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Mathbox for Glauco Siliprandi |
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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 2760 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ -𝐶) = (𝑥 ∈ 𝐴 ↦ -𝐶) | |
3 | eqid 2760 | . . 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 10591 | . . 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 40400 | . . 3 ⊢ (𝜑 → -𝐼 ∈ ((𝑥 ∈ 𝐴 ↦ -𝐶) limℂ 𝐷)) |
11 | 1, 2, 3, 4, 6, 7, 10 | addlimc 40401 | . 2 ⊢ (𝜑 → (𝐸 + -𝐼) ∈ ((𝑥 ∈ 𝐴 ↦ (𝐵 + -𝐶)) limℂ 𝐷)) |
12 | limccl 23858 | . . . . 5 ⊢ (𝐹 limℂ 𝐷) ⊆ ℂ | |
13 | 12, 7 | sseldi 3742 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ ℂ) |
14 | limccl 23858 | . . . . 5 ⊢ (𝐺 limℂ 𝐷) ⊆ ℂ | |
15 | 14, 9 | sseldi 3742 | . . . 4 ⊢ (𝜑 → 𝐼 ∈ ℂ) |
16 | 13, 15 | negsubd 10610 | . . 3 ⊢ (𝜑 → (𝐸 + -𝐼) = (𝐸 − 𝐼)) |
17 | 16 | eqcomd 2766 | . 2 ⊢ (𝜑 → (𝐸 − 𝐼) = (𝐸 + -𝐼)) |
18 | sublimc.3 | . . . 4 ⊢ 𝐻 = (𝑥 ∈ 𝐴 ↦ (𝐵 − 𝐶)) | |
19 | 4, 5 | negsubd 10610 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 + -𝐶) = (𝐵 − 𝐶)) |
20 | 19 | eqcomd 2766 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 − 𝐶) = (𝐵 + -𝐶)) |
21 | 20 | mpteq2dva 4896 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 − 𝐶)) = (𝑥 ∈ 𝐴 ↦ (𝐵 + -𝐶))) |
22 | 18, 21 | syl5eq 2806 | . . 3 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐴 ↦ (𝐵 + -𝐶))) |
23 | 22 | oveq1d 6829 | . 2 ⊢ (𝜑 → (𝐻 limℂ 𝐷) = ((𝑥 ∈ 𝐴 ↦ (𝐵 + -𝐶)) limℂ 𝐷)) |
24 | 11, 17, 23 | 3eltr4d 2854 | 1 ⊢ (𝜑 → (𝐸 − 𝐼) ∈ (𝐻 limℂ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ↦ cmpt 4881 (class class class)co 6814 ℂcc 10146 + caddc 10151 − cmin 10478 -cneg 10479 limℂ climc 23845 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 ax-pre-mulgt0 10225 ax-pre-sup 10226 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-om 7232 df-1st 7334 df-2nd 7335 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-1o 7730 df-oadd 7734 df-er 7913 df-map 8027 df-pm 8028 df-en 8124 df-dom 8125 df-sdom 8126 df-fin 8127 df-fi 8484 df-sup 8515 df-inf 8516 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-sub 10480 df-neg 10481 df-div 10897 df-nn 11233 df-2 11291 df-3 11292 df-4 11293 df-5 11294 df-6 11295 df-7 11296 df-8 11297 df-9 11298 df-n0 11505 df-z 11590 df-dec 11706 df-uz 11900 df-q 12002 df-rp 12046 df-xneg 12159 df-xadd 12160 df-xmul 12161 df-fz 12540 df-seq 13016 df-exp 13075 df-cj 14058 df-re 14059 df-im 14060 df-sqrt 14194 df-abs 14195 df-struct 16081 df-ndx 16082 df-slot 16083 df-base 16085 df-plusg 16176 df-mulr 16177 df-starv 16178 df-tset 16182 df-ple 16183 df-ds 16186 df-unif 16187 df-rest 16305 df-topn 16306 df-topgen 16326 df-psmet 19960 df-xmet 19961 df-met 19962 df-bl 19963 df-mopn 19964 df-cnfld 19969 df-top 20921 df-topon 20938 df-topsp 20959 df-bases 20972 df-cnp 21254 df-xms 22346 df-ms 22347 df-limc 23849 |
This theorem is referenced by: fourierdlem60 40904 fourierdlem61 40905 fourierdlem74 40918 fourierdlem75 40919 fourierdlem76 40920 |
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