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Theorem jensenlem1 26968

Description: Lemma for jensen 26970. (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 21369	. . . . 5 ⊢ ℂ_fld ∈ Ring
4		ringcmn 20254	. . . . 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 4122	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
9	6, 8	ssfid 9169	. . . 4 ⊢ (𝜑 → 𝐵 ∈ Fin)
10		rge0ssre 13400	. . . . . 6 ⊢ (0[,)+∞) ⊆ ℝ
11		ax-resscn 11086	. . . . . 6 ⊢ ℝ ⊆ ℂ
12	10, 11	sstri 3924	. . . . 5 ⊢ (0[,)+∞) ⊆ ℂ
13	8	sselda 3915	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐴)
14		jensen.5	. . . . . . 7 ⊢ (𝜑 → 𝑇:𝐴⟶(0[,)+∞))
15	14	ffvelcdmda 7025	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑇‘𝑥) ∈ (0[,)+∞))
16	13, 15	syldan 597	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑇‘𝑥) ∈ (0[,)+∞))
17	12, 16	sselid 3913	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑇‘𝑥) ∈ ℂ)
18	7	unssbd 4123	. . . . 5 ⊢ (𝜑 → {𝑧} ⊆ 𝐴)
19		vex 3435	. . . . . 6 ⊢ 𝑧 ∈ V
20	19	snss 4716	. . . . 5 ⊢ (𝑧 ∈ 𝐴 ↔ {𝑧} ⊆ 𝐴)
21	18, 20	sylibr 235	. . . 4 ⊢ (𝜑 → 𝑧 ∈ 𝐴)
22		jensenlem.1	. . . 4 ⊢ (𝜑 → ¬ 𝑧 ∈ 𝐵)
23	14, 21	ffvelcdmd 7026	. . . . 5 ⊢ (𝜑 → (𝑇‘𝑧) ∈ (0[,)+∞))
24	12, 23	sselid 3913	. . . 4 ⊢ (𝜑 → (𝑇‘𝑧) ∈ ℂ)
25		fveq2 6827	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑇‘𝑥) = (𝑇‘𝑧))
26	1, 2, 5, 9, 17, 21, 22, 24, 25	gsumunsn 19926	. . 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 6896	. . . 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 6896	. . . . 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 2784	. 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 2799	1 ⊢ (𝜑 → 𝐿 = (𝑆 + (𝑇‘𝑧)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ∪ cun 3881 ⊆ wss 3883 {csn 4555 class class class wbr 5072 ↦ cmpt 5153 ↾ cres 5620 ⟶wf 6481 ‘cfv 6485 (class class class)co 7356 Fincfn 8883 ℂcc 11027 ℝcr 11028 0cc0 11029 1c1 11030 + caddc 11032 · cmul 11034 +∞cpnf 11167 < clt 11170 ≤ cle 11171 − cmin 11368 [,)cico 13291 [,]cicc 13292 Σ_g cgsu 17394 CMndccmn 19746 Ringcrg 20205 ℂ_fldccnfld 21347

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-addf 11108

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-iin 4924 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-se 5572 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-isom 6494 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 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-oi 9415 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-ico 13295 df-fz 13453 df-fzo 13600 df-seq 13955 df-hash 14284 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-0g 17395 df-gsum 17396 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-grp 18903 df-minusg 18904 df-mulg 19035 df-cntz 19283 df-cmn 19748 df-abl 19749 df-mgp 20113 df-ur 20154 df-ring 20207 df-cring 20208 df-cnfld 21348

This theorem is referenced by: jensenlem2 26969

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