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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 13038 | . . . . 5 ⊢ 1 ∈ ℝ+ | |
| 3 | 2 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → 1 ∈ ℝ+) |
| 4 | simpr 484 | . . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → 𝑣 ∈ 𝐴) | |
| 5 | vex 3484 | . . . . . . . . . . . . . . . 16 ⊢ 𝑣 ∈ V | |
| 6 | nfcv 2905 | . . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥𝐵 | |
| 7 | csbtt 3916 | . . . . . . . . . . . . . . . 16 ⊢ ((𝑣 ∈ V ∧ Ⅎ𝑥𝐵) → ⦋𝑣 / 𝑥⦌𝐵 = 𝐵) | |
| 8 | 5, 6, 7 | mp2an 692 | . . . . . . . . . . . . . . 15 ⊢ ⦋𝑣 / 𝑥⦌𝐵 = 𝐵 |
| 9 | 8, 1 | eqeltrid 2845 | . . . . . . . . . . . . . 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 2793 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → ((𝐹‘𝑣) − 𝐵) = (𝐵 − 𝐵)) |
| 17 | 16 | fveq2d 6910 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (abs‘((𝐹‘𝑣) − 𝐵)) = (abs‘(𝐵 − 𝐵))) |
| 18 | 1 | subidd 11608 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 − 𝐵) = 0) |
| 19 | 18 | fveq2d 6910 | . . . . . . . . . 10 ⊢ (𝜑 → (abs‘(𝐵 − 𝐵)) = (abs‘0)) |
| 20 | 19 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (abs‘(𝐵 − 𝐵)) = (abs‘0)) |
| 21 | abs0 15324 | . . . . . . . . . 10 ⊢ (abs‘0) = 0 | |
| 22 | 21 | a1i 11 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (abs‘0) = 0) |
| 23 | 17, 20, 22 | 3eqtrd 2781 | . . . . . . . 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 13047 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ+ → 0 < 𝑦) | |
| 26 | 25 | ad2antlr 727 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ+) ∧ 𝑣 ∈ 𝐴) → 0 < 𝑦) |
| 27 | 24, 26 | eqbrtrd 5165 | . . . . . 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 3146 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐶 ∧ (abs‘(𝑣 − 𝐶)) < 1) → (abs‘((𝐹‘𝑣) − 𝐵)) < 𝑦)) |
| 30 | brimralrspcev 5204 | . . . 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 3146 | . 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 25914 | . 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 1540 ∈ wcel 2108 Ⅎwnfc 2890 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 Vcvv 3480 ⦋csb 3899 ⊆ wss 3951 class class class wbr 5143 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 0cc0 11155 1c1 11156 < clt 11295 − cmin 11492 ℝ+crp 13034 abscabs 15273 limℂ climc 25897 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fi 9451 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-fz 13548 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-struct 17184 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-mulr 17311 df-starv 17312 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-rest 17467 df-topn 17468 df-topgen 17488 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cnp 23236 df-xms 24330 df-ms 24331 df-limc 25901 |
| This theorem is referenced by: reclimc 45668 fourierdlem53 46174 fourierdlem60 46181 fourierdlem61 46182 fourierdlem73 46194 fourierdlem74 46195 fourierdlem75 46196 fourierdlem76 46197 fouriersw 46246 |
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