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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > divlimc | Structured version Visualization version GIF version |
Description: Limit of the quotient of two functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
divlimc.f | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
divlimc.g | ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ 𝐶) |
divlimc.h | ⊢ 𝐻 = (𝑥 ∈ 𝐴 ↦ (𝐵 / 𝐶)) |
divlimc.b | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) |
divlimc.c | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ (ℂ ∖ {0})) |
divlimc.x | ⊢ (𝜑 → 𝑋 ∈ (𝐹 limℂ 𝐷)) |
divlimc.y | ⊢ (𝜑 → 𝑌 ∈ (𝐺 limℂ 𝐷)) |
divlimc.yne0 | ⊢ (𝜑 → 𝑌 ≠ 0) |
divlimc.cne0 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ≠ 0) |
Ref | Expression |
---|---|
divlimc | ⊢ (𝜑 → (𝑋 / 𝑌) ∈ (𝐻 limℂ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | divlimc.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
2 | eqid 2726 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ (1 / 𝐶)) = (𝑥 ∈ 𝐴 ↦ (1 / 𝐶)) | |
3 | eqid 2726 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐵 · (1 / 𝐶))) = (𝑥 ∈ 𝐴 ↦ (𝐵 · (1 / 𝐶))) | |
4 | divlimc.b | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) | |
5 | divlimc.c | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ (ℂ ∖ {0})) | |
6 | 5 | eldifad 3959 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ) |
7 | divlimc.cne0 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ≠ 0) | |
8 | 6, 7 | reccld 12026 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (1 / 𝐶) ∈ ℂ) |
9 | divlimc.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝐹 limℂ 𝐷)) | |
10 | divlimc.g | . . . 4 ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
11 | divlimc.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝐺 limℂ 𝐷)) | |
12 | divlimc.yne0 | . . . 4 ⊢ (𝜑 → 𝑌 ≠ 0) | |
13 | 10, 2, 5, 11, 12 | reclimc 45308 | . . 3 ⊢ (𝜑 → (1 / 𝑌) ∈ ((𝑥 ∈ 𝐴 ↦ (1 / 𝐶)) limℂ 𝐷)) |
14 | 1, 2, 3, 4, 8, 9, 13 | mullimc 45271 | . 2 ⊢ (𝜑 → (𝑋 · (1 / 𝑌)) ∈ ((𝑥 ∈ 𝐴 ↦ (𝐵 · (1 / 𝐶))) limℂ 𝐷)) |
15 | limccl 25890 | . . . 4 ⊢ (𝐹 limℂ 𝐷) ⊆ ℂ | |
16 | 15, 9 | sselid 3977 | . . 3 ⊢ (𝜑 → 𝑋 ∈ ℂ) |
17 | limccl 25890 | . . . 4 ⊢ (𝐺 limℂ 𝐷) ⊆ ℂ | |
18 | 17, 11 | sselid 3977 | . . 3 ⊢ (𝜑 → 𝑌 ∈ ℂ) |
19 | 16, 18, 12 | divrecd 12036 | . 2 ⊢ (𝜑 → (𝑋 / 𝑌) = (𝑋 · (1 / 𝑌))) |
20 | divlimc.h | . . . 4 ⊢ 𝐻 = (𝑥 ∈ 𝐴 ↦ (𝐵 / 𝐶)) | |
21 | 4, 6, 7 | divrecd 12036 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 / 𝐶) = (𝐵 · (1 / 𝐶))) |
22 | 21 | mpteq2dva 5244 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 / 𝐶)) = (𝑥 ∈ 𝐴 ↦ (𝐵 · (1 / 𝐶)))) |
23 | 20, 22 | eqtrid 2778 | . . 3 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐴 ↦ (𝐵 · (1 / 𝐶)))) |
24 | 23 | oveq1d 7429 | . 2 ⊢ (𝜑 → (𝐻 limℂ 𝐷) = ((𝑥 ∈ 𝐴 ↦ (𝐵 · (1 / 𝐶))) limℂ 𝐷)) |
25 | 14, 19, 24 | 3eltr4d 2841 | 1 ⊢ (𝜑 → (𝑋 / 𝑌) ∈ (𝐻 limℂ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∖ cdif 3944 {csn 4624 ↦ cmpt 5227 (class class class)co 7414 ℂcc 11145 0cc0 11147 1c1 11148 · cmul 11152 / cdiv 11910 limℂ climc 25877 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-cnex 11203 ax-resscn 11204 ax-1cn 11205 ax-icn 11206 ax-addcl 11207 ax-addrcl 11208 ax-mulcl 11209 ax-mulrcl 11210 ax-mulcom 11211 ax-addass 11212 ax-mulass 11213 ax-distr 11214 ax-i2m1 11215 ax-1ne0 11216 ax-1rid 11217 ax-rnegex 11218 ax-rrecex 11219 ax-cnre 11220 ax-pre-lttri 11221 ax-pre-lttrn 11222 ax-pre-ltadd 11223 ax-pre-mulgt0 11224 ax-pre-sup 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3365 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4907 df-int 4948 df-iun 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6303 df-ord 6369 df-on 6370 df-lim 6371 df-suc 6372 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-map 8847 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fi 9445 df-sup 9476 df-inf 9477 df-pnf 11289 df-mnf 11290 df-xr 11291 df-ltxr 11292 df-le 11293 df-sub 11485 df-neg 11486 df-div 11911 df-nn 12257 df-2 12319 df-3 12320 df-4 12321 df-5 12322 df-6 12323 df-7 12324 df-8 12325 df-9 12326 df-n0 12517 df-z 12603 df-dec 12722 df-uz 12867 df-q 12977 df-rp 13021 df-xneg 13138 df-xadd 13139 df-xmul 13140 df-fz 13531 df-seq 14014 df-exp 14074 df-cj 15097 df-re 15098 df-im 15099 df-sqrt 15233 df-abs 15234 df-struct 17142 df-slot 17177 df-ndx 17189 df-base 17207 df-plusg 17272 df-mulr 17273 df-starv 17274 df-tset 17278 df-ple 17279 df-ds 17281 df-unif 17282 df-rest 17430 df-topn 17431 df-topgen 17451 df-psmet 21329 df-xmet 21330 df-met 21331 df-bl 21332 df-mopn 21333 df-cnfld 21338 df-top 22882 df-topon 22899 df-topsp 22921 df-bases 22935 df-cnp 23218 df-xms 24312 df-ms 24313 df-limc 25881 |
This theorem is referenced by: fourierdlem74 45835 fourierdlem75 45836 fourierdlem76 45837 |
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