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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 13036 | . . . . 5 ⊢ 1 ∈ ℝ+ | |
3 | 2 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → 1 ∈ ℝ+) |
4 | simpr 484 | . . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → 𝑣 ∈ 𝐴) | |
5 | vex 3482 | . . . . . . . . . . . . . . . 16 ⊢ 𝑣 ∈ V | |
6 | nfcv 2903 | . . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥𝐵 | |
7 | csbtt 3925 | . . . . . . . . . . . . . . . 16 ⊢ ((𝑣 ∈ V ∧ Ⅎ𝑥𝐵) → ⦋𝑣 / 𝑥⦌𝐵 = 𝐵) | |
8 | 5, 6, 7 | mp2an 692 | . . . . . . . . . . . . . . 15 ⊢ ⦋𝑣 / 𝑥⦌𝐵 = 𝐵 |
9 | 8, 1 | eqeltrid 2843 | . . . . . . . . . . . . . 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 7019 | . . . . . . . . . . . . 13 ⊢ ((𝑣 ∈ 𝐴 ∧ ⦋𝑣 / 𝑥⦌𝐵 ∈ ℂ) → (𝐹‘𝑣) = ⦋𝑣 / 𝑥⦌𝐵) |
13 | 4, 10, 12 | syl2anc 584 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝐹‘𝑣) = ⦋𝑣 / 𝑥⦌𝐵) |
14 | 13 | oveq1d 7446 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → ((𝐹‘𝑣) − 𝐵) = (⦋𝑣 / 𝑥⦌𝐵 − 𝐵)) |
15 | 8 | oveq1i 7441 | . . . . . . . . . . 11 ⊢ (⦋𝑣 / 𝑥⦌𝐵 − 𝐵) = (𝐵 − 𝐵) |
16 | 14, 15 | eqtrdi 2791 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → ((𝐹‘𝑣) − 𝐵) = (𝐵 − 𝐵)) |
17 | 16 | fveq2d 6911 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (abs‘((𝐹‘𝑣) − 𝐵)) = (abs‘(𝐵 − 𝐵))) |
18 | 1 | subidd 11606 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 − 𝐵) = 0) |
19 | 18 | fveq2d 6911 | . . . . . . . . . 10 ⊢ (𝜑 → (abs‘(𝐵 − 𝐵)) = (abs‘0)) |
20 | 19 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (abs‘(𝐵 − 𝐵)) = (abs‘0)) |
21 | abs0 15321 | . . . . . . . . . 10 ⊢ (abs‘0) = 0 | |
22 | 21 | a1i 11 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (abs‘0) = 0) |
23 | 17, 20, 22 | 3eqtrd 2779 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (abs‘((𝐹‘𝑣) − 𝐵)) = 0) |
24 | 23 | adantlr 715 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ+) ∧ 𝑣 ∈ 𝐴) → (abs‘((𝐹‘𝑣) − 𝐵)) = 0) |
25 | rpgt0 13045 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ+ → 0 < 𝑦) | |
26 | 25 | ad2antlr 727 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ+) ∧ 𝑣 ∈ 𝐴) → 0 < 𝑦) |
27 | 24, 26 | eqbrtrd 5170 | . . . . . 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 3144 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐶 ∧ (abs‘(𝑣 − 𝐶)) < 1) → (abs‘((𝐹‘𝑣) − 𝐵)) < 𝑦)) |
30 | brimralrspcev 5209 | . . . 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 584 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → ∃𝑤 ∈ ℝ+ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐶 ∧ (abs‘(𝑣 − 𝐶)) < 𝑤) → (abs‘((𝐹‘𝑣) − 𝐵)) < 𝑦)) |
32 | 31 | ralrimiva 3144 | . 2 ⊢ (𝜑 → ∀𝑦 ∈ ℝ+ ∃𝑤 ∈ ℝ+ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐶 ∧ (abs‘(𝑣 − 𝐶)) < 𝑤) → (abs‘((𝐹‘𝑣) − 𝐵)) < 𝑦)) |
33 | 1 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) |
34 | 33, 11 | fmptd 7134 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) |
35 | constlimc.a | . . 3 ⊢ (𝜑 → 𝐴 ⊆ ℂ) | |
36 | constlimc.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
37 | 34, 35, 36 | ellimc3 25929 | . 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 713 | 1 ⊢ (𝜑 → 𝐵 ∈ (𝐹 limℂ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Ⅎwnfc 2888 ≠ wne 2938 ∀wral 3059 ∃wrex 3068 Vcvv 3478 ⦋csb 3908 ⊆ wss 3963 class class class wbr 5148 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 0cc0 11153 1c1 11154 < clt 11293 − cmin 11490 ℝ+crp 13032 abscabs 15270 limℂ climc 25912 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fi 9449 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-fz 13545 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-struct 17181 df-slot 17216 df-ndx 17228 df-base 17246 df-plusg 17311 df-mulr 17312 df-starv 17313 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-rest 17469 df-topn 17470 df-topgen 17490 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cnp 23252 df-xms 24346 df-ms 24347 df-limc 25916 |
This theorem is referenced by: reclimc 45609 fourierdlem53 46115 fourierdlem60 46122 fourierdlem61 46123 fourierdlem73 46135 fourierdlem74 46136 fourierdlem75 46137 fourierdlem76 46138 fouriersw 46187 |
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