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 2736 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ -𝐶) = (𝑥 ∈ 𝐴 ↦ -𝐶) | |
3 | eqid 2736 | . . 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 11141 | . . 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 42806 | . . 3 ⊢ (𝜑 → -𝐼 ∈ ((𝑥 ∈ 𝐴 ↦ -𝐶) limℂ 𝐷)) |
11 | 1, 2, 3, 4, 6, 7, 10 | addlimc 42807 | . 2 ⊢ (𝜑 → (𝐸 + -𝐼) ∈ ((𝑥 ∈ 𝐴 ↦ (𝐵 + -𝐶)) limℂ 𝐷)) |
12 | limccl 24726 | . . . . 5 ⊢ (𝐹 limℂ 𝐷) ⊆ ℂ | |
13 | 12, 7 | sseldi 3885 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ ℂ) |
14 | limccl 24726 | . . . . 5 ⊢ (𝐺 limℂ 𝐷) ⊆ ℂ | |
15 | 14, 9 | sseldi 3885 | . . . 4 ⊢ (𝜑 → 𝐼 ∈ ℂ) |
16 | 13, 15 | negsubd 11160 | . . 3 ⊢ (𝜑 → (𝐸 + -𝐼) = (𝐸 − 𝐼)) |
17 | 16 | eqcomd 2742 | . 2 ⊢ (𝜑 → (𝐸 − 𝐼) = (𝐸 + -𝐼)) |
18 | sublimc.3 | . . . 4 ⊢ 𝐻 = (𝑥 ∈ 𝐴 ↦ (𝐵 − 𝐶)) | |
19 | 4, 5 | negsubd 11160 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 + -𝐶) = (𝐵 − 𝐶)) |
20 | 19 | eqcomd 2742 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 − 𝐶) = (𝐵 + -𝐶)) |
21 | 20 | mpteq2dva 5135 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 − 𝐶)) = (𝑥 ∈ 𝐴 ↦ (𝐵 + -𝐶))) |
22 | 18, 21 | syl5eq 2783 | . . 3 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐴 ↦ (𝐵 + -𝐶))) |
23 | 22 | oveq1d 7206 | . 2 ⊢ (𝜑 → (𝐻 limℂ 𝐷) = ((𝑥 ∈ 𝐴 ↦ (𝐵 + -𝐶)) limℂ 𝐷)) |
24 | 11, 17, 23 | 3eltr4d 2846 | 1 ⊢ (𝜑 → (𝐸 − 𝐼) ∈ (𝐻 limℂ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ↦ cmpt 5120 (class class class)co 7191 ℂcc 10692 + caddc 10697 − cmin 11027 -cneg 11028 limℂ climc 24713 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-map 8488 df-pm 8489 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-fi 9005 df-sup 9036 df-inf 9037 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-q 12510 df-rp 12552 df-xneg 12669 df-xadd 12670 df-xmul 12671 df-fz 13061 df-seq 13540 df-exp 13601 df-cj 14627 df-re 14628 df-im 14629 df-sqrt 14763 df-abs 14764 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-plusg 16762 df-mulr 16763 df-starv 16764 df-tset 16768 df-ple 16769 df-ds 16771 df-unif 16772 df-rest 16881 df-topn 16882 df-topgen 16902 df-psmet 20309 df-xmet 20310 df-met 20311 df-bl 20312 df-mopn 20313 df-cnfld 20318 df-top 21745 df-topon 21762 df-topsp 21784 df-bases 21797 df-cnp 22079 df-xms 23172 df-ms 23173 df-limc 24717 |
This theorem is referenced by: fourierdlem60 43325 fourierdlem61 43326 fourierdlem74 43339 fourierdlem75 43340 fourierdlem76 43341 |
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