴) → 0 < 𝑦) |
27 | 24, 26 | eqbrtrd 4605 | . . . . . 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 2949 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐶 ∧ (abs‘(𝑣 − 𝐶)) < 1) → (abs‘((𝐹‘𝑣) − 𝐵)) < 𝑦)) |
30 | breq2 4587 | . . . . . . . 8 ⊢ (𝑤 = 1 → ((abs‘(𝑣 − 𝐶)) < 𝑤 ↔ (abs‘(𝑣 − 𝐶)) < 1)) | |
31 | 30 | anbi2d 736 | . . . . . . 7 ⊢ (𝑤 = 1 → ((𝑣 ≠ 𝐶 ∧ (abs‘(𝑣 − 𝐶)) < 𝑤) ↔ (𝑣 ≠ 𝐶 ∧ (abs‘(𝑣 − 𝐶)) < 1))) |
32 | 31 | imbi1d 330 | . . . . . 6 ⊢ (𝑤 = 1 → (((𝑣 ≠ 𝐶 ∧ (abs‘(𝑣 − 𝐶)) < 𝑤) → (abs‘((𝐹‘𝑣) − 𝐵)) < 𝑦) ↔ ((𝑣 ≠ 𝐶 ∧ (abs‘(𝑣 − 𝐶)) < 1) → (abs‘((𝐹‘𝑣) − 𝐵)) < 𝑦))) |
33 | 32 | ralbidv 2969 | . . . . 5 ⊢ (𝑤 = 1 → (∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐶 ∧ (abs‘(𝑣 − 𝐶)) < 𝑤) → (abs‘((𝐹‘𝑣) − 𝐵)) < 𝑦) ↔ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐶 ∧ (abs‘(𝑣 − 𝐶)) < 1) → (abs‘((𝐹‘𝑣) − 𝐵)) < 𝑦))) |
34 | 33 | rspcev 3282 | . . . 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)) |
35 | 3, 29, 34 | syl2anc 691 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → ∃𝑤 ∈ ℝ+ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐶 ∧ (abs‘(𝑣 − 𝐶)) < 𝑤) → (abs‘((𝐹‘𝑣) − 𝐵)) < 𝑦)) |
36 | 35 | ralrimiva 2949 | . 2 ⊢ (𝜑 → ∀𝑦 ∈ ℝ+ ∃𝑤 ∈ ℝ+ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐶 ∧ (abs‘(𝑣 − 𝐶)) < 𝑤) → (abs‘((𝐹‘𝑣) − 𝐵)) < 𝑦)) |
37 | 1 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) |
38 | 37, 11 | fmptd 6292 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) |
39 | constlimc.a | . . 3 ⊢ (𝜑 → 𝐴 ⊆ ℂ) | |
40 | constlimc.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
41 | 38, 39, 40 | ellimc3 23449 | . 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)))) |
42 | 1, 36, 41 | mpbir2and 959 | 1 ⊢ (𝜑 → 𝐵 ∈ (𝐹 limℂ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Ⅎwnfc 2738 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 Vcvv 3173 ⦋csb 3499 ⊆ wss 3540 class class class wbr 4583 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 0cc0 9815 1c1 9816 < clt 9953 − cmin 10145 ℝ+crp 11708 abscabs 13822 limℂ climc 23432 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fi 8200 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-fz 12198 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-plusg 15781 df-mulr 15782 df-starv 15783 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-rest 15906 df-topn 15907 df-topgen 15927 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cnp 20842 df-xms 21935 df-ms 21936 df-limc 23436 |
This theorem is referenced by: reclimc 38720 fourierdlem53 39052 fourierdlem60 39059 fourierdlem61 39060 fourierdlem73 39072 fourierdlem74 39073 fourierdlem75 39074 fourierdlem76 39075 fouriersw 39124 |
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