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Mirrors > Home > MPE Home > Th. List > jensenlem1 | Structured version Visualization version GIF version |
Description: Lemma for jensen 25574. (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 20095 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
2 | cnfldadd 20096 | . . . 4 ⊢ + = (+g‘ℂfld) | |
3 | cnring 20113 | . . . . 5 ⊢ ℂfld ∈ Ring | |
4 | ringcmn 19327 | . . . . 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 4114 | . . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
9 | 6, 8 | ssfid 8725 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ Fin) |
10 | rge0ssre 12834 | . . . . . 6 ⊢ (0[,)+∞) ⊆ ℝ | |
11 | ax-resscn 10583 | . . . . . 6 ⊢ ℝ ⊆ ℂ | |
12 | 10, 11 | sstri 3924 | . . . . 5 ⊢ (0[,)+∞) ⊆ ℂ |
13 | 8 | sselda 3915 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐴) |
14 | jensen.5 | . . . . . . 7 ⊢ (𝜑 → 𝑇:𝐴⟶(0[,)+∞)) | |
15 | 14 | ffvelrnda 6828 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑇‘𝑥) ∈ (0[,)+∞)) |
16 | 13, 15 | syldan 594 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑇‘𝑥) ∈ (0[,)+∞)) |
17 | 12, 16 | sseldi 3913 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑇‘𝑥) ∈ ℂ) |
18 | 7 | unssbd 4115 | . . . . 5 ⊢ (𝜑 → {𝑧} ⊆ 𝐴) |
19 | vex 3444 | . . . . . 6 ⊢ 𝑧 ∈ V | |
20 | 19 | snss 4679 | . . . . 5 ⊢ (𝑧 ∈ 𝐴 ↔ {𝑧} ⊆ 𝐴) |
21 | 18, 20 | sylibr 237 | . . . 4 ⊢ (𝜑 → 𝑧 ∈ 𝐴) |
22 | jensenlem.1 | . . . 4 ⊢ (𝜑 → ¬ 𝑧 ∈ 𝐵) | |
23 | 14, 21 | ffvelrnd 6829 | . . . . 5 ⊢ (𝜑 → (𝑇‘𝑧) ∈ (0[,)+∞)) |
24 | 12, 23 | sseldi 3913 | . . . 4 ⊢ (𝜑 → (𝑇‘𝑧) ∈ ℂ) |
25 | fveq2 6645 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝑇‘𝑥) = (𝑇‘𝑧)) | |
26 | 1, 2, 5, 9, 17, 21, 22, 24, 25 | gsumunsn 19073 | . . 3 ⊢ (𝜑 → (ℂfld Σg (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ (𝑇‘𝑥))) = ((ℂfld Σg (𝑥 ∈ 𝐵 ↦ (𝑇‘𝑥))) + (𝑇‘𝑧))) |
27 | 14, 7 | feqresmpt 6709 | . . . 4 ⊢ (𝜑 → (𝑇 ↾ (𝐵 ∪ {𝑧})) = (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ (𝑇‘𝑥))) |
28 | 27 | oveq2d 7151 | . . 3 ⊢ (𝜑 → (ℂfld Σg (𝑇 ↾ (𝐵 ∪ {𝑧}))) = (ℂfld Σg (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ (𝑇‘𝑥)))) |
29 | 14, 8 | feqresmpt 6709 | . . . . 5 ⊢ (𝜑 → (𝑇 ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ (𝑇‘𝑥))) |
30 | 29 | oveq2d 7151 | . . . 4 ⊢ (𝜑 → (ℂfld Σg (𝑇 ↾ 𝐵)) = (ℂfld Σg (𝑥 ∈ 𝐵 ↦ (𝑇‘𝑥)))) |
31 | 30 | oveq1d 7150 | . . 3 ⊢ (𝜑 → ((ℂfld Σg (𝑇 ↾ 𝐵)) + (𝑇‘𝑧)) = ((ℂfld Σg (𝑥 ∈ 𝐵 ↦ (𝑇‘𝑥))) + (𝑇‘𝑧))) |
32 | 26, 28, 31 | 3eqtr4d 2843 | . 2 ⊢ (𝜑 → (ℂfld Σg (𝑇 ↾ (𝐵 ∪ {𝑧}))) = ((ℂfld Σg (𝑇 ↾ 𝐵)) + (𝑇‘𝑧))) |
33 | jensenlem.l | . 2 ⊢ 𝐿 = (ℂfld Σg (𝑇 ↾ (𝐵 ∪ {𝑧}))) | |
34 | jensenlem.s | . . 3 ⊢ 𝑆 = (ℂfld Σg (𝑇 ↾ 𝐵)) | |
35 | 34 | oveq1i 7145 | . 2 ⊢ (𝑆 + (𝑇‘𝑧)) = ((ℂfld Σg (𝑇 ↾ 𝐵)) + (𝑇‘𝑧)) |
36 | 32, 33, 35 | 3eqtr4g 2858 | 1 ⊢ (𝜑 → 𝐿 = (𝑆 + (𝑇‘𝑧))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∪ cun 3879 ⊆ wss 3881 {csn 4525 class class class wbr 5030 ↦ cmpt 5110 ↾ cres 5521 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 Fincfn 8492 ℂcc 10524 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 +∞cpnf 10661 < clt 10664 ≤ cle 10665 − cmin 10859 [,)cico 12728 [,]cicc 12729 Σg cgsu 16706 CMndccmn 18898 Ringcrg 19290 ℂfldccnfld 20091 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-ico 12732 df-fz 12886 df-fzo 13029 df-seq 13365 df-hash 13687 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-0g 16707 df-gsum 16708 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-grp 18098 df-minusg 18099 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-cnfld 20092 |
This theorem is referenced by: jensenlem2 25573 |
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