Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > jensenlem1 | Structured version Visualization version GIF version |
Description: Lemma for jensen 25568. (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 20551 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
2 | cnfldadd 20552 | . . . 4 ⊢ + = (+g‘ℂfld) | |
3 | cnring 20569 | . . . . 5 ⊢ ℂfld ∈ Ring | |
4 | ringcmn 19333 | . . . . 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 4165 | . . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
9 | 6, 8 | ssfid 8743 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ Fin) |
10 | rge0ssre 12847 | . . . . . 6 ⊢ (0[,)+∞) ⊆ ℝ | |
11 | ax-resscn 10596 | . . . . . 6 ⊢ ℝ ⊆ ℂ | |
12 | 10, 11 | sstri 3978 | . . . . 5 ⊢ (0[,)+∞) ⊆ ℂ |
13 | 8 | sselda 3969 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐴) |
14 | jensen.5 | . . . . . . 7 ⊢ (𝜑 → 𝑇:𝐴⟶(0[,)+∞)) | |
15 | 14 | ffvelrnda 6853 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑇‘𝑥) ∈ (0[,)+∞)) |
16 | 13, 15 | syldan 593 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑇‘𝑥) ∈ (0[,)+∞)) |
17 | 12, 16 | sseldi 3967 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑇‘𝑥) ∈ ℂ) |
18 | 7 | unssbd 4166 | . . . . 5 ⊢ (𝜑 → {𝑧} ⊆ 𝐴) |
19 | vex 3499 | . . . . . 6 ⊢ 𝑧 ∈ V | |
20 | 19 | snss 4720 | . . . . 5 ⊢ (𝑧 ∈ 𝐴 ↔ {𝑧} ⊆ 𝐴) |
21 | 18, 20 | sylibr 236 | . . . 4 ⊢ (𝜑 → 𝑧 ∈ 𝐴) |
22 | jensenlem.1 | . . . 4 ⊢ (𝜑 → ¬ 𝑧 ∈ 𝐵) | |
23 | 14, 21 | ffvelrnd 6854 | . . . . 5 ⊢ (𝜑 → (𝑇‘𝑧) ∈ (0[,)+∞)) |
24 | 12, 23 | sseldi 3967 | . . . 4 ⊢ (𝜑 → (𝑇‘𝑧) ∈ ℂ) |
25 | fveq2 6672 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝑇‘𝑥) = (𝑇‘𝑧)) | |
26 | 1, 2, 5, 9, 17, 21, 22, 24, 25 | gsumunsn 19082 | . . 3 ⊢ (𝜑 → (ℂfld Σg (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ (𝑇‘𝑥))) = ((ℂfld Σg (𝑥 ∈ 𝐵 ↦ (𝑇‘𝑥))) + (𝑇‘𝑧))) |
27 | 14, 7 | feqresmpt 6736 | . . . 4 ⊢ (𝜑 → (𝑇 ↾ (𝐵 ∪ {𝑧})) = (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ (𝑇‘𝑥))) |
28 | 27 | oveq2d 7174 | . . 3 ⊢ (𝜑 → (ℂfld Σg (𝑇 ↾ (𝐵 ∪ {𝑧}))) = (ℂfld Σg (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ (𝑇‘𝑥)))) |
29 | 14, 8 | feqresmpt 6736 | . . . . 5 ⊢ (𝜑 → (𝑇 ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ (𝑇‘𝑥))) |
30 | 29 | oveq2d 7174 | . . . 4 ⊢ (𝜑 → (ℂfld Σg (𝑇 ↾ 𝐵)) = (ℂfld Σg (𝑥 ∈ 𝐵 ↦ (𝑇‘𝑥)))) |
31 | 30 | oveq1d 7173 | . . 3 ⊢ (𝜑 → ((ℂfld Σg (𝑇 ↾ 𝐵)) + (𝑇‘𝑧)) = ((ℂfld Σg (𝑥 ∈ 𝐵 ↦ (𝑇‘𝑥))) + (𝑇‘𝑧))) |
32 | 26, 28, 31 | 3eqtr4d 2868 | . 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 7168 | . 2 ⊢ (𝑆 + (𝑇‘𝑧)) = ((ℂfld Σg (𝑇 ↾ 𝐵)) + (𝑇‘𝑧)) |
36 | 32, 33, 35 | 3eqtr4g 2883 | 1 ⊢ (𝜑 → 𝐿 = (𝑆 + (𝑇‘𝑧))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∪ cun 3936 ⊆ wss 3938 {csn 4569 class class class wbr 5068 ↦ cmpt 5148 ↾ cres 5559 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 Fincfn 8511 ℂcc 10537 ℝcr 10538 0cc0 10539 1c1 10540 + caddc 10542 · cmul 10544 +∞cpnf 10674 < clt 10677 ≤ cle 10678 − cmin 10872 [,)cico 12743 [,]cicc 12744 Σg cgsu 16716 CMndccmn 18908 Ringcrg 19299 ℂfldccnfld 20547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-addf 10618 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-ico 12747 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-0g 16717 df-gsum 16718 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-grp 18108 df-minusg 18109 df-mulg 18227 df-cntz 18449 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-cring 19302 df-cnfld 20548 |
This theorem is referenced by: jensenlem2 25567 |
Copyright terms: Public domain | W3C validator |