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Theorem jensenlem1 26897

Description: Lemma for jensen 26899. (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 21268	. . . 4 ⊢ ℂ = (Base‘ℂ_fld)
2		cnfldadd 21270	. . . 4 ⊢ + = (+_g‘ℂ_fld)
3		cnring 21302	. . . . 5 ⊢ ℂ_fld ∈ Ring
4		ringcmn 20191	. . . . 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 4156	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
9	6, 8	ssfid 9212	. . . 4 ⊢ (𝜑 → 𝐵 ∈ Fin)
10		rge0ssre 13417	. . . . . 6 ⊢ (0[,)+∞) ⊆ ℝ
11		ax-resscn 11125	. . . . . 6 ⊢ ℝ ⊆ ℂ
12	10, 11	sstri 3956	. . . . 5 ⊢ (0[,)+∞) ⊆ ℂ
13	8	sselda 3946	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐴)
14		jensen.5	. . . . . . 7 ⊢ (𝜑 → 𝑇:𝐴⟶(0[,)+∞))
15	14	ffvelcdmda 7056	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑇‘𝑥) ∈ (0[,)+∞))
16	13, 15	syldan 591	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑇‘𝑥) ∈ (0[,)+∞))
17	12, 16	sselid 3944	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑇‘𝑥) ∈ ℂ)
18	7	unssbd 4157	. . . . 5 ⊢ (𝜑 → {𝑧} ⊆ 𝐴)
19		vex 3451	. . . . . 6 ⊢ 𝑧 ∈ V
20	19	snss 4749	. . . . 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 7057	. . . . 5 ⊢ (𝜑 → (𝑇‘𝑧) ∈ (0[,)+∞))
24	12, 23	sselid 3944	. . . 4 ⊢ (𝜑 → (𝑇‘𝑧) ∈ ℂ)
25		fveq2 6858	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑇‘𝑥) = (𝑇‘𝑧))
26	1, 2, 5, 9, 17, 21, 22, 24, 25	gsumunsn 19890	. . 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 6930	. . . 4 ⊢ (𝜑 → (𝑇 ↾ (𝐵 ∪ {𝑧})) = (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ (𝑇‘𝑥)))
28	27	oveq2d 7403	. . 3 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑇 ↾ (𝐵 ∪ {𝑧}))) = (ℂ_fld Σ_g (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ (𝑇‘𝑥))))
29	14, 8	feqresmpt 6930	. . . . 5 ⊢ (𝜑 → (𝑇 ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ (𝑇‘𝑥)))
30	29	oveq2d 7403	. . . 4 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑇 ↾ 𝐵)) = (ℂ_fld Σ_g (𝑥 ∈ 𝐵 ↦ (𝑇‘𝑥))))
31	30	oveq1d 7402	. . 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 2774	. 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 7397	. 2 ⊢ (𝑆 + (𝑇‘𝑧)) = ((ℂ_fld Σ_g (𝑇 ↾ 𝐵)) + (𝑇‘𝑧))
36	32, 33, 35	3eqtr4g 2789	1 ⊢ (𝜑 → 𝐿 = (𝑆 + (𝑇‘𝑧)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∪ cun 3912 ⊆ wss 3914 {csn 4589 class class class wbr 5107 ↦ cmpt 5188 ↾ cres 5640 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 Fincfn 8918 ℂcc 11066 ℝcr 11067 0cc0 11068 1c1 11069 + caddc 11071 · cmul 11073 +∞cpnf 11205 < clt 11208 ≤ cle 11209 − cmin 11405 [,)cico 13308 [,]cicc 13309 Σ_g cgsu 17403 CMndccmn 19710 Ringcrg 20142 ℂ_fldccnfld 21264

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-addf 11147

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-oi 9463 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-ico 13312 df-fz 13469 df-fzo 13616 df-seq 13967 df-hash 14296 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-0g 17404 df-gsum 17405 df-mre 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-grp 18868 df-minusg 18869 df-mulg 19000 df-cntz 19249 df-cmn 19712 df-abl 19713 df-mgp 20050 df-ur 20091 df-ring 20144 df-cring 20145 df-cnfld 21265

This theorem is referenced by: jensenlem2 26898

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