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Theorem jensenlem1 27223

Description: Lemma for jensen 27225. (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 21589	. . . 4 ⊢ ℂ = (Base‘ℂ_fld)
2		cnfldadd 21591	. . . 4 ⊢ + = (+_g‘ℂ_fld)
3		cnring 21607	. . . . 5 ⊢ ℂ_fld ∈ Ring
4		ringcmn 20423	. . . . 5 ⊢ (ℂ_fld ∈ Ring → ℂ_fld ∈ CMnd)
5	3, 4	mp1i 14	. . . 4 ⊢ (𝜑 → ℂ_fld ∈ CMnd)
6		jensen.4	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ Fin)
7		jensenlem.2	. . . . . 6 ⊢ (𝜑 → (𝐵 ∪ {𝑧}) ⊆ 𝐴)
8	7	unssad 4139	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
9	6, 8	ssfid 9239	. . . 4 ⊢ (𝜑 → 𝐵 ∈ Fin)
10		rge0ssre 13509	. . . . . 6 ⊢ (0[,)+∞) ⊆ ℝ
11		ax-resscn 11181	. . . . . 6 ⊢ ℝ ⊆ ℂ
12	10, 11	sstri 3940	. . . . 5 ⊢ (0[,)+∞) ⊆ ℂ
13	8	sselda 3931	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐴)
14		jensen.5	. . . . . . 7 ⊢ (𝜑 → 𝑇:𝐴⟶(0[,)+∞))
15	14	ffvelcdmda 7077	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑇‘𝑥) ∈ (0[,)+∞))
16	13, 15	syldan 603	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑇‘𝑥) ∈ (0[,)+∞))
17	12, 16	sselid 3929	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑇‘𝑥) ∈ ℂ)
18	7	unssbd 4140	. . . . 5 ⊢ (𝜑 → {𝑧} ⊆ 𝐴)
19		vex 3454	. . . . . 6 ⊢ 𝑧 ∈ V
20	19	snss 4745	. . . . 5 ⊢ (𝑧 ∈ 𝐴 ↔ {𝑧} ⊆ 𝐴)
21	18, 20	sylibr 237	. . . 4 ⊢ (𝜑 → 𝑧 ∈ 𝐴)
22		jensenlem.1	. . . 4 ⊢ (𝜑 → ¬ 𝑧 ∈ 𝐵)
23	14, 21	ffvelcdmd 7078	. . . . 5 ⊢ (𝜑 → (𝑇‘𝑧) ∈ (0[,)+∞))
24	12, 23	sselid 3929	. . . 4 ⊢ (𝜑 → (𝑇‘𝑧) ∈ ℂ)
25		fveq2 6878	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑇‘𝑥) = (𝑇‘𝑧))
26	1, 2, 5, 9, 17, 21, 22, 24, 25	gsumunsn 20087	. . 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 6947	. . . 4 ⊢ (𝜑 → (𝑇 ↾ (𝐵 ∪ {𝑧})) = (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ (𝑇‘𝑥)))
28	27	oveq2d 7429	. . 3 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑇 ↾ (𝐵 ∪ {𝑧}))) = (ℂ_fld Σ_g (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ (𝑇‘𝑥))))
29	14, 8	feqresmpt 6947	. . . . 5 ⊢ (𝜑 → (𝑇 ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ (𝑇‘𝑥)))
30	29	oveq2d 7429	. . . 4 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑇 ↾ 𝐵)) = (ℂ_fld Σ_g (𝑥 ∈ 𝐵 ↦ (𝑇‘𝑥))))
31	30	oveq1d 7428	. . 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 2805	. 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 7423	. 2 ⊢ (𝑆 + (𝑇‘𝑧)) = ((ℂ_fld Σ_g (𝑇 ↾ 𝐵)) + (𝑇‘𝑧))
36	32, 33, 35	3eqtr4g 2820	1 ⊢ (𝜑 → 𝐿 = (𝑆 + (𝑇‘𝑧)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∪ cun 3897 ⊆ wss 3899 {csn 4584 class class class wbr 5103 ↦ cmpt 5186 ↾ cres 5657 ⟶wf 6529 ‘cfv 6533 (class class class)co 7413 Fincfn 8952 ℂcc 11122 ℝcr 11123 0cc0 11124 1c1 11125 + caddc 11127 · cmul 11129 +∞cpnf 11264 < clt 11267 ≤ cle 11268 − cmin 11465 [,)cico 13400 [,]cicc 13401 Σ_g cgsu 17525 CMndccmn 19907 Ringcrg 20372 ℂ_fldccnfld 21585

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-addf 11203

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7678 df-om 7863 df-1st 7986 df-2nd 7987 df-supp 8159 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-er 8696 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fsupp 9332 df-oi 9482 df-card 9944 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-nn 12258 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12529 df-z 12616 df-dec 12737 df-uz 12888 df-ico 13404 df-fz 13562 df-fzo 13710 df-seq 14066 df-hash 14395 df-struct 17239 df-sets 17256 df-slot 17274 df-ndx 17286 df-base 17302 df-ress 17323 df-plusg 17355 df-mulr 17356 df-starv 17357 df-tset 17361 df-ple 17362 df-ds 17364 df-unif 17365 df-0g 17526 df-gsum 17527 df-mre 17670 df-mrc 17671 df-acs 17673 df-mgm 18730 df-sgrp 18821 df-mnd 18837 df-submnd 18892 df-grp 19060 df-minusg 19061 df-mulg 19191 df-cntz 19444 df-cmn 19909 df-abl 19910 df-mgp 20274 df-ur 20321 df-ring 20374 df-cring 20375 df-cnfld 21586

This theorem is used by: jensenlem2 27224

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