esumpad2 - Mathbox for Thierry Arnoux

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Theorem esumpad2 30398

Description: Remove zeroes from an extended sum. (Contributed by Thierry Arnoux, 5-Jun-2020.)

**Hypotheses**
Ref	Expression
esumpad.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
esumpad.2	⊢ (𝜑 → 𝐵 ∈ 𝑊)
esumpad.3	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (0[,]+∞))
esumpad.4	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐶 = 0)

**Assertion**
Ref	Expression
esumpad2	⊢ (𝜑 → Σ^𝑘 ∈ (𝐴 ∖ 𝐵)𝐶 = Σ^𝑘 ∈ 𝐴𝐶)

Distinct variable groups: 𝐴,𝑘 𝐵,𝑘 𝑘,𝑉 𝜑,𝑘 Allowed substitution hints: 𝐶(𝑘) 𝑊(𝑘)

**Proof of Theorem esumpad2**
Step	Hyp	Ref	Expression
1		nfv 1980	. . . 4 ⊢ Ⅎ𝑘𝜑
2		esumpad.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		esumpad.3	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (0[,]+∞))
4		difssd 3869	. . . 4 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ 𝐴)
5	1, 2, 3, 4	esummono 30396	. . 3 ⊢ (𝜑 → Σ^𝑘 ∈ (𝐴 ∖ 𝐵)𝐶 ≤ Σ^𝑘 ∈ 𝐴𝐶)
6		esumpad.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
7		unexg 7112	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)
8	2, 6, 7	syl2anc 696	. . . . 5 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ V)
9		elun 3884	. . . . . 6 ⊢ (𝑘 ∈ (𝐴 ∪ 𝐵) ↔ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵))
10		esumpad.4	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐶 = 0)
11		0e0iccpnf 12447	. . . . . . . 8 ⊢ 0 ∈ (0[,]+∞)
12	10, 11	syl6eqel 2835	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ (0[,]+∞))
13	3, 12	jaodan 861	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵)) → 𝐶 ∈ (0[,]+∞))
14	9, 13	sylan2b 493	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ 𝐵)) → 𝐶 ∈ (0[,]+∞))
15		ssun1 3907	. . . . . 6 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
16	15	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ (𝐴 ∪ 𝐵))
17	1, 8, 14, 16	esummono 30396	. . . 4 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐶 ≤ Σ^𝑘 ∈ (𝐴 ∪ 𝐵)𝐶)
18		undif1 4175	. . . . . 6 ⊢ ((𝐴 ∖ 𝐵) ∪ 𝐵) = (𝐴 ∪ 𝐵)
19		esumeq1 30376	. . . . . 6 ⊢ (((𝐴 ∖ 𝐵) ∪ 𝐵) = (𝐴 ∪ 𝐵) → Σ^𝑘 ∈ ((𝐴 ∖ 𝐵) ∪ 𝐵)𝐶 = Σ^𝑘 ∈ (𝐴 ∪ 𝐵)𝐶)
20	18, 19	ax-mp 5	. . . . 5 ⊢ Σ^𝑘 ∈ ((𝐴 ∖ 𝐵) ∪ 𝐵)𝐶 = Σ^𝑘 ∈ (𝐴 ∪ 𝐵)𝐶
21		difexg 4948	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝐵) ∈ V)
22	2, 21	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ V)
23	4	sselda 3732	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐵)) → 𝑘 ∈ 𝐴)
24	23, 3	syldan 488	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐵)) → 𝐶 ∈ (0[,]+∞))
25	22, 6, 24, 10	esumpad 30397	. . . . 5 ⊢ (𝜑 → Σ^𝑘 ∈ ((𝐴 ∖ 𝐵) ∪ 𝐵)𝐶 = Σ^𝑘 ∈ (𝐴 ∖ 𝐵)𝐶)
26	20, 25	syl5eqr 2796	. . . 4 ⊢ (𝜑 → Σ^𝑘 ∈ (𝐴 ∪ 𝐵)𝐶 = Σ^𝑘 ∈ (𝐴 ∖ 𝐵)𝐶)
27	17, 26	breqtrd 4818	. . 3 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐶 ≤ Σ^𝑘 ∈ (𝐴 ∖ 𝐵)𝐶)
28	5, 27	jca 555	. 2 ⊢ (𝜑 → (Σ^𝑘 ∈ (𝐴 ∖ 𝐵)𝐶 ≤ Σ^𝑘 ∈ 𝐴𝐶 ∧ Σ^𝑘 ∈ 𝐴𝐶 ≤ Σ^𝑘 ∈ (𝐴 ∖ 𝐵)𝐶))
29		iccssxr 12420	. . . 4 ⊢ (0[,]+∞) ⊆ ℝ^*
30	24	ralrimiva 3092	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ (𝐴 ∖ 𝐵)𝐶 ∈ (0[,]+∞))
31		nfcv 2890	. . . . . 6 ⊢ Ⅎ𝑘(𝐴 ∖ 𝐵)
32	31	esumcl 30372	. . . . 5 ⊢ (((𝐴 ∖ 𝐵) ∈ V ∧ ∀𝑘 ∈ (𝐴 ∖ 𝐵)𝐶 ∈ (0[,]+∞)) → Σ^*𝑘 ∈ (𝐴 ∖ 𝐵)𝐶 ∈ (0[,]+∞))
33	22, 30, 32	syl2anc 696	. . . 4 ⊢ (𝜑 → Σ^*𝑘 ∈ (𝐴 ∖ 𝐵)𝐶 ∈ (0[,]+∞))
34	29, 33	sseldi 3730	. . 3 ⊢ (𝜑 → Σ^𝑘 ∈ (𝐴 ∖ 𝐵)𝐶 ∈ ℝ^)
35	3	ralrimiva 3092	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 ∈ (0[,]+∞))
36		nfcv 2890	. . . . . 6 ⊢ Ⅎ𝑘𝐴
37	36	esumcl 30372	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐶 ∈ (0[,]+∞)) → Σ^*𝑘 ∈ 𝐴𝐶 ∈ (0[,]+∞))
38	2, 35, 37	syl2anc 696	. . . 4 ⊢ (𝜑 → Σ^*𝑘 ∈ 𝐴𝐶 ∈ (0[,]+∞))
39	29, 38	sseldi 3730	. . 3 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐶 ∈ ℝ^)
40		xrletri3 12149	. . 3 ⊢ ((Σ^𝑘 ∈ (𝐴 ∖ 𝐵)𝐶 ∈ ℝ^ ∧ Σ^𝑘 ∈ 𝐴𝐶 ∈ ℝ^) → (Σ^𝑘 ∈ (𝐴 ∖ 𝐵)𝐶 = Σ^𝑘 ∈ 𝐴𝐶 ↔ (Σ^𝑘 ∈ (𝐴 ∖ 𝐵)𝐶 ≤ Σ^𝑘 ∈ 𝐴𝐶 ∧ Σ^𝑘 ∈ 𝐴𝐶 ≤ Σ^𝑘 ∈ (𝐴 ∖ 𝐵)𝐶)))
41	34, 39, 40	syl2anc 696	. 2 ⊢ (𝜑 → (Σ^𝑘 ∈ (𝐴 ∖ 𝐵)𝐶 = Σ^𝑘 ∈ 𝐴𝐶 ↔ (Σ^𝑘 ∈ (𝐴 ∖ 𝐵)𝐶 ≤ Σ^𝑘 ∈ 𝐴𝐶 ∧ Σ^𝑘 ∈ 𝐴𝐶 ≤ Σ^𝑘 ∈ (𝐴 ∖ 𝐵)𝐶)))
42	28, 41	mpbird 247	1 ⊢ (𝜑 → Σ^𝑘 ∈ (𝐴 ∖ 𝐵)𝐶 = Σ^𝑘 ∈ 𝐴𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∨ wo 382 ∧ wa 383 = wceq 1620 ∈ wcel 2127 ∀wral 3038 Vcvv 3328 ∖ cdif 3700 ∪ cun 3701 ⊆ wss 3703 class class class wbr 4792 (class class class)co 6801 0cc0 10099 +∞cpnf 10234 ℝ^*cxr 10236 ≤ cle 10238 [,]cicc 12342 Σ^*cesum 30369

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1859 ax-4 1874 ax-5 1976 ax-6 2042 ax-7 2078 ax-8 2129 ax-9 2136 ax-10 2156 ax-11 2171 ax-12 2184 ax-13 2379 ax-ext 2728 ax-rep 4911 ax-sep 4921 ax-nul 4929 ax-pow 4980 ax-pr 5043 ax-un 7102 ax-inf2 8699 ax-cnex 10155 ax-resscn 10156 ax-1cn 10157 ax-icn 10158 ax-addcl 10159 ax-addrcl 10160 ax-mulcl 10161 ax-mulrcl 10162 ax-mulcom 10163 ax-addass 10164 ax-mulass 10165 ax-distr 10166 ax-i2m1 10167 ax-1ne0 10168 ax-1rid 10169 ax-rnegex 10170 ax-rrecex 10171 ax-cnre 10172 ax-pre-lttri 10173 ax-pre-lttrn 10174 ax-pre-ltadd 10175 ax-pre-mulgt0 10176 ax-pre-sup 10177 ax-addf 10178 ax-mulf 10179

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1623 df-fal 1626 df-ex 1842 df-nf 1847 df-sb 2035 df-eu 2599 df-mo 2600 df-clab 2735 df-cleq 2741 df-clel 2744 df-nfc 2879 df-ne 2921 df-nel 3024 df-ral 3043 df-rex 3044 df-reu 3045 df-rmo 3046 df-rab 3047 df-v 3330 df-sbc 3565 df-csb 3663 df-dif 3706 df-un 3708 df-in 3710 df-ss 3717 df-pss 3719 df-nul 4047 df-if 4219 df-pw 4292 df-sn 4310 df-pr 4312 df-tp 4314 df-op 4316 df-uni 4577 df-int 4616 df-iun 4662 df-iin 4663 df-br 4793 df-opab 4853 df-mpt 4870 df-tr 4893 df-id 5162 df-eprel 5167 df-po 5175 df-so 5176 df-fr 5213 df-se 5214 df-we 5215 df-xp 5260 df-rel 5261 df-cnv 5262 df-co 5263 df-dm 5264 df-rn 5265 df-res 5266 df-ima 5267 df-pred 5829 df-ord 5875 df-on 5876 df-lim 5877 df-suc 5878 df-iota 6000 df-fun 6039 df-fn 6040 df-f 6041 df-f1 6042 df-fo 6043 df-f1o 6044 df-fv 6045 df-isom 6046 df-riota 6762 df-ov 6804 df-oprab 6805 df-mpt2 6806 df-of 7050 df-om 7219 df-1st 7321 df-2nd 7322 df-supp 7452 df-wrecs 7564 df-recs 7625 df-rdg 7663 df-1o 7717 df-2o 7718 df-oadd 7721 df-er 7899 df-map 8013 df-pm 8014 df-ixp 8063 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-fsupp 8429 df-fi 8470 df-sup 8501 df-inf 8502 df-oi 8568 df-card 8926 df-cda 9153 df-pnf 10239 df-mnf 10240 df-xr 10241 df-ltxr 10242 df-le 10243 df-sub 10431 df-neg 10432 df-div 10848 df-nn 11184 df-2 11242 df-3 11243 df-4 11244 df-5 11245 df-6 11246 df-7 11247 df-8 11248 df-9 11249 df-n0 11456 df-z 11541 df-dec 11657 df-uz 11851 df-q 11953 df-rp 11997 df-xneg 12110 df-xadd 12111 df-xmul 12112 df-ioo 12343 df-ioc 12344 df-ico 12345 df-icc 12346 df-fz 12491 df-fzo 12631 df-fl 12758 df-mod 12834 df-seq 12967 df-exp 13026 df-fac 13226 df-bc 13255 df-hash 13283 df-shft 13977 df-cj 14009 df-re 14010 df-im 14011 df-sqrt 14145 df-abs 14146 df-limsup 14372 df-clim 14389 df-rlim 14390 df-sum 14587 df-ef 14968 df-sin 14970 df-cos 14971 df-pi 14973 df-struct 16032 df-ndx 16033 df-slot 16034 df-base 16036 df-sets 16037 df-ress 16038 df-plusg 16127 df-mulr 16128 df-starv 16129 df-sca 16130 df-vsca 16131 df-ip 16132 df-tset 16133 df-ple 16134 df-ds 16137 df-unif 16138 df-hom 16139 df-cco 16140 df-rest 16256 df-topn 16257 df-0g 16275 df-gsum 16276 df-topgen 16277 df-pt 16278 df-prds 16281 df-ordt 16334 df-xrs 16335 df-qtop 16340 df-imas 16341 df-xps 16343 df-mre 16419 df-mrc 16420 df-acs 16422 df-ps 17372 df-tsr 17373 df-plusf 17413 df-mgm 17414 df-sgrp 17456 df-mnd 17467 df-mhm 17507 df-submnd 17508 df-grp 17597 df-minusg 17598 df-sbg 17599 df-mulg 17713 df-subg 17763 df-cntz 17921 df-cmn 18366 df-abl 18367 df-mgp 18661 df-ur 18673 df-ring 18720 df-cring 18721 df-subrg 18951 df-abv 18990 df-lmod 19038 df-scaf 19039 df-sra 19345 df-rgmod 19346 df-psmet 19911 df-xmet 19912 df-met 19913 df-bl 19914 df-mopn 19915 df-fbas 19916 df-fg 19917 df-cnfld 19920 df-top 20872 df-topon 20889 df-topsp 20910 df-bases 20923 df-cld 20996 df-ntr 20997 df-cls 20998 df-nei 21075 df-lp 21113 df-perf 21114 df-cn 21204 df-cnp 21205 df-haus 21292 df-tx 21538 df-hmeo 21731 df-fil 21822 df-fm 21914 df-flim 21915 df-flf 21916 df-tmd 22048 df-tgp 22049 df-tsms 22102 df-trg 22135 df-xms 22297 df-ms 22298 df-tms 22299 df-nm 22559 df-ngp 22560 df-nrg 22562 df-nlm 22563 df-ii 22852 df-cncf 22853 df-limc 23800 df-dv 23801 df-log 24473 df-esum 30370

This theorem is referenced by: omsmeas 30665

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