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| Mirrors > Home > MPE Home > Th. List > jensenlem1 | Structured version Visualization version GIF version | ||
| Description: Lemma for jensen 27032. (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 21368 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
| 2 | cnfldadd 21370 | . . . 4 ⊢ + = (+g‘ℂfld) | |
| 3 | cnring 21403 | . . . . 5 ⊢ ℂfld ∈ Ring | |
| 4 | ringcmn 20279 | . . . . 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 4193 | . . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
| 9 | 6, 8 | ssfid 9301 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ Fin) |
| 10 | rge0ssre 13496 | . . . . . 6 ⊢ (0[,)+∞) ⊆ ℝ | |
| 11 | ax-resscn 11212 | . . . . . 6 ⊢ ℝ ⊆ ℂ | |
| 12 | 10, 11 | sstri 3993 | . . . . 5 ⊢ (0[,)+∞) ⊆ ℂ |
| 13 | 8 | sselda 3983 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐴) |
| 14 | jensen.5 | . . . . . . 7 ⊢ (𝜑 → 𝑇:𝐴⟶(0[,)+∞)) | |
| 15 | 14 | ffvelcdmda 7104 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑇‘𝑥) ∈ (0[,)+∞)) |
| 16 | 13, 15 | syldan 591 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑇‘𝑥) ∈ (0[,)+∞)) |
| 17 | 12, 16 | sselid 3981 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑇‘𝑥) ∈ ℂ) |
| 18 | 7 | unssbd 4194 | . . . . 5 ⊢ (𝜑 → {𝑧} ⊆ 𝐴) |
| 19 | vex 3484 | . . . . . 6 ⊢ 𝑧 ∈ V | |
| 20 | 19 | snss 4785 | . . . . 5 ⊢ (𝑧 ∈ 𝐴 ↔ {𝑧} ⊆ 𝐴) |
| 21 | 18, 20 | sylibr 234 | . . . 4 ⊢ (𝜑 → 𝑧 ∈ 𝐴) |
| 22 | jensenlem.1 | . . . 4 ⊢ (𝜑 → ¬ 𝑧 ∈ 𝐵) | |
| 23 | 14, 21 | ffvelcdmd 7105 | . . . . 5 ⊢ (𝜑 → (𝑇‘𝑧) ∈ (0[,)+∞)) |
| 24 | 12, 23 | sselid 3981 | . . . 4 ⊢ (𝜑 → (𝑇‘𝑧) ∈ ℂ) |
| 25 | fveq2 6906 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝑇‘𝑥) = (𝑇‘𝑧)) | |
| 26 | 1, 2, 5, 9, 17, 21, 22, 24, 25 | gsumunsn 19978 | . . 3 ⊢ (𝜑 → (ℂfld Σg (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ (𝑇‘𝑥))) = ((ℂfld Σg (𝑥 ∈ 𝐵 ↦ (𝑇‘𝑥))) + (𝑇‘𝑧))) |
| 27 | 14, 7 | feqresmpt 6978 | . . . 4 ⊢ (𝜑 → (𝑇 ↾ (𝐵 ∪ {𝑧})) = (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ (𝑇‘𝑥))) |
| 28 | 27 | oveq2d 7447 | . . 3 ⊢ (𝜑 → (ℂfld Σg (𝑇 ↾ (𝐵 ∪ {𝑧}))) = (ℂfld Σg (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ (𝑇‘𝑥)))) |
| 29 | 14, 8 | feqresmpt 6978 | . . . . 5 ⊢ (𝜑 → (𝑇 ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ (𝑇‘𝑥))) |
| 30 | 29 | oveq2d 7447 | . . . 4 ⊢ (𝜑 → (ℂfld Σg (𝑇 ↾ 𝐵)) = (ℂfld Σg (𝑥 ∈ 𝐵 ↦ (𝑇‘𝑥)))) |
| 31 | 30 | oveq1d 7446 | . . 3 ⊢ (𝜑 → ((ℂfld Σg (𝑇 ↾ 𝐵)) + (𝑇‘𝑧)) = ((ℂfld Σg (𝑥 ∈ 𝐵 ↦ (𝑇‘𝑥))) + (𝑇‘𝑧))) |
| 32 | 26, 28, 31 | 3eqtr4d 2787 | . 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 7441 | . 2 ⊢ (𝑆 + (𝑇‘𝑧)) = ((ℂfld Σg (𝑇 ↾ 𝐵)) + (𝑇‘𝑧)) |
| 36 | 32, 33, 35 | 3eqtr4g 2802 | 1 ⊢ (𝜑 → 𝐿 = (𝑆 + (𝑇‘𝑧))) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∪ cun 3949 ⊆ wss 3951 {csn 4626 class class class wbr 5143 ↦ cmpt 5225 ↾ cres 5687 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 +∞cpnf 11292 < clt 11295 ≤ cle 11296 − cmin 11492 [,)cico 13389 [,]cicc 13390 Σg cgsu 17485 CMndccmn 19798 Ringcrg 20230 ℂfldccnfld 21364 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-addf 11234 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-ico 13393 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-0g 17486 df-gsum 17487 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-grp 18954 df-minusg 18955 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-ur 20179 df-ring 20232 df-cring 20233 df-cnfld 21365 |
| This theorem is referenced by: jensenlem2 27031 |
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