Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
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 2823 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ (1 / 𝐶)) = (𝑥 ∈ 𝐴 ↦ (1 / 𝐶)) | |
3 | eqid 2823 | . . 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 3950 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ) |
7 | divlimc.cne0 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ≠ 0) | |
8 | 6, 7 | reccld 11411 | . . 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 41941 | . . 3 ⊢ (𝜑 → (1 / 𝑌) ∈ ((𝑥 ∈ 𝐴 ↦ (1 / 𝐶)) limℂ 𝐷)) |
14 | 1, 2, 3, 4, 8, 9, 13 | mullimc 41904 | . 2 ⊢ (𝜑 → (𝑋 · (1 / 𝑌)) ∈ ((𝑥 ∈ 𝐴 ↦ (𝐵 · (1 / 𝐶))) limℂ 𝐷)) |
15 | limccl 24475 | . . . 4 ⊢ (𝐹 limℂ 𝐷) ⊆ ℂ | |
16 | 15, 9 | sseldi 3967 | . . 3 ⊢ (𝜑 → 𝑋 ∈ ℂ) |
17 | limccl 24475 | . . . 4 ⊢ (𝐺 limℂ 𝐷) ⊆ ℂ | |
18 | 17, 11 | sseldi 3967 | . . 3 ⊢ (𝜑 → 𝑌 ∈ ℂ) |
19 | 16, 18, 12 | divrecd 11421 | . 2 ⊢ (𝜑 → (𝑋 / 𝑌) = (𝑋 · (1 / 𝑌))) |
20 | divlimc.h | . . . 4 ⊢ 𝐻 = (𝑥 ∈ 𝐴 ↦ (𝐵 / 𝐶)) | |
21 | 4, 6, 7 | divrecd 11421 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 / 𝐶) = (𝐵 · (1 / 𝐶))) |
22 | 21 | mpteq2dva 5163 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 / 𝐶)) = (𝑥 ∈ 𝐴 ↦ (𝐵 · (1 / 𝐶)))) |
23 | 20, 22 | syl5eq 2870 | . . 3 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐴 ↦ (𝐵 · (1 / 𝐶)))) |
24 | 23 | oveq1d 7173 | . 2 ⊢ (𝜑 → (𝐻 limℂ 𝐷) = ((𝑥 ∈ 𝐴 ↦ (𝐵 · (1 / 𝐶))) limℂ 𝐷)) |
25 | 14, 19, 24 | 3eltr4d 2930 | 1 ⊢ (𝜑 → (𝑋 / 𝑌) ∈ (𝐻 limℂ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∖ cdif 3935 {csn 4569 ↦ cmpt 5148 (class class class)co 7158 ℂcc 10537 0cc0 10539 1c1 10540 · cmul 10544 / cdiv 11299 limℂ climc 24462 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fi 8877 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-fz 12896 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-plusg 16580 df-mulr 16581 df-starv 16582 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-rest 16698 df-topn 16699 df-topgen 16719 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-cnfld 20548 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cnp 21838 df-xms 22932 df-ms 22933 df-limc 24466 |
This theorem is referenced by: fourierdlem74 42472 fourierdlem75 42473 fourierdlem76 42474 |
Copyright terms: Public domain | W3C validator |