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Theorem jensenlem1 26951

Description: Lemma for jensen 26953. (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 21311	. . . 4 ⊢ ℂ = (Base‘ℂ_fld)
2		cnfldadd 21313	. . . 4 ⊢ + = (+_g‘ℂ_fld)
3		cnring 21343	. . . . 5 ⊢ ℂ_fld ∈ Ring
4		ringcmn 20215	. . . . 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 4143	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
9	6, 8	ssfid 9167	. . . 4 ⊢ (𝜑 → 𝐵 ∈ Fin)
10		rge0ssre 13370	. . . . . 6 ⊢ (0[,)+∞) ⊆ ℝ
11		ax-resscn 11081	. . . . . 6 ⊢ ℝ ⊆ ℂ
12	10, 11	sstri 3941	. . . . 5 ⊢ (0[,)+∞) ⊆ ℂ
13	8	sselda 3931	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐴)
14		jensen.5	. . . . . . 7 ⊢ (𝜑 → 𝑇:𝐴⟶(0[,)+∞))
15	14	ffvelcdmda 7027	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑇‘𝑥) ∈ (0[,)+∞))
16	13, 15	syldan 591	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑇‘𝑥) ∈ (0[,)+∞))
17	12, 16	sselid 3929	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑇‘𝑥) ∈ ℂ)
18	7	unssbd 4144	. . . . 5 ⊢ (𝜑 → {𝑧} ⊆ 𝐴)
19		vex 3442	. . . . . 6 ⊢ 𝑧 ∈ V
20	19	snss 4739	. . . . 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 7028	. . . . 5 ⊢ (𝜑 → (𝑇‘𝑧) ∈ (0[,)+∞))
24	12, 23	sselid 3929	. . . 4 ⊢ (𝜑 → (𝑇‘𝑧) ∈ ℂ)
25		fveq2 6832	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑇‘𝑥) = (𝑇‘𝑧))
26	1, 2, 5, 9, 17, 21, 22, 24, 25	gsumunsn 19887	. . 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 6901	. . . 4 ⊢ (𝜑 → (𝑇 ↾ (𝐵 ∪ {𝑧})) = (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ (𝑇‘𝑥)))
28	27	oveq2d 7372	. . 3 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑇 ↾ (𝐵 ∪ {𝑧}))) = (ℂ_fld Σ_g (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ (𝑇‘𝑥))))
29	14, 8	feqresmpt 6901	. . . . 5 ⊢ (𝜑 → (𝑇 ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ (𝑇‘𝑥)))
30	29	oveq2d 7372	. . . 4 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑇 ↾ 𝐵)) = (ℂ_fld Σ_g (𝑥 ∈ 𝐵 ↦ (𝑇‘𝑥))))
31	30	oveq1d 7371	. . 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 2779	. 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 7366	. 2 ⊢ (𝑆 + (𝑇‘𝑧)) = ((ℂ_fld Σ_g (𝑇 ↾ 𝐵)) + (𝑇‘𝑧))
36	32, 33, 35	3eqtr4g 2794	1 ⊢ (𝜑 → 𝐿 = (𝑆 + (𝑇‘𝑧)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ∪ cun 3897 ⊆ wss 3899 {csn 4578 class class class wbr 5096 ↦ cmpt 5177 ↾ cres 5624 ⟶wf 6486 ‘cfv 6490 (class class class)co 7356 Fincfn 8881 ℂcc 11022 ℝcr 11023 0cc0 11024 1c1 11025 + caddc 11027 · cmul 11029 +∞cpnf 11161 < clt 11164 ≤ cle 11165 − cmin 11362 [,)cico 13261 [,]cicc 13262 Σ_g cgsu 17358 CMndccmn 19707 Ringcrg 20166 ℂ_fldccnfld 21307

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-addf 11103

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-iin 4947 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8633 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-fsupp 9263 df-oi 9413 df-card 9849 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-5 12209 df-6 12210 df-7 12211 df-8 12212 df-9 12213 df-n0 12400 df-z 12487 df-dec 12606 df-uz 12750 df-ico 13265 df-fz 13422 df-fzo 13569 df-seq 13923 df-hash 14252 df-struct 17072 df-sets 17089 df-slot 17107 df-ndx 17119 df-base 17135 df-ress 17156 df-plusg 17188 df-mulr 17189 df-starv 17190 df-tset 17194 df-ple 17195 df-ds 17197 df-unif 17198 df-0g 17359 df-gsum 17360 df-mre 17503 df-mrc 17504 df-acs 17506 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-submnd 18707 df-grp 18864 df-minusg 18865 df-mulg 18996 df-cntz 19244 df-cmn 19709 df-abl 19710 df-mgp 20074 df-ur 20115 df-ring 20168 df-cring 20169 df-cnfld 21308

This theorem is referenced by: jensenlem2 26952

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