![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > sublimc | Structured version Visualization version GIF version |
Description: Subtraction of two limits. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
sublimc.f | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
sublimc.2 | ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ 𝐶) |
sublimc.3 | ⊢ 𝐻 = (𝑥 ∈ 𝐴 ↦ (𝐵 − 𝐶)) |
sublimc.4 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) |
sublimc.5 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ) |
sublimc.6 | ⊢ (𝜑 → 𝐸 ∈ (𝐹 limℂ 𝐷)) |
sublimc.7 | ⊢ (𝜑 → 𝐼 ∈ (𝐺 limℂ 𝐷)) |
Ref | Expression |
---|---|
sublimc | ⊢ (𝜑 → (𝐸 − 𝐼) ∈ (𝐻 limℂ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sublimc.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
2 | eqid 2731 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ -𝐶) = (𝑥 ∈ 𝐴 ↦ -𝐶) | |
3 | eqid 2731 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐵 + -𝐶)) = (𝑥 ∈ 𝐴 ↦ (𝐵 + -𝐶)) | |
4 | sublimc.4 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) | |
5 | sublimc.5 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ) | |
6 | 5 | negcld 11537 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐶 ∈ ℂ) |
7 | sublimc.6 | . . 3 ⊢ (𝜑 → 𝐸 ∈ (𝐹 limℂ 𝐷)) | |
8 | sublimc.2 | . . . 4 ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
9 | sublimc.7 | . . . 4 ⊢ (𝜑 → 𝐼 ∈ (𝐺 limℂ 𝐷)) | |
10 | 8, 2, 5, 9 | neglimc 44122 | . . 3 ⊢ (𝜑 → -𝐼 ∈ ((𝑥 ∈ 𝐴 ↦ -𝐶) limℂ 𝐷)) |
11 | 1, 2, 3, 4, 6, 7, 10 | addlimc 44123 | . 2 ⊢ (𝜑 → (𝐸 + -𝐼) ∈ ((𝑥 ∈ 𝐴 ↦ (𝐵 + -𝐶)) limℂ 𝐷)) |
12 | limccl 25316 | . . . . 5 ⊢ (𝐹 limℂ 𝐷) ⊆ ℂ | |
13 | 12, 7 | sselid 3973 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ ℂ) |
14 | limccl 25316 | . . . . 5 ⊢ (𝐺 limℂ 𝐷) ⊆ ℂ | |
15 | 14, 9 | sselid 3973 | . . . 4 ⊢ (𝜑 → 𝐼 ∈ ℂ) |
16 | 13, 15 | negsubd 11556 | . . 3 ⊢ (𝜑 → (𝐸 + -𝐼) = (𝐸 − 𝐼)) |
17 | 16 | eqcomd 2737 | . 2 ⊢ (𝜑 → (𝐸 − 𝐼) = (𝐸 + -𝐼)) |
18 | sublimc.3 | . . . 4 ⊢ 𝐻 = (𝑥 ∈ 𝐴 ↦ (𝐵 − 𝐶)) | |
19 | 4, 5 | negsubd 11556 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 + -𝐶) = (𝐵 − 𝐶)) |
20 | 19 | eqcomd 2737 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 − 𝐶) = (𝐵 + -𝐶)) |
21 | 20 | mpteq2dva 5238 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 − 𝐶)) = (𝑥 ∈ 𝐴 ↦ (𝐵 + -𝐶))) |
22 | 18, 21 | eqtrid 2783 | . . 3 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐴 ↦ (𝐵 + -𝐶))) |
23 | 22 | oveq1d 7405 | . 2 ⊢ (𝜑 → (𝐻 limℂ 𝐷) = ((𝑥 ∈ 𝐴 ↦ (𝐵 + -𝐶)) limℂ 𝐷)) |
24 | 11, 17, 23 | 3eltr4d 2847 | 1 ⊢ (𝜑 → (𝐸 − 𝐼) ∈ (𝐻 limℂ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ↦ cmpt 5221 (class class class)co 7390 ℂcc 11087 + caddc 11092 − cmin 11423 -cneg 11424 limℂ climc 25303 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7705 ax-cnex 11145 ax-resscn 11146 ax-1cn 11147 ax-icn 11148 ax-addcl 11149 ax-addrcl 11150 ax-mulcl 11151 ax-mulrcl 11152 ax-mulcom 11153 ax-addass 11154 ax-mulass 11155 ax-distr 11156 ax-i2m1 11157 ax-1ne0 11158 ax-1rid 11159 ax-rnegex 11160 ax-rrecex 11161 ax-cnre 11162 ax-pre-lttri 11163 ax-pre-lttrn 11164 ax-pre-ltadd 11165 ax-pre-mulgt0 11166 ax-pre-sup 11167 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3430 df-v 3472 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4520 df-pw 4595 df-sn 4620 df-pr 4622 df-tp 4624 df-op 4626 df-uni 4899 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6286 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6531 df-fn 6532 df-f 6533 df-f1 6534 df-fo 6535 df-f1o 6536 df-fv 6537 df-riota 7346 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7836 df-1st 7954 df-2nd 7955 df-frecs 8245 df-wrecs 8276 df-recs 8350 df-rdg 8389 df-1o 8445 df-er 8683 df-map 8802 df-pm 8803 df-en 8920 df-dom 8921 df-sdom 8922 df-fin 8923 df-fi 9385 df-sup 9416 df-inf 9417 df-pnf 11229 df-mnf 11230 df-xr 11231 df-ltxr 11232 df-le 11233 df-sub 11425 df-neg 11426 df-div 11851 df-nn 12192 df-2 12254 df-3 12255 df-4 12256 df-5 12257 df-6 12258 df-7 12259 df-8 12260 df-9 12261 df-n0 12452 df-z 12538 df-dec 12657 df-uz 12802 df-q 12912 df-rp 12954 df-xneg 13071 df-xadd 13072 df-xmul 13073 df-fz 13464 df-seq 13946 df-exp 14007 df-cj 15025 df-re 15026 df-im 15027 df-sqrt 15161 df-abs 15162 df-struct 17059 df-slot 17094 df-ndx 17106 df-base 17124 df-plusg 17189 df-mulr 17190 df-starv 17191 df-tset 17195 df-ple 17196 df-ds 17198 df-unif 17199 df-rest 17347 df-topn 17348 df-topgen 17368 df-psmet 20865 df-xmet 20866 df-met 20867 df-bl 20868 df-mopn 20869 df-cnfld 20874 df-top 22320 df-topon 22337 df-topsp 22359 df-bases 22373 df-cnp 22656 df-xms 23750 df-ms 23751 df-limc 25307 |
This theorem is referenced by: fourierdlem60 44641 fourierdlem61 44642 fourierdlem74 44655 fourierdlem75 44656 fourierdlem76 44657 |
Copyright terms: Public domain | W3C validator |