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Mirrors > Home > MPE Home > Th. List > jensenlem1 | Structured version Visualization version GIF version |
Description: Lemma for jensen 24715. (Contributed by Mario Carneiro, 4-Jun-2016.) |
Ref | Expression |
---|---|
jensen.1 | ⊢ (𝜑 → 𝐷 ⊆ ℝ) |
jensen.2 | ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) |
jensen.3 | ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐷 ∧ 𝑏 ∈ 𝐷)) → (𝑎[,]𝑏) ⊆ 𝐷) |
jensen.4 | ⊢ (𝜑 → 𝐴 ∈ Fin) |
jensen.5 | ⊢ (𝜑 → 𝑇:𝐴⟶(0[,)+∞)) |
jensen.6 | ⊢ (𝜑 → 𝑋:𝐴⟶𝐷) |
jensen.7 | ⊢ (𝜑 → 0 < (ℂfld Σg 𝑇)) |
jensen.8 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷 ∧ 𝑡 ∈ (0[,]1))) → (𝐹‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ≤ ((𝑡 · (𝐹‘𝑥)) + ((1 − 𝑡) · (𝐹‘𝑦)))) |
jensenlem.1 | ⊢ (𝜑 → ¬ 𝑧 ∈ 𝐵) |
jensenlem.2 | ⊢ (𝜑 → (𝐵 ∪ {𝑧}) ⊆ 𝐴) |
jensenlem.s | ⊢ 𝑆 = (ℂfld Σg (𝑇 ↾ 𝐵)) |
jensenlem.l | ⊢ 𝐿 = (ℂfld Σg (𝑇 ↾ (𝐵 ∪ {𝑧}))) |
Ref | Expression |
---|---|
jensenlem1 | ⊢ (𝜑 → 𝐿 = (𝑆 + (𝑇‘𝑧))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnfldbas 19750 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
2 | cnfldadd 19751 | . . . 4 ⊢ + = (+g‘ℂfld) | |
3 | cnring 19768 | . . . . 5 ⊢ ℂfld ∈ Ring | |
4 | ringcmn 18581 | . . . . 5 ⊢ (ℂfld ∈ Ring → ℂfld ∈ CMnd) | |
5 | 3, 4 | mp1i 13 | . . . 4 ⊢ (𝜑 → ℂfld ∈ CMnd) |
6 | jensen.4 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
7 | jensenlem.2 | . . . . . 6 ⊢ (𝜑 → (𝐵 ∪ {𝑧}) ⊆ 𝐴) | |
8 | 7 | unssad 3790 | . . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
9 | ssfi 8180 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ Fin) | |
10 | 6, 8, 9 | syl2anc 693 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ Fin) |
11 | rge0ssre 12280 | . . . . . 6 ⊢ (0[,)+∞) ⊆ ℝ | |
12 | ax-resscn 9993 | . . . . . 6 ⊢ ℝ ⊆ ℂ | |
13 | 11, 12 | sstri 3612 | . . . . 5 ⊢ (0[,)+∞) ⊆ ℂ |
14 | 8 | sselda 3603 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐴) |
15 | jensen.5 | . . . . . . 7 ⊢ (𝜑 → 𝑇:𝐴⟶(0[,)+∞)) | |
16 | 15 | ffvelrnda 6359 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑇‘𝑥) ∈ (0[,)+∞)) |
17 | 14, 16 | syldan 487 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑇‘𝑥) ∈ (0[,)+∞)) |
18 | 13, 17 | sseldi 3601 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑇‘𝑥) ∈ ℂ) |
19 | 7 | unssbd 3791 | . . . . 5 ⊢ (𝜑 → {𝑧} ⊆ 𝐴) |
20 | vex 3203 | . . . . . 6 ⊢ 𝑧 ∈ V | |
21 | 20 | snss 4316 | . . . . 5 ⊢ (𝑧 ∈ 𝐴 ↔ {𝑧} ⊆ 𝐴) |
22 | 19, 21 | sylibr 224 | . . . 4 ⊢ (𝜑 → 𝑧 ∈ 𝐴) |
23 | jensenlem.1 | . . . 4 ⊢ (𝜑 → ¬ 𝑧 ∈ 𝐵) | |
24 | 15, 22 | ffvelrnd 6360 | . . . . 5 ⊢ (𝜑 → (𝑇‘𝑧) ∈ (0[,)+∞)) |
25 | 13, 24 | sseldi 3601 | . . . 4 ⊢ (𝜑 → (𝑇‘𝑧) ∈ ℂ) |
26 | fveq2 6191 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝑇‘𝑥) = (𝑇‘𝑧)) | |
27 | 1, 2, 5, 10, 18, 22, 23, 25, 26 | gsumunsn 18359 | . . 3 ⊢ (𝜑 → (ℂfld Σg (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ (𝑇‘𝑥))) = ((ℂfld Σg (𝑥 ∈ 𝐵 ↦ (𝑇‘𝑥))) + (𝑇‘𝑧))) |
28 | 15, 7 | feqresmpt 6250 | . . . 4 ⊢ (𝜑 → (𝑇 ↾ (𝐵 ∪ {𝑧})) = (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ (𝑇‘𝑥))) |
29 | 28 | oveq2d 6666 | . . 3 ⊢ (𝜑 → (ℂfld Σg (𝑇 ↾ (𝐵 ∪ {𝑧}))) = (ℂfld Σg (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ (𝑇‘𝑥)))) |
30 | 15, 8 | feqresmpt 6250 | . . . . 5 ⊢ (𝜑 → (𝑇 ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ (𝑇‘𝑥))) |
31 | 30 | oveq2d 6666 | . . . 4 ⊢ (𝜑 → (ℂfld Σg (𝑇 ↾ 𝐵)) = (ℂfld Σg (𝑥 ∈ 𝐵 ↦ (𝑇‘𝑥)))) |
32 | 31 | oveq1d 6665 | . . 3 ⊢ (𝜑 → ((ℂfld Σg (𝑇 ↾ 𝐵)) + (𝑇‘𝑧)) = ((ℂfld Σg (𝑥 ∈ 𝐵 ↦ (𝑇‘𝑥))) + (𝑇‘𝑧))) |
33 | 27, 29, 32 | 3eqtr4d 2666 | . 2 ⊢ (𝜑 → (ℂfld Σg (𝑇 ↾ (𝐵 ∪ {𝑧}))) = ((ℂfld Σg (𝑇 ↾ 𝐵)) + (𝑇‘𝑧))) |
34 | jensenlem.l | . 2 ⊢ 𝐿 = (ℂfld Σg (𝑇 ↾ (𝐵 ∪ {𝑧}))) | |
35 | jensenlem.s | . . 3 ⊢ 𝑆 = (ℂfld Σg (𝑇 ↾ 𝐵)) | |
36 | 35 | oveq1i 6660 | . 2 ⊢ (𝑆 + (𝑇‘𝑧)) = ((ℂfld Σg (𝑇 ↾ 𝐵)) + (𝑇‘𝑧)) |
37 | 33, 34, 36 | 3eqtr4g 2681 | 1 ⊢ (𝜑 → 𝐿 = (𝑆 + (𝑇‘𝑧))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∪ cun 3572 ⊆ wss 3574 {csn 4177 class class class wbr 4653 ↦ cmpt 4729 ↾ cres 5116 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 +∞cpnf 10071 < clt 10074 ≤ cle 10075 − cmin 10266 [,)cico 12177 [,]cicc 12178 Σg cgsu 16101 CMndccmn 18193 Ringcrg 18547 ℂfldccnfld 19746 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-ico 12181 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-0g 16102 df-gsum 16103 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-cnfld 19747 |
This theorem is referenced by: jensenlem2 24714 |
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