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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ellimciota | Structured version Visualization version GIF version |
Description: An explicit value for the limit, when the limit exists at a limit point. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
ellimciota.f | ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) |
ellimciota.a | ⊢ (𝜑 → 𝐴 ⊆ ℂ) |
ellimciota.b | ⊢ (𝜑 → 𝐵 ∈ ((limPt‘𝐾)‘𝐴)) |
ellimciota.4 | ⊢ (𝜑 → (𝐹 limℂ 𝐵) ≠ ∅) |
ellimciota.k | ⊢ 𝐾 = (TopOpen‘ℂfld) |
Ref | Expression |
---|---|
ellimciota | ⊢ (𝜑 → (℩𝑥𝑥 ∈ (𝐹 limℂ 𝐵)) ∈ (𝐹 limℂ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2676 | . . 3 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ (𝐹 limℂ 𝐵) ↔ 𝑦 ∈ (𝐹 limℂ 𝐵))) | |
2 | 1 | cbviotav 5774 | . 2 ⊢ (℩𝑥𝑥 ∈ (𝐹 limℂ 𝐵)) = (℩𝑦𝑦 ∈ (𝐹 limℂ 𝐵)) |
3 | iotaex 5785 | . . . 4 ⊢ (℩𝑦𝑦 ∈ (𝐹 limℂ 𝐵)) ∈ V | |
4 | ellimciota.4 | . . . . . 6 ⊢ (𝜑 → (𝐹 limℂ 𝐵) ≠ ∅) | |
5 | n0 3890 | . . . . . 6 ⊢ ((𝐹 limℂ 𝐵) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐹 limℂ 𝐵)) | |
6 | 4, 5 | sylib 207 | . . . . 5 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ (𝐹 limℂ 𝐵)) |
7 | ellimciota.f | . . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) | |
8 | ellimciota.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ℂ) | |
9 | ellimciota.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ((limPt‘𝐾)‘𝐴)) | |
10 | ellimciota.k | . . . . . 6 ⊢ 𝐾 = (TopOpen‘ℂfld) | |
11 | 7, 8, 9, 10 | limcmo 23452 | . . . . 5 ⊢ (𝜑 → ∃*𝑥 𝑥 ∈ (𝐹 limℂ 𝐵)) |
12 | eu5 2484 | . . . . 5 ⊢ (∃!𝑥 𝑥 ∈ (𝐹 limℂ 𝐵) ↔ (∃𝑥 𝑥 ∈ (𝐹 limℂ 𝐵) ∧ ∃*𝑥 𝑥 ∈ (𝐹 limℂ 𝐵))) | |
13 | 6, 11, 12 | sylanbrc 695 | . . . 4 ⊢ (𝜑 → ∃!𝑥 𝑥 ∈ (𝐹 limℂ 𝐵)) |
14 | eleq1 2676 | . . . . 5 ⊢ (𝑥 = (℩𝑦𝑦 ∈ (𝐹 limℂ 𝐵)) → (𝑥 ∈ (𝐹 limℂ 𝐵) ↔ (℩𝑦𝑦 ∈ (𝐹 limℂ 𝐵)) ∈ (𝐹 limℂ 𝐵))) | |
15 | 14 | iota2 5794 | . . . 4 ⊢ (((℩𝑦𝑦 ∈ (𝐹 limℂ 𝐵)) ∈ V ∧ ∃!𝑥 𝑥 ∈ (𝐹 limℂ 𝐵)) → ((℩𝑦𝑦 ∈ (𝐹 limℂ 𝐵)) ∈ (𝐹 limℂ 𝐵) ↔ (℩𝑥𝑥 ∈ (𝐹 limℂ 𝐵)) = (℩𝑦𝑦 ∈ (𝐹 limℂ 𝐵)))) |
16 | 3, 13, 15 | sylancr 694 | . . 3 ⊢ (𝜑 → ((℩𝑦𝑦 ∈ (𝐹 limℂ 𝐵)) ∈ (𝐹 limℂ 𝐵) ↔ (℩𝑥𝑥 ∈ (𝐹 limℂ 𝐵)) = (℩𝑦𝑦 ∈ (𝐹 limℂ 𝐵)))) |
17 | 2, 16 | mpbiri 247 | . 2 ⊢ (𝜑 → (℩𝑦𝑦 ∈ (𝐹 limℂ 𝐵)) ∈ (𝐹 limℂ 𝐵)) |
18 | 2, 17 | syl5eqel 2692 | 1 ⊢ (𝜑 → (℩𝑥𝑥 ∈ (𝐹 limℂ 𝐵)) ∈ (𝐹 limℂ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ∃!weu 2458 ∃*wmo 2459 ≠ wne 2780 Vcvv 3173 ⊆ wss 3540 ∅c0 3874 ℩cio 5766 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 TopOpenctopn 15905 ℂfldccnfld 19567 limPtclp 20748 limℂ climc 23432 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fi 8200 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-icc 12053 df-fz 12198 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-plusg 15781 df-mulr 15782 df-starv 15783 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-rest 15906 df-topn 15907 df-topgen 15927 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-cnp 20842 df-haus 20929 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-xms 21935 df-ms 21936 df-limc 23436 |
This theorem is referenced by: fourierdlem94 39093 fourierdlem113 39112 |
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