jensenlem1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > jensenlem1		Structured version Visualization version GIF version

Theorem jensenlem1 26967

Description: Lemma for jensen 26969. (Contributed by Mario Carneiro, 4-Jun-2016.)

**Hypotheses**
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 < (ℂ_fld Σ_g 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 = (ℂ_fld Σ_g (𝑇 ↾ 𝐵))
jensenlem.l	⊢ 𝐿 = (ℂ_fld Σ_g (𝑇 ↾ (𝐵 ∪ {𝑧})))

**Assertion**
Ref	Expression
jensenlem1	⊢ (𝜑 → 𝐿 = (𝑆 + (𝑇‘𝑧)))

Distinct variable groups: 𝑎,𝑏,𝑡,𝑥,𝑦,𝐴 𝐷,𝑎,𝑏,𝑡,𝑥,𝑦 𝜑,𝑎,𝑏,𝑡,𝑥,𝑦 𝐹,𝑎,𝑏,𝑡,𝑥,𝑦 𝑇,𝑎,𝑏,𝑡,𝑥,𝑦 𝑋,𝑎,𝑏,𝑡,𝑥,𝑦 𝑧,𝑎,𝐵,𝑏,𝑡,𝑥,𝑦 𝑡,𝐿,𝑥,𝑦 𝑆,𝑎,𝑏,𝑡,𝑥,𝑦 Allowed substitution hints: 𝜑(𝑧) 𝐴(𝑧) 𝐷(𝑧) 𝑆(𝑧) 𝑇(𝑧) 𝐹(𝑧) 𝐿(𝑧,𝑎,𝑏) 𝑋(𝑧)

**Proof of Theorem jensenlem1**
Step	Hyp	Ref	Expression
1		cnfldbas 21351	. . . 4 ⊢ ℂ = (Base‘ℂ_fld)
2		cnfldadd 21353	. . . 4 ⊢ + = (+_g‘ℂ_fld)
3		cnring 21383	. . . . 5 ⊢ ℂ_fld ∈ Ring
4		ringcmn 20257	. . . . 5 ⊢ (ℂ_fld ∈ Ring → ℂ_fld ∈ CMnd)
5	3, 4	mp1i 13	. . . 4 ⊢ (𝜑 → ℂ_fld ∈ CMnd)
6		jensen.4	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ Fin)
7		jensenlem.2	. . . . . 6 ⊢ (𝜑 → (𝐵 ∪ {𝑧}) ⊆ 𝐴)
8	7	unssad 4134	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
9	6, 8	ssfid 9173	. . . 4 ⊢ (𝜑 → 𝐵 ∈ Fin)
10		rge0ssre 13403	. . . . . 6 ⊢ (0[,)+∞) ⊆ ℝ
11		ax-resscn 11089	. . . . . 6 ⊢ ℝ ⊆ ℂ
12	10, 11	sstri 3932	. . . . 5 ⊢ (0[,)+∞) ⊆ ℂ
13	8	sselda 3922	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐴)
14		jensen.5	. . . . . . 7 ⊢ (𝜑 → 𝑇:𝐴⟶(0[,)+∞))
15	14	ffvelcdmda 7031	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑇‘𝑥) ∈ (0[,)+∞))
16	13, 15	syldan 592	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑇‘𝑥) ∈ (0[,)+∞))
17	12, 16	sselid 3920	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑇‘𝑥) ∈ ℂ)
18	7	unssbd 4135	. . . . 5 ⊢ (𝜑 → {𝑧} ⊆ 𝐴)
19		vex 3434	. . . . . 6 ⊢ 𝑧 ∈ V
20	19	snss 4729	. . . . 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 7032	. . . . 5 ⊢ (𝜑 → (𝑇‘𝑧) ∈ (0[,)+∞))
24	12, 23	sselid 3920	. . . 4 ⊢ (𝜑 → (𝑇‘𝑧) ∈ ℂ)
25		fveq2 6835	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑇‘𝑥) = (𝑇‘𝑧))
26	1, 2, 5, 9, 17, 21, 22, 24, 25	gsumunsn 19929	. . 3 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ (𝑇‘𝑥))) = ((ℂ_fld Σ_g (𝑥 ∈ 𝐵 ↦ (𝑇‘𝑥))) + (𝑇‘𝑧)))
27	14, 7	feqresmpt 6904	. . . 4 ⊢ (𝜑 → (𝑇 ↾ (𝐵 ∪ {𝑧})) = (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ (𝑇‘𝑥)))
28	27	oveq2d 7377	. . 3 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑇 ↾ (𝐵 ∪ {𝑧}))) = (ℂ_fld Σ_g (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ (𝑇‘𝑥))))
29	14, 8	feqresmpt 6904	. . . . 5 ⊢ (𝜑 → (𝑇 ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ (𝑇‘𝑥)))
30	29	oveq2d 7377	. . . 4 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑇 ↾ 𝐵)) = (ℂ_fld Σ_g (𝑥 ∈ 𝐵 ↦ (𝑇‘𝑥))))
31	30	oveq1d 7376	. . 3 ⊢ (𝜑 → ((ℂ_fld Σ_g (𝑇 ↾ 𝐵)) + (𝑇‘𝑧)) = ((ℂ_fld Σ_g (𝑥 ∈ 𝐵 ↦ (𝑇‘𝑥))) + (𝑇‘𝑧)))
32	26, 28, 31	3eqtr4d 2782	. 2 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑇 ↾ (𝐵 ∪ {𝑧}))) = ((ℂ_fld Σ_g (𝑇 ↾ 𝐵)) + (𝑇‘𝑧)))
33		jensenlem.l	. 2 ⊢ 𝐿 = (ℂ_fld Σ_g (𝑇 ↾ (𝐵 ∪ {𝑧})))
34		jensenlem.s	. . 3 ⊢ 𝑆 = (ℂ_fld Σ_g (𝑇 ↾ 𝐵))
35	34	oveq1i 7371	. 2 ⊢ (𝑆 + (𝑇‘𝑧)) = ((ℂ_fld Σ_g (𝑇 ↾ 𝐵)) + (𝑇‘𝑧))
36	32, 33, 35	3eqtr4g 2797	1 ⊢ (𝜑 → 𝐿 = (𝑆 + (𝑇‘𝑧)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∪ cun 3888 ⊆ wss 3890 {csn 4568 class class class wbr 5086 ↦ cmpt 5167 ↾ cres 5627 ⟶wf 6489 ‘cfv 6493 (class class class)co 7361 Fincfn 8887 ℂcc 11030 ℝcr 11031 0cc0 11032 1c1 11033 + caddc 11035 · cmul 11037 +∞cpnf 11170 < clt 11173 ≤ cle 11174 − cmin 11371 [,)cico 13294 [,]cicc 13295 Σ_g cgsu 17397 CMndccmn 19749 Ringcrg 20208 ℂ_fldccnfld 21347

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109 ax-addf 11111

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7625 df-om 7812 df-1st 7936 df-2nd 7937 df-supp 8105 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-oi 9419 df-card 9857 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-nn 12169 df-2 12238 df-3 12239 df-4 12240 df-5 12241 df-6 12242 df-7 12243 df-8 12244 df-9 12245 df-n0 12432 df-z 12519 df-dec 12639 df-uz 12783 df-ico 13298 df-fz 13456 df-fzo 13603 df-seq 13958 df-hash 14287 df-struct 17111 df-sets 17128 df-slot 17146 df-ndx 17158 df-base 17174 df-ress 17195 df-plusg 17227 df-mulr 17228 df-starv 17229 df-tset 17233 df-ple 17234 df-ds 17236 df-unif 17237 df-0g 17398 df-gsum 17399 df-mre 17542 df-mrc 17543 df-acs 17545 df-mgm 18602 df-sgrp 18681 df-mnd 18697 df-submnd 18746 df-grp 18906 df-minusg 18907 df-mulg 19038 df-cntz 19286 df-cmn 19751 df-abl 19752 df-mgp 20116 df-ur 20157 df-ring 20210 df-cring 20211 df-cnfld 21348

This theorem is referenced by: jensenlem2 26968

W3C validator