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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 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 11508 | . . 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 44008 | . . 3 ⊢ (𝜑 → -𝐼 ∈ ((𝑥 ∈ 𝐴 ↦ -𝐶) limℂ 𝐷)) |
11 | 1, 2, 3, 4, 6, 7, 10 | addlimc 44009 | . 2 ⊢ (𝜑 → (𝐸 + -𝐼) ∈ ((𝑥 ∈ 𝐴 ↦ (𝐵 + -𝐶)) limℂ 𝐷)) |
12 | limccl 25276 | . . . . 5 ⊢ (𝐹 limℂ 𝐷) ⊆ ℂ | |
13 | 12, 7 | sselid 3945 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ ℂ) |
14 | limccl 25276 | . . . . 5 ⊢ (𝐺 limℂ 𝐷) ⊆ ℂ | |
15 | 14, 9 | sselid 3945 | . . . 4 ⊢ (𝜑 → 𝐼 ∈ ℂ) |
16 | 13, 15 | negsubd 11527 | . . 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 11527 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 + -𝐶) = (𝐵 − 𝐶)) |
20 | 19 | eqcomd 2737 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 − 𝐶) = (𝐵 + -𝐶)) |
21 | 20 | mpteq2dva 5210 | . . . 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 7377 | . 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 5193 (class class class)co 7362 ℂcc 11058 + caddc 11063 − cmin 11394 -cneg 11395 limℂ climc 25263 |
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 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11116 ax-resscn 11117 ax-1cn 11118 ax-icn 11119 ax-addcl 11120 ax-addrcl 11121 ax-mulcl 11122 ax-mulrcl 11123 ax-mulcom 11124 ax-addass 11125 ax-mulass 11126 ax-distr 11127 ax-i2m1 11128 ax-1ne0 11129 ax-1rid 11130 ax-rnegex 11131 ax-rrecex 11132 ax-cnre 11133 ax-pre-lttri 11134 ax-pre-lttrn 11135 ax-pre-ltadd 11136 ax-pre-mulgt0 11137 ax-pre-sup 11138 |
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 3351 df-reu 3352 df-rab 3406 df-v 3448 df-sbc 3743 df-csb 3859 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-map 8774 df-pm 8775 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fi 9356 df-sup 9387 df-inf 9388 df-pnf 11200 df-mnf 11201 df-xr 11202 df-ltxr 11203 df-le 11204 df-sub 11396 df-neg 11397 df-div 11822 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12423 df-z 12509 df-dec 12628 df-uz 12773 df-q 12883 df-rp 12925 df-xneg 13042 df-xadd 13043 df-xmul 13044 df-fz 13435 df-seq 13917 df-exp 13978 df-cj 14996 df-re 14997 df-im 14998 df-sqrt 15132 df-abs 15133 df-struct 17030 df-slot 17065 df-ndx 17077 df-base 17095 df-plusg 17160 df-mulr 17161 df-starv 17162 df-tset 17166 df-ple 17167 df-ds 17169 df-unif 17170 df-rest 17318 df-topn 17319 df-topgen 17339 df-psmet 20825 df-xmet 20826 df-met 20827 df-bl 20828 df-mopn 20829 df-cnfld 20834 df-top 22280 df-topon 22297 df-topsp 22319 df-bases 22333 df-cnp 22616 df-xms 23710 df-ms 23711 df-limc 25267 |
This theorem is referenced by: fourierdlem60 44527 fourierdlem61 44528 fourierdlem74 44541 fourierdlem75 44542 fourierdlem76 44543 |
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