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Mirrors > Home > MPE Home > Th. List > jensenlem1 | Structured version Visualization version GIF version |
Description: Lemma for jensen 26941. (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 21290 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
2 | cnfldadd 21292 | . . . 4 ⊢ + = (+g‘ℂfld) | |
3 | cnring 21325 | . . . . 5 ⊢ ℂfld ∈ Ring | |
4 | ringcmn 20225 | . . . . 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 4189 | . . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
9 | 6, 8 | ssfid 9298 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ Fin) |
10 | rge0ssre 13473 | . . . . . 6 ⊢ (0[,)+∞) ⊆ ℝ | |
11 | ax-resscn 11203 | . . . . . 6 ⊢ ℝ ⊆ ℂ | |
12 | 10, 11 | sstri 3991 | . . . . 5 ⊢ (0[,)+∞) ⊆ ℂ |
13 | 8 | sselda 3982 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐴) |
14 | jensen.5 | . . . . . . 7 ⊢ (𝜑 → 𝑇:𝐴⟶(0[,)+∞)) | |
15 | 14 | ffvelcdmda 7099 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑇‘𝑥) ∈ (0[,)+∞)) |
16 | 13, 15 | syldan 589 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑇‘𝑥) ∈ (0[,)+∞)) |
17 | 12, 16 | sselid 3980 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑇‘𝑥) ∈ ℂ) |
18 | 7 | unssbd 4190 | . . . . 5 ⊢ (𝜑 → {𝑧} ⊆ 𝐴) |
19 | vex 3477 | . . . . . 6 ⊢ 𝑧 ∈ V | |
20 | 19 | snss 4794 | . . . . 5 ⊢ (𝑧 ∈ 𝐴 ↔ {𝑧} ⊆ 𝐴) |
21 | 18, 20 | sylibr 233 | . . . 4 ⊢ (𝜑 → 𝑧 ∈ 𝐴) |
22 | jensenlem.1 | . . . 4 ⊢ (𝜑 → ¬ 𝑧 ∈ 𝐵) | |
23 | 14, 21 | ffvelcdmd 7100 | . . . . 5 ⊢ (𝜑 → (𝑇‘𝑧) ∈ (0[,)+∞)) |
24 | 12, 23 | sselid 3980 | . . . 4 ⊢ (𝜑 → (𝑇‘𝑧) ∈ ℂ) |
25 | fveq2 6902 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝑇‘𝑥) = (𝑇‘𝑧)) | |
26 | 1, 2, 5, 9, 17, 21, 22, 24, 25 | gsumunsn 19922 | . . 3 ⊢ (𝜑 → (ℂfld Σg (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ (𝑇‘𝑥))) = ((ℂfld Σg (𝑥 ∈ 𝐵 ↦ (𝑇‘𝑥))) + (𝑇‘𝑧))) |
27 | 14, 7 | feqresmpt 6973 | . . . 4 ⊢ (𝜑 → (𝑇 ↾ (𝐵 ∪ {𝑧})) = (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ (𝑇‘𝑥))) |
28 | 27 | oveq2d 7442 | . . 3 ⊢ (𝜑 → (ℂfld Σg (𝑇 ↾ (𝐵 ∪ {𝑧}))) = (ℂfld Σg (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ (𝑇‘𝑥)))) |
29 | 14, 8 | feqresmpt 6973 | . . . . 5 ⊢ (𝜑 → (𝑇 ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ (𝑇‘𝑥))) |
30 | 29 | oveq2d 7442 | . . . 4 ⊢ (𝜑 → (ℂfld Σg (𝑇 ↾ 𝐵)) = (ℂfld Σg (𝑥 ∈ 𝐵 ↦ (𝑇‘𝑥)))) |
31 | 30 | oveq1d 7441 | . . 3 ⊢ (𝜑 → ((ℂfld Σg (𝑇 ↾ 𝐵)) + (𝑇‘𝑧)) = ((ℂfld Σg (𝑥 ∈ 𝐵 ↦ (𝑇‘𝑥))) + (𝑇‘𝑧))) |
32 | 26, 28, 31 | 3eqtr4d 2778 | . 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 7436 | . 2 ⊢ (𝑆 + (𝑇‘𝑧)) = ((ℂfld Σg (𝑇 ↾ 𝐵)) + (𝑇‘𝑧)) |
36 | 32, 33, 35 | 3eqtr4g 2793 | 1 ⊢ (𝜑 → 𝐿 = (𝑆 + (𝑇‘𝑧))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∪ cun 3947 ⊆ wss 3949 {csn 4632 class class class wbr 5152 ↦ cmpt 5235 ↾ cres 5684 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 Fincfn 8970 ℂcc 11144 ℝcr 11145 0cc0 11146 1c1 11147 + caddc 11149 · cmul 11151 +∞cpnf 11283 < clt 11286 ≤ cle 11287 − cmin 11482 [,)cico 13366 [,]cicc 13367 Σg cgsu 17429 CMndccmn 19742 Ringcrg 20180 ℂfldccnfld 21286 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-addf 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-ico 13370 df-fz 13525 df-fzo 13668 df-seq 14007 df-hash 14330 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-starv 17255 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-0g 17430 df-gsum 17431 df-mre 17573 df-mrc 17574 df-acs 17576 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-submnd 18748 df-grp 18900 df-minusg 18901 df-mulg 19031 df-cntz 19275 df-cmn 19744 df-abl 19745 df-mgp 20082 df-ur 20129 df-ring 20182 df-cring 20183 df-cnfld 21287 |
This theorem is referenced by: jensenlem2 26940 |
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