esumpad2 - Mathbox for Thierry Arnoux

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Theorem esumpad2 31317

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 1915	. . . 4 ⊢ Ⅎ𝑘𝜑
2		esumpad.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		esumpad.3	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (0[,]+∞))
4		difssd 4111	. . . 4 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ 𝐴)
5	1, 2, 3, 4	esummono 31315	. . 3 ⊢ (𝜑 → Σ^𝑘 ∈ (𝐴 ∖ 𝐵)𝐶 ≤ Σ^𝑘 ∈ 𝐴𝐶)
6		esumpad.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
7		unexg 7474	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)
8	2, 6, 7	syl2anc 586	. . . . 5 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ V)
9		elun 4127	. . . . . 6 ⊢ (𝑘 ∈ (𝐴 ∪ 𝐵) ↔ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵))
10		esumpad.4	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐶 = 0)
11		0e0iccpnf 12850	. . . . . . . 8 ⊢ 0 ∈ (0[,]+∞)
12	10, 11	eqeltrdi 2923	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ (0[,]+∞))
13	3, 12	jaodan 954	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵)) → 𝐶 ∈ (0[,]+∞))
14	9, 13	sylan2b 595	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ 𝐵)) → 𝐶 ∈ (0[,]+∞))
15		ssun1 4150	. . . . . 6 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
16	15	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ (𝐴 ∪ 𝐵))
17	1, 8, 14, 16	esummono 31315	. . . 4 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐶 ≤ Σ^𝑘 ∈ (𝐴 ∪ 𝐵)𝐶)
18		undif1 4426	. . . . . 6 ⊢ ((𝐴 ∖ 𝐵) ∪ 𝐵) = (𝐴 ∪ 𝐵)
19		esumeq1 31295	. . . . . 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 5233	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝐵) ∈ V)
22	2, 21	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ V)
23	4	sselda 3969	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐵)) → 𝑘 ∈ 𝐴)
24	23, 3	syldan 593	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐵)) → 𝐶 ∈ (0[,]+∞))
25	22, 6, 24, 10	esumpad 31316	. . . . 5 ⊢ (𝜑 → Σ^𝑘 ∈ ((𝐴 ∖ 𝐵) ∪ 𝐵)𝐶 = Σ^𝑘 ∈ (𝐴 ∖ 𝐵)𝐶)
26	20, 25	syl5eqr 2872	. . . 4 ⊢ (𝜑 → Σ^𝑘 ∈ (𝐴 ∪ 𝐵)𝐶 = Σ^𝑘 ∈ (𝐴 ∖ 𝐵)𝐶)
27	17, 26	breqtrd 5094	. . 3 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐶 ≤ Σ^𝑘 ∈ (𝐴 ∖ 𝐵)𝐶)
28	5, 27	jca 514	. 2 ⊢ (𝜑 → (Σ^𝑘 ∈ (𝐴 ∖ 𝐵)𝐶 ≤ Σ^𝑘 ∈ 𝐴𝐶 ∧ Σ^𝑘 ∈ 𝐴𝐶 ≤ Σ^𝑘 ∈ (𝐴 ∖ 𝐵)𝐶))
29		iccssxr 12822	. . . 4 ⊢ (0[,]+∞) ⊆ ℝ^*
30	24	ralrimiva 3184	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ (𝐴 ∖ 𝐵)𝐶 ∈ (0[,]+∞))
31		nfcv 2979	. . . . . 6 ⊢ Ⅎ𝑘(𝐴 ∖ 𝐵)
32	31	esumcl 31291	. . . . 5 ⊢ (((𝐴 ∖ 𝐵) ∈ V ∧ ∀𝑘 ∈ (𝐴 ∖ 𝐵)𝐶 ∈ (0[,]+∞)) → Σ^*𝑘 ∈ (𝐴 ∖ 𝐵)𝐶 ∈ (0[,]+∞))
33	22, 30, 32	syl2anc 586	. . . 4 ⊢ (𝜑 → Σ^*𝑘 ∈ (𝐴 ∖ 𝐵)𝐶 ∈ (0[,]+∞))
34	29, 33	sseldi 3967	. . 3 ⊢ (𝜑 → Σ^𝑘 ∈ (𝐴 ∖ 𝐵)𝐶 ∈ ℝ^)
35	3	ralrimiva 3184	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 ∈ (0[,]+∞))
36		nfcv 2979	. . . . . 6 ⊢ Ⅎ𝑘𝐴
37	36	esumcl 31291	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐶 ∈ (0[,]+∞)) → Σ^*𝑘 ∈ 𝐴𝐶 ∈ (0[,]+∞))
38	2, 35, 37	syl2anc 586	. . . 4 ⊢ (𝜑 → Σ^*𝑘 ∈ 𝐴𝐶 ∈ (0[,]+∞))
39	29, 38	sseldi 3967	. . 3 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐶 ∈ ℝ^)
40		xrletri3 12550	. . 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 586	. 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 259	1 ⊢ (𝜑 → Σ^𝑘 ∈ (𝐴 ∖ 𝐵)𝐶 = Σ^𝑘 ∈ 𝐴𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ∀wral 3140 Vcvv 3496 ∖ cdif 3935 ∪ cun 3936 ⊆ wss 3938 class class class wbr 5068 (class class class)co 7158 0cc0 10539 +∞cpnf 10674 ℝ^*cxr 10676 ≤ cle 10678 [,]cicc 12744 Σ^*cesum 31288

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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ioc 12746 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-fac 13637 df-bc 13666 df-hash 13694 df-shft 14428 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-limsup 14830 df-clim 14847 df-rlim 14848 df-sum 15045 df-ef 15423 df-sin 15425 df-cos 15426 df-pi 15428 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-hom 16591 df-cco 16592 df-rest 16698 df-topn 16699 df-0g 16717 df-gsum 16718 df-topgen 16719 df-pt 16720 df-prds 16723 df-ordt 16776 df-xrs 16777 df-qtop 16782 df-imas 16783 df-xps 16785 df-mre 16859 df-mrc 16860 df-acs 16862 df-ps 17812 df-tsr 17813 df-plusf 17853 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-submnd 17959 df-grp 18108 df-minusg 18109 df-sbg 18110 df-mulg 18227 df-subg 18278 df-cntz 18449 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-cring 19302 df-subrg 19535 df-abv 19590 df-lmod 19638 df-scaf 19639 df-sra 19946 df-rgmod 19947 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-fbas 20544 df-fg 20545 df-cnfld 20548 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cld 21629 df-ntr 21630 df-cls 21631 df-nei 21708 df-lp 21746 df-perf 21747 df-cn 21837 df-cnp 21838 df-haus 21925 df-tx 22172 df-hmeo 22365 df-fil 22456 df-fm 22548 df-flim 22549 df-flf 22550 df-tmd 22682 df-tgp 22683 df-tsms 22737 df-trg 22770 df-xms 22932 df-ms 22933 df-tms 22934 df-nm 23194 df-ngp 23195 df-nrg 23197 df-nlm 23198 df-ii 23487 df-cncf 23488 df-limc 24466 df-dv 24467 df-log 25142 df-esum 31289

This theorem is referenced by: omsmeas 31583

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