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Mirrors > Home > MPE Home > Th. List > Mathboxes > iprodefisum | Structured version Visualization version GIF version |
Description: Applying the exponential function to an infinite sum yields an infinite product. (Contributed by Scott Fenton, 11-Feb-2018.) |
Ref | Expression |
---|---|
iprodefisum.1 | β’ π = (β€β₯βπ) |
iprodefisum.2 | β’ (π β π β β€) |
iprodefisum.3 | β’ ((π β§ π β π) β (πΉβπ) = π΅) |
iprodefisum.4 | β’ ((π β§ π β π) β π΅ β β) |
iprodefisum.5 | β’ (π β seqπ( + , πΉ) β dom β ) |
Ref | Expression |
---|---|
iprodefisum | β’ (π β βπ β π (expβπ΅) = (expβΞ£π β π π΅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iprodefisum.1 | . 2 β’ π = (β€β₯βπ) | |
2 | iprodefisum.2 | . 2 β’ (π β π β β€) | |
3 | iprodefisum.3 | . . . 4 β’ ((π β§ π β π) β (πΉβπ) = π΅) | |
4 | iprodefisum.4 | . . . 4 β’ ((π β§ π β π) β π΅ β β) | |
5 | iprodefisum.5 | . . . 4 β’ (π β seqπ( + , πΉ) β dom β ) | |
6 | 1, 2, 3, 4, 5 | isumcl 15672 | . . 3 β’ (π β Ξ£π β π π΅ β β) |
7 | efne0 16005 | . . 3 β’ (Ξ£π β π π΅ β β β (expβΞ£π β π π΅) β 0) | |
8 | 6, 7 | syl 17 | . 2 β’ (π β (expβΞ£π β π π΅) β 0) |
9 | efcn 25854 | . . . . 5 β’ exp β (ββcnββ) | |
10 | 9 | a1i 11 | . . . 4 β’ (π β exp β (ββcnββ)) |
11 | fveq2 6862 | . . . . . . . 8 β’ (π = π β (πΉβπ) = (πΉβπ)) | |
12 | eqid 2731 | . . . . . . . 8 β’ (π β π β¦ (πΉβπ)) = (π β π β¦ (πΉβπ)) | |
13 | fvex 6875 | . . . . . . . 8 β’ (πΉβπ) β V | |
14 | 11, 12, 13 | fvmpt 6968 | . . . . . . 7 β’ (π β π β ((π β π β¦ (πΉβπ))βπ) = (πΉβπ)) |
15 | 14 | adantl 482 | . . . . . 6 β’ ((π β§ π β π) β ((π β π β¦ (πΉβπ))βπ) = (πΉβπ)) |
16 | 3, 4 | eqeltrd 2832 | . . . . . 6 β’ ((π β§ π β π) β (πΉβπ) β β) |
17 | 15, 16 | eqeltrd 2832 | . . . . 5 β’ ((π β§ π β π) β ((π β π β¦ (πΉβπ))βπ) β β) |
18 | 1, 2, 17 | serf 13961 | . . . 4 β’ (π β seqπ( + , (π β π β¦ (πΉβπ))):πβΆβ) |
19 | 1 | eqcomi 2740 | . . . . . . . 8 β’ (β€β₯βπ) = π |
20 | 14, 19 | eleq2s 2850 | . . . . . . 7 β’ (π β (β€β₯βπ) β ((π β π β¦ (πΉβπ))βπ) = (πΉβπ)) |
21 | 20 | adantl 482 | . . . . . 6 β’ ((π β§ π β (β€β₯βπ)) β ((π β π β¦ (πΉβπ))βπ) = (πΉβπ)) |
22 | 2, 21 | seqfeq 13958 | . . . . 5 β’ (π β seqπ( + , (π β π β¦ (πΉβπ))) = seqπ( + , πΉ)) |
23 | climdm 15463 | . . . . . 6 β’ (seqπ( + , πΉ) β dom β β seqπ( + , πΉ) β ( β βseqπ( + , πΉ))) | |
24 | 5, 23 | sylib 217 | . . . . 5 β’ (π β seqπ( + , πΉ) β ( β βseqπ( + , πΉ))) |
25 | 22, 24 | eqbrtrd 5147 | . . . 4 β’ (π β seqπ( + , (π β π β¦ (πΉβπ))) β ( β βseqπ( + , πΉ))) |
26 | climcl 15408 | . . . . 5 β’ (seqπ( + , πΉ) β ( β βseqπ( + , πΉ)) β ( β βseqπ( + , πΉ)) β β) | |
27 | 24, 26 | syl 17 | . . . 4 β’ (π β ( β βseqπ( + , πΉ)) β β) |
28 | 1, 2, 10, 18, 25, 27 | climcncf 24315 | . . 3 β’ (π β (exp β seqπ( + , (π β π β¦ (πΉβπ)))) β (expβ( β βseqπ( + , πΉ)))) |
29 | 11 | cbvmptv 5238 | . . . . 5 β’ (π β π β¦ (πΉβπ)) = (π β π β¦ (πΉβπ)) |
30 | 16, 29 | fmptd 7082 | . . . 4 β’ (π β (π β π β¦ (πΉβπ)):πβΆβ) |
31 | 1, 2, 30 | iprodefisumlem 34433 | . . 3 β’ (π β seqπ( Β· , (exp β (π β π β¦ (πΉβπ)))) = (exp β seqπ( + , (π β π β¦ (πΉβπ))))) |
32 | 1, 2, 3, 4 | isum 15630 | . . . 4 β’ (π β Ξ£π β π π΅ = ( β βseqπ( + , πΉ))) |
33 | 32 | fveq2d 6866 | . . 3 β’ (π β (expβΞ£π β π π΅) = (expβ( β βseqπ( + , πΉ)))) |
34 | 28, 31, 33 | 3brtr4d 5157 | . 2 β’ (π β seqπ( Β· , (exp β (π β π β¦ (πΉβπ)))) β (expβΞ£π β π π΅)) |
35 | fvco3 6960 | . . . 4 β’ (((π β π β¦ (πΉβπ)):πβΆβ β§ π β π) β ((exp β (π β π β¦ (πΉβπ)))βπ) = (expβ((π β π β¦ (πΉβπ))βπ))) | |
36 | 30, 35 | sylan 580 | . . 3 β’ ((π β§ π β π) β ((exp β (π β π β¦ (πΉβπ)))βπ) = (expβ((π β π β¦ (πΉβπ))βπ))) |
37 | 15 | fveq2d 6866 | . . 3 β’ ((π β§ π β π) β (expβ((π β π β¦ (πΉβπ))βπ)) = (expβ(πΉβπ))) |
38 | 3 | fveq2d 6866 | . . 3 β’ ((π β§ π β π) β (expβ(πΉβπ)) = (expβπ΅)) |
39 | 36, 37, 38 | 3eqtrd 2775 | . 2 β’ ((π β§ π β π) β ((exp β (π β π β¦ (πΉβπ)))βπ) = (expβπ΅)) |
40 | efcl 15991 | . . 3 β’ (π΅ β β β (expβπ΅) β β) | |
41 | 4, 40 | syl 17 | . 2 β’ ((π β§ π β π) β (expβπ΅) β β) |
42 | 1, 2, 8, 34, 39, 41 | iprodn0 15849 | 1 β’ (π β βπ β π (expβπ΅) = (expβΞ£π β π π΅)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 β wne 2939 class class class wbr 5125 β¦ cmpt 5208 dom cdm 5653 β ccom 5657 βΆwf 6512 βcfv 6516 (class class class)co 7377 βcc 11073 0cc0 11075 + caddc 11078 Β· cmul 11080 β€cz 12523 β€β₯cuz 12787 seqcseq 13931 β cli 15393 Ξ£csu 15597 βcprod 15814 expce 15970 βcnβccncf 24291 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5262 ax-sep 5276 ax-nul 5283 ax-pow 5340 ax-pr 5404 ax-un 7692 ax-inf2 9601 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 ax-addf 11154 ax-mulf 11155 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3364 df-reu 3365 df-rab 3419 df-v 3461 df-sbc 3758 df-csb 3874 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-pss 3947 df-nul 4303 df-if 4507 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4886 df-int 4928 df-iun 4976 df-iin 4977 df-br 5126 df-opab 5188 df-mpt 5209 df-tr 5243 df-id 5551 df-eprel 5557 df-po 5565 df-so 5566 df-fr 5608 df-se 5609 df-we 5610 df-xp 5659 df-rel 5660 df-cnv 5661 df-co 5662 df-dm 5663 df-rn 5664 df-res 5665 df-ima 5666 df-pred 6273 df-ord 6340 df-on 6341 df-lim 6342 df-suc 6343 df-iota 6468 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-isom 6525 df-riota 7333 df-ov 7380 df-oprab 7381 df-mpo 7382 df-of 7637 df-om 7823 df-1st 7941 df-2nd 7942 df-supp 8113 df-frecs 8232 df-wrecs 8263 df-recs 8337 df-rdg 8376 df-1o 8432 df-2o 8433 df-er 8670 df-map 8789 df-pm 8790 df-ixp 8858 df-en 8906 df-dom 8907 df-sdom 8908 df-fin 8909 df-fsupp 9328 df-fi 9371 df-sup 9402 df-inf 9403 df-oi 9470 df-card 9899 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11411 df-neg 11412 df-div 11837 df-nn 12178 df-2 12240 df-3 12241 df-4 12242 df-5 12243 df-6 12244 df-7 12245 df-8 12246 df-9 12247 df-n0 12438 df-z 12524 df-dec 12643 df-uz 12788 df-q 12898 df-rp 12940 df-xneg 13057 df-xadd 13058 df-xmul 13059 df-ico 13295 df-icc 13296 df-fz 13450 df-fzo 13593 df-fl 13722 df-seq 13932 df-exp 13993 df-fac 14199 df-bc 14228 df-hash 14256 df-shft 14979 df-cj 15011 df-re 15012 df-im 15013 df-sqrt 15147 df-abs 15148 df-limsup 15380 df-clim 15397 df-rlim 15398 df-sum 15598 df-prod 15815 df-ef 15976 df-struct 17045 df-sets 17062 df-slot 17080 df-ndx 17092 df-base 17110 df-ress 17139 df-plusg 17175 df-mulr 17176 df-starv 17177 df-sca 17178 df-vsca 17179 df-ip 17180 df-tset 17181 df-ple 17182 df-ds 17184 df-unif 17185 df-hom 17186 df-cco 17187 df-rest 17333 df-topn 17334 df-0g 17352 df-gsum 17353 df-topgen 17354 df-pt 17355 df-prds 17358 df-xrs 17413 df-qtop 17418 df-imas 17419 df-xps 17421 df-mre 17495 df-mrc 17496 df-acs 17498 df-mgm 18526 df-sgrp 18575 df-mnd 18586 df-submnd 18631 df-mulg 18902 df-cntz 19126 df-cmn 19593 df-psmet 20840 df-xmet 20841 df-met 20842 df-bl 20843 df-mopn 20844 df-fbas 20845 df-fg 20846 df-cnfld 20849 df-top 22295 df-topon 22312 df-topsp 22334 df-bases 22348 df-cld 22422 df-ntr 22423 df-cls 22424 df-nei 22501 df-lp 22539 df-perf 22540 df-cn 22630 df-cnp 22631 df-haus 22718 df-tx 22965 df-hmeo 23158 df-fil 23249 df-fm 23341 df-flim 23342 df-flf 23343 df-xms 23725 df-ms 23726 df-tms 23727 df-cncf 24293 df-limc 25282 df-dv 25283 |
This theorem is referenced by: iprodgam 34435 |
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