esumpad2 - Mathbox for Thierry Arnoux

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Theorem esumpad2 33669

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 1910	. . . 4 ⊢ Ⅎ𝑘𝜑
2		esumpad.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		esumpad.3	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (0[,]+∞))
4		difssd 4128	. . . 4 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ 𝐴)
5	1, 2, 3, 4	esummono 33667	. . 3 ⊢ (𝜑 → Σ^𝑘 ∈ (𝐴 ∖ 𝐵)𝐶 ≤ Σ^𝑘 ∈ 𝐴𝐶)
6		esumpad.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
7		unexg 7745	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)
8	2, 6, 7	syl2anc 583	. . . . 5 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ V)
9		elun 4144	. . . . . 6 ⊢ (𝑘 ∈ (𝐴 ∪ 𝐵) ↔ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵))
10		esumpad.4	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐶 = 0)
11		0e0iccpnf 13462	. . . . . . . 8 ⊢ 0 ∈ (0[,]+∞)
12	10, 11	eqeltrdi 2837	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ (0[,]+∞))
13	3, 12	jaodan 956	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵)) → 𝐶 ∈ (0[,]+∞))
14	9, 13	sylan2b 593	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ 𝐵)) → 𝐶 ∈ (0[,]+∞))
15		ssun1 4168	. . . . . 6 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
16	15	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ (𝐴 ∪ 𝐵))
17	1, 8, 14, 16	esummono 33667	. . . 4 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐶 ≤ Σ^𝑘 ∈ (𝐴 ∪ 𝐵)𝐶)
18		undif1 4471	. . . . . 6 ⊢ ((𝐴 ∖ 𝐵) ∪ 𝐵) = (𝐴 ∪ 𝐵)
19		esumeq1 33647	. . . . . 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	2	difexd 5325	. . . . . 6 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ V)
22	4	sselda 3978	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐵)) → 𝑘 ∈ 𝐴)
23	22, 3	syldan 590	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐵)) → 𝐶 ∈ (0[,]+∞))
24	21, 6, 23, 10	esumpad 33668	. . . . 5 ⊢ (𝜑 → Σ^𝑘 ∈ ((𝐴 ∖ 𝐵) ∪ 𝐵)𝐶 = Σ^𝑘 ∈ (𝐴 ∖ 𝐵)𝐶)
25	20, 24	eqtr3id 2782	. . . 4 ⊢ (𝜑 → Σ^𝑘 ∈ (𝐴 ∪ 𝐵)𝐶 = Σ^𝑘 ∈ (𝐴 ∖ 𝐵)𝐶)
26	17, 25	breqtrd 5168	. . 3 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐶 ≤ Σ^𝑘 ∈ (𝐴 ∖ 𝐵)𝐶)
27	5, 26	jca 511	. 2 ⊢ (𝜑 → (Σ^𝑘 ∈ (𝐴 ∖ 𝐵)𝐶 ≤ Σ^𝑘 ∈ 𝐴𝐶 ∧ Σ^𝑘 ∈ 𝐴𝐶 ≤ Σ^𝑘 ∈ (𝐴 ∖ 𝐵)𝐶))
28		iccssxr 13433	. . . 4 ⊢ (0[,]+∞) ⊆ ℝ^*
29	23	ralrimiva 3142	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ (𝐴 ∖ 𝐵)𝐶 ∈ (0[,]+∞))
30		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑘(𝐴 ∖ 𝐵)
31	30	esumcl 33643	. . . . 5 ⊢ (((𝐴 ∖ 𝐵) ∈ V ∧ ∀𝑘 ∈ (𝐴 ∖ 𝐵)𝐶 ∈ (0[,]+∞)) → Σ^*𝑘 ∈ (𝐴 ∖ 𝐵)𝐶 ∈ (0[,]+∞))
32	21, 29, 31	syl2anc 583	. . . 4 ⊢ (𝜑 → Σ^*𝑘 ∈ (𝐴 ∖ 𝐵)𝐶 ∈ (0[,]+∞))
33	28, 32	sselid 3976	. . 3 ⊢ (𝜑 → Σ^𝑘 ∈ (𝐴 ∖ 𝐵)𝐶 ∈ ℝ^)
34	3	ralrimiva 3142	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 ∈ (0[,]+∞))
35		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑘𝐴
36	35	esumcl 33643	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐶 ∈ (0[,]+∞)) → Σ^*𝑘 ∈ 𝐴𝐶 ∈ (0[,]+∞))
37	2, 34, 36	syl2anc 583	. . . 4 ⊢ (𝜑 → Σ^*𝑘 ∈ 𝐴𝐶 ∈ (0[,]+∞))
38	28, 37	sselid 3976	. . 3 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐶 ∈ ℝ^)
39		xrletri3 13159	. . 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)))
40	33, 38, 39	syl2anc 583	. 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)))
41	27, 40	mpbird 257	1 ⊢ (𝜑 → Σ^𝑘 ∈ (𝐴 ∖ 𝐵)𝐶 = Σ^𝑘 ∈ 𝐴𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 846 = wceq 1534 ∈ wcel 2099 ∀wral 3057 Vcvv 3470 ∖ cdif 3942 ∪ cun 3943 ⊆ wss 3945 class class class wbr 5142 (class class class)co 7414 0cc0 11132 +∞cpnf 11269 ℝ^*cxr 11271 ≤ cle 11273 [,]cicc 13353 Σ^*cesum 33640

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9658 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210 ax-addf 11211 ax-mulf 11212

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8718 df-map 8840 df-pm 8841 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9380 df-fi 9428 df-sup 9459 df-inf 9460 df-oi 9527 df-card 9956 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-z 12583 df-dec 12702 df-uz 12847 df-q 12957 df-rp 13001 df-xneg 13118 df-xadd 13119 df-xmul 13120 df-ioo 13354 df-ioc 13355 df-ico 13356 df-icc 13357 df-fz 13511 df-fzo 13654 df-fl 13783 df-mod 13861 df-seq 13993 df-exp 14053 df-fac 14259 df-bc 14288 df-hash 14316 df-shft 15040 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-limsup 15441 df-clim 15458 df-rlim 15459 df-sum 15659 df-ef 16037 df-sin 16039 df-cos 16040 df-pi 16042 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203 df-plusg 17239 df-mulr 17240 df-starv 17241 df-sca 17242 df-vsca 17243 df-ip 17244 df-tset 17245 df-ple 17246 df-ds 17248 df-unif 17249 df-hom 17250 df-cco 17251 df-rest 17397 df-topn 17398 df-0g 17416 df-gsum 17417 df-topgen 17418 df-pt 17419 df-prds 17422 df-ordt 17476 df-xrs 17477 df-qtop 17482 df-imas 17483 df-xps 17485 df-mre 17559 df-mrc 17560 df-acs 17562 df-ps 18551 df-tsr 18552 df-plusf 18592 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-mhm 18733 df-submnd 18734 df-grp 18886 df-minusg 18887 df-sbg 18888 df-mulg 19017 df-subg 19071 df-cntz 19261 df-cmn 19730 df-abl 19731 df-mgp 20068 df-rng 20086 df-ur 20115 df-ring 20168 df-cring 20169 df-subrng 20476 df-subrg 20501 df-abv 20690 df-lmod 20738 df-scaf 20739 df-sra 21051 df-rgmod 21052 df-psmet 21264 df-xmet 21265 df-met 21266 df-bl 21267 df-mopn 21268 df-fbas 21269 df-fg 21270 df-cnfld 21273 df-top 22789 df-topon 22806 df-topsp 22828 df-bases 22842 df-cld 22916 df-ntr 22917 df-cls 22918 df-nei 22995 df-lp 23033 df-perf 23034 df-cn 23124 df-cnp 23125 df-haus 23212 df-tx 23459 df-hmeo 23652 df-fil 23743 df-fm 23835 df-flim 23836 df-flf 23837 df-tmd 23969 df-tgp 23970 df-tsms 24024 df-trg 24057 df-xms 24219 df-ms 24220 df-tms 24221 df-nm 24484 df-ngp 24485 df-nrg 24487 df-nlm 24488 df-ii 24790 df-cncf 24791 df-limc 25788 df-dv 25789 df-log 26483 df-esum 33641

This theorem is referenced by: omsmeas 33937

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