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Mirrors > Home > MPE Home > Th. List > limcco | Structured version Visualization version GIF version |
Description: Composition of two limits. (Contributed by Mario Carneiro, 29-Dec-2016.) |
Ref | Expression |
---|---|
limcco.r | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑅 ≠ 𝐶)) → 𝑅 ∈ 𝐵) |
limcco.s | ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑆 ∈ ℂ) |
limcco.c | ⊢ (𝜑 → 𝐶 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑅) limℂ 𝑋)) |
limcco.d | ⊢ (𝜑 → 𝐷 ∈ ((𝑦 ∈ 𝐵 ↦ 𝑆) limℂ 𝐶)) |
limcco.1 | ⊢ (𝑦 = 𝑅 → 𝑆 = 𝑇) |
limcco.2 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑅 = 𝐶)) → 𝑇 = 𝐷) |
Ref | Expression |
---|---|
limcco | ⊢ (𝜑 → 𝐷 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑇) limℂ 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limcco.r | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑅 ≠ 𝐶)) → 𝑅 ∈ 𝐵) | |
2 | 1 | expr 459 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑅 ≠ 𝐶 → 𝑅 ∈ 𝐵)) |
3 | 2 | necon1bd 3034 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (¬ 𝑅 ∈ 𝐵 → 𝑅 = 𝐶)) |
4 | limccl 24473 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝑅) limℂ 𝑋) ⊆ ℂ | |
5 | limcco.c | . . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑅) limℂ 𝑋)) | |
6 | 4, 5 | sseldi 3965 | . . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
7 | 6 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ) |
8 | elsn2g 4603 | . . . . . . . 8 ⊢ (𝐶 ∈ ℂ → (𝑅 ∈ {𝐶} ↔ 𝑅 = 𝐶)) | |
9 | 7, 8 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑅 ∈ {𝐶} ↔ 𝑅 = 𝐶)) |
10 | 3, 9 | sylibrd 261 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (¬ 𝑅 ∈ 𝐵 → 𝑅 ∈ {𝐶})) |
11 | 10 | orrd 859 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑅 ∈ 𝐵 ∨ 𝑅 ∈ {𝐶})) |
12 | elun 4125 | . . . . 5 ⊢ (𝑅 ∈ (𝐵 ∪ {𝐶}) ↔ (𝑅 ∈ 𝐵 ∨ 𝑅 ∈ {𝐶})) | |
13 | 11, 12 | sylibr 236 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ (𝐵 ∪ {𝐶})) |
14 | 13 | fmpttd 6879 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑅):𝐴⟶(𝐵 ∪ {𝐶})) |
15 | eqid 2821 | . . . . . 6 ⊢ (𝑦 ∈ 𝐵 ↦ 𝑆) = (𝑦 ∈ 𝐵 ↦ 𝑆) | |
16 | limcco.s | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑆 ∈ ℂ) | |
17 | 15, 16 | dmmptd 6493 | . . . . 5 ⊢ (𝜑 → dom (𝑦 ∈ 𝐵 ↦ 𝑆) = 𝐵) |
18 | limcco.d | . . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ ((𝑦 ∈ 𝐵 ↦ 𝑆) limℂ 𝐶)) | |
19 | limcrcl 24472 | . . . . . . 7 ⊢ (𝐷 ∈ ((𝑦 ∈ 𝐵 ↦ 𝑆) limℂ 𝐶) → ((𝑦 ∈ 𝐵 ↦ 𝑆):dom (𝑦 ∈ 𝐵 ↦ 𝑆)⟶ℂ ∧ dom (𝑦 ∈ 𝐵 ↦ 𝑆) ⊆ ℂ ∧ 𝐶 ∈ ℂ)) | |
20 | 18, 19 | syl 17 | . . . . . 6 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ 𝑆):dom (𝑦 ∈ 𝐵 ↦ 𝑆)⟶ℂ ∧ dom (𝑦 ∈ 𝐵 ↦ 𝑆) ⊆ ℂ ∧ 𝐶 ∈ ℂ)) |
21 | 20 | simp2d 1139 | . . . . 5 ⊢ (𝜑 → dom (𝑦 ∈ 𝐵 ↦ 𝑆) ⊆ ℂ) |
22 | 17, 21 | eqsstrrd 4006 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℂ) |
23 | 6 | snssd 4742 | . . . 4 ⊢ (𝜑 → {𝐶} ⊆ ℂ) |
24 | 22, 23 | unssd 4162 | . . 3 ⊢ (𝜑 → (𝐵 ∪ {𝐶}) ⊆ ℂ) |
25 | eqid 2821 | . . 3 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
26 | eqid 2821 | . . 3 ⊢ ((TopOpen‘ℂfld) ↾t (𝐵 ∪ {𝐶})) = ((TopOpen‘ℂfld) ↾t (𝐵 ∪ {𝐶})) | |
27 | 22, 6, 16, 26, 25 | limcmpt 24481 | . . . 4 ⊢ (𝜑 → (𝐷 ∈ ((𝑦 ∈ 𝐵 ↦ 𝑆) limℂ 𝐶) ↔ (𝑦 ∈ (𝐵 ∪ {𝐶}) ↦ if(𝑦 = 𝐶, 𝐷, 𝑆)) ∈ ((((TopOpen‘ℂfld) ↾t (𝐵 ∪ {𝐶})) CnP (TopOpen‘ℂfld))‘𝐶))) |
28 | 18, 27 | mpbid 234 | . . 3 ⊢ (𝜑 → (𝑦 ∈ (𝐵 ∪ {𝐶}) ↦ if(𝑦 = 𝐶, 𝐷, 𝑆)) ∈ ((((TopOpen‘ℂfld) ↾t (𝐵 ∪ {𝐶})) CnP (TopOpen‘ℂfld))‘𝐶)) |
29 | 14, 24, 25, 26, 5, 28 | limccnp 24489 | . 2 ⊢ (𝜑 → ((𝑦 ∈ (𝐵 ∪ {𝐶}) ↦ if(𝑦 = 𝐶, 𝐷, 𝑆))‘𝐶) ∈ (((𝑦 ∈ (𝐵 ∪ {𝐶}) ↦ if(𝑦 = 𝐶, 𝐷, 𝑆)) ∘ (𝑥 ∈ 𝐴 ↦ 𝑅)) limℂ 𝑋)) |
30 | eqid 2821 | . . 3 ⊢ (𝑦 ∈ (𝐵 ∪ {𝐶}) ↦ if(𝑦 = 𝐶, 𝐷, 𝑆)) = (𝑦 ∈ (𝐵 ∪ {𝐶}) ↦ if(𝑦 = 𝐶, 𝐷, 𝑆)) | |
31 | iftrue 4473 | . . 3 ⊢ (𝑦 = 𝐶 → if(𝑦 = 𝐶, 𝐷, 𝑆) = 𝐷) | |
32 | ssun2 4149 | . . . 4 ⊢ {𝐶} ⊆ (𝐵 ∪ {𝐶}) | |
33 | snssg 4717 | . . . . 5 ⊢ (𝐶 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑅) limℂ 𝑋) → (𝐶 ∈ (𝐵 ∪ {𝐶}) ↔ {𝐶} ⊆ (𝐵 ∪ {𝐶}))) | |
34 | 5, 33 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐶 ∈ (𝐵 ∪ {𝐶}) ↔ {𝐶} ⊆ (𝐵 ∪ {𝐶}))) |
35 | 32, 34 | mpbiri 260 | . . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐵 ∪ {𝐶})) |
36 | 30, 31, 35, 18 | fvmptd3 6791 | . 2 ⊢ (𝜑 → ((𝑦 ∈ (𝐵 ∪ {𝐶}) ↦ if(𝑦 = 𝐶, 𝐷, 𝑆))‘𝐶) = 𝐷) |
37 | eqidd 2822 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑅) = (𝑥 ∈ 𝐴 ↦ 𝑅)) | |
38 | eqidd 2822 | . . . . 5 ⊢ (𝜑 → (𝑦 ∈ (𝐵 ∪ {𝐶}) ↦ if(𝑦 = 𝐶, 𝐷, 𝑆)) = (𝑦 ∈ (𝐵 ∪ {𝐶}) ↦ if(𝑦 = 𝐶, 𝐷, 𝑆))) | |
39 | eqeq1 2825 | . . . . . 6 ⊢ (𝑦 = 𝑅 → (𝑦 = 𝐶 ↔ 𝑅 = 𝐶)) | |
40 | limcco.1 | . . . . . 6 ⊢ (𝑦 = 𝑅 → 𝑆 = 𝑇) | |
41 | 39, 40 | ifbieq2d 4492 | . . . . 5 ⊢ (𝑦 = 𝑅 → if(𝑦 = 𝐶, 𝐷, 𝑆) = if(𝑅 = 𝐶, 𝐷, 𝑇)) |
42 | 13, 37, 38, 41 | fmptco 6891 | . . . 4 ⊢ (𝜑 → ((𝑦 ∈ (𝐵 ∪ {𝐶}) ↦ if(𝑦 = 𝐶, 𝐷, 𝑆)) ∘ (𝑥 ∈ 𝐴 ↦ 𝑅)) = (𝑥 ∈ 𝐴 ↦ if(𝑅 = 𝐶, 𝐷, 𝑇))) |
43 | ifid 4506 | . . . . . 6 ⊢ if(𝑅 = 𝐶, 𝑇, 𝑇) = 𝑇 | |
44 | limcco.2 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑅 = 𝐶)) → 𝑇 = 𝐷) | |
45 | 44 | anassrs 470 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑅 = 𝐶) → 𝑇 = 𝐷) |
46 | 45 | ifeq1da 4497 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑅 = 𝐶, 𝑇, 𝑇) = if(𝑅 = 𝐶, 𝐷, 𝑇)) |
47 | 43, 46 | syl5reqr 2871 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑅 = 𝐶, 𝐷, 𝑇) = 𝑇) |
48 | 47 | mpteq2dva 5161 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(𝑅 = 𝐶, 𝐷, 𝑇)) = (𝑥 ∈ 𝐴 ↦ 𝑇)) |
49 | 42, 48 | eqtrd 2856 | . . 3 ⊢ (𝜑 → ((𝑦 ∈ (𝐵 ∪ {𝐶}) ↦ if(𝑦 = 𝐶, 𝐷, 𝑆)) ∘ (𝑥 ∈ 𝐴 ↦ 𝑅)) = (𝑥 ∈ 𝐴 ↦ 𝑇)) |
50 | 49 | oveq1d 7171 | . 2 ⊢ (𝜑 → (((𝑦 ∈ (𝐵 ∪ {𝐶}) ↦ if(𝑦 = 𝐶, 𝐷, 𝑆)) ∘ (𝑥 ∈ 𝐴 ↦ 𝑅)) limℂ 𝑋) = ((𝑥 ∈ 𝐴 ↦ 𝑇) limℂ 𝑋)) |
51 | 29, 36, 50 | 3eltr3d 2927 | 1 ⊢ (𝜑 → 𝐷 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑇) limℂ 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∪ cun 3934 ⊆ wss 3936 ifcif 4467 {csn 4567 ↦ cmpt 5146 dom cdm 5555 ∘ ccom 5559 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 ↾t crest 16694 TopOpenctopn 16695 ℂfldccnfld 20545 CnP ccnp 21833 limℂ climc 24460 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fi 8875 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-fz 12894 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-plusg 16578 df-mulr 16579 df-starv 16580 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-rest 16696 df-topn 16697 df-topgen 16717 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cnp 21836 df-xms 22930 df-ms 22931 df-limc 24464 |
This theorem is referenced by: dvcobr 24543 dvcnvlem 24573 lhop2 24612 fourierdlem60 42471 fourierdlem61 42472 fourierdlem62 42473 fourierdlem73 42484 fourierdlem76 42487 |
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