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Mirrors > Home > MPE Home > Th. List > jensenlem1 | Structured version Visualization version GIF version |
Description: Lemma for jensen 26361. (Contributed by Mario Carneiro, 4-Jun-2016.) |
Ref | Expression |
---|---|
jensen.1 | β’ (π β π· β β) |
jensen.2 | β’ (π β πΉ:π·βΆβ) |
jensen.3 | β’ ((π β§ (π β π· β§ π β π·)) β (π[,]π) β π·) |
jensen.4 | β’ (π β π΄ β Fin) |
jensen.5 | β’ (π β π:π΄βΆ(0[,)+β)) |
jensen.6 | β’ (π β π:π΄βΆπ·) |
jensen.7 | β’ (π β 0 < (βfld Ξ£g π)) |
jensen.8 | β’ ((π β§ (π₯ β π· β§ π¦ β π· β§ π‘ β (0[,]1))) β (πΉβ((π‘ Β· π₯) + ((1 β π‘) Β· π¦))) β€ ((π‘ Β· (πΉβπ₯)) + ((1 β π‘) Β· (πΉβπ¦)))) |
jensenlem.1 | β’ (π β Β¬ π§ β π΅) |
jensenlem.2 | β’ (π β (π΅ βͺ {π§}) β π΄) |
jensenlem.s | β’ π = (βfld Ξ£g (π βΎ π΅)) |
jensenlem.l | β’ πΏ = (βfld Ξ£g (π βΎ (π΅ βͺ {π§}))) |
Ref | Expression |
---|---|
jensenlem1 | β’ (π β πΏ = (π + (πβπ§))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnfldbas 20823 | . . . 4 β’ β = (Baseββfld) | |
2 | cnfldadd 20824 | . . . 4 β’ + = (+gββfld) | |
3 | cnring 20842 | . . . . 5 β’ βfld β Ring | |
4 | ringcmn 20011 | . . . . 5 β’ (βfld β Ring β βfld β CMnd) | |
5 | 3, 4 | mp1i 13 | . . . 4 β’ (π β βfld β CMnd) |
6 | jensen.4 | . . . . 5 β’ (π β π΄ β Fin) | |
7 | jensenlem.2 | . . . . . 6 β’ (π β (π΅ βͺ {π§}) β π΄) | |
8 | 7 | unssad 4151 | . . . . 5 β’ (π β π΅ β π΄) |
9 | 6, 8 | ssfid 9217 | . . . 4 β’ (π β π΅ β Fin) |
10 | rge0ssre 13382 | . . . . . 6 β’ (0[,)+β) β β | |
11 | ax-resscn 11116 | . . . . . 6 β’ β β β | |
12 | 10, 11 | sstri 3957 | . . . . 5 β’ (0[,)+β) β β |
13 | 8 | sselda 3948 | . . . . . 6 β’ ((π β§ π₯ β π΅) β π₯ β π΄) |
14 | jensen.5 | . . . . . . 7 β’ (π β π:π΄βΆ(0[,)+β)) | |
15 | 14 | ffvelcdmda 7039 | . . . . . 6 β’ ((π β§ π₯ β π΄) β (πβπ₯) β (0[,)+β)) |
16 | 13, 15 | syldan 592 | . . . . 5 β’ ((π β§ π₯ β π΅) β (πβπ₯) β (0[,)+β)) |
17 | 12, 16 | sselid 3946 | . . . 4 β’ ((π β§ π₯ β π΅) β (πβπ₯) β β) |
18 | 7 | unssbd 4152 | . . . . 5 β’ (π β {π§} β π΄) |
19 | vex 3451 | . . . . . 6 β’ π§ β V | |
20 | 19 | snss 4750 | . . . . 5 β’ (π§ β π΄ β {π§} β π΄) |
21 | 18, 20 | sylibr 233 | . . . 4 β’ (π β π§ β π΄) |
22 | jensenlem.1 | . . . 4 β’ (π β Β¬ π§ β π΅) | |
23 | 14, 21 | ffvelcdmd 7040 | . . . . 5 β’ (π β (πβπ§) β (0[,)+β)) |
24 | 12, 23 | sselid 3946 | . . . 4 β’ (π β (πβπ§) β β) |
25 | fveq2 6846 | . . . 4 β’ (π₯ = π§ β (πβπ₯) = (πβπ§)) | |
26 | 1, 2, 5, 9, 17, 21, 22, 24, 25 | gsumunsn 19745 | . . 3 β’ (π β (βfld Ξ£g (π₯ β (π΅ βͺ {π§}) β¦ (πβπ₯))) = ((βfld Ξ£g (π₯ β π΅ β¦ (πβπ₯))) + (πβπ§))) |
27 | 14, 7 | feqresmpt 6915 | . . . 4 β’ (π β (π βΎ (π΅ βͺ {π§})) = (π₯ β (π΅ βͺ {π§}) β¦ (πβπ₯))) |
28 | 27 | oveq2d 7377 | . . 3 β’ (π β (βfld Ξ£g (π βΎ (π΅ βͺ {π§}))) = (βfld Ξ£g (π₯ β (π΅ βͺ {π§}) β¦ (πβπ₯)))) |
29 | 14, 8 | feqresmpt 6915 | . . . . 5 β’ (π β (π βΎ π΅) = (π₯ β π΅ β¦ (πβπ₯))) |
30 | 29 | oveq2d 7377 | . . . 4 β’ (π β (βfld Ξ£g (π βΎ π΅)) = (βfld Ξ£g (π₯ β π΅ β¦ (πβπ₯)))) |
31 | 30 | oveq1d 7376 | . . 3 β’ (π β ((βfld Ξ£g (π βΎ π΅)) + (πβπ§)) = ((βfld Ξ£g (π₯ β π΅ β¦ (πβπ₯))) + (πβπ§))) |
32 | 26, 28, 31 | 3eqtr4d 2783 | . 2 β’ (π β (βfld Ξ£g (π βΎ (π΅ βͺ {π§}))) = ((βfld Ξ£g (π βΎ π΅)) + (πβπ§))) |
33 | jensenlem.l | . 2 β’ πΏ = (βfld Ξ£g (π βΎ (π΅ βͺ {π§}))) | |
34 | jensenlem.s | . . 3 β’ π = (βfld Ξ£g (π βΎ π΅)) | |
35 | 34 | oveq1i 7371 | . 2 β’ (π + (πβπ§)) = ((βfld Ξ£g (π βΎ π΅)) + (πβπ§)) |
36 | 32, 33, 35 | 3eqtr4g 2798 | 1 β’ (π β πΏ = (π + (πβπ§))) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 βͺ cun 3912 β wss 3914 {csn 4590 class class class wbr 5109 β¦ cmpt 5192 βΎ cres 5639 βΆwf 6496 βcfv 6500 (class class class)co 7361 Fincfn 8889 βcc 11057 βcr 11058 0cc0 11059 1c1 11060 + caddc 11062 Β· cmul 11064 +βcpnf 11194 < clt 11197 β€ cle 11198 β cmin 11393 [,)cico 13275 [,]cicc 13276 Ξ£g cgsu 17330 CMndccmn 19570 Ringcrg 19972 βfldccnfld 20819 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-addf 11138 ax-mulf 11139 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7621 df-om 7807 df-1st 7925 df-2nd 7926 df-supp 8097 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-fsupp 9312 df-oi 9454 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-z 12508 df-dec 12627 df-uz 12772 df-ico 13279 df-fz 13434 df-fzo 13577 df-seq 13916 df-hash 14240 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-mulr 17155 df-starv 17156 df-tset 17160 df-ple 17161 df-ds 17163 df-unif 17164 df-0g 17331 df-gsum 17332 df-mre 17474 df-mrc 17475 df-acs 17477 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-submnd 18610 df-grp 18759 df-minusg 18760 df-mulg 18881 df-cntz 19105 df-cmn 19572 df-abl 19573 df-mgp 19905 df-ur 19922 df-ring 19974 df-cring 19975 df-cnfld 20820 |
This theorem is referenced by: jensenlem2 26360 |
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