esumnul - Mathbox for Thierry Arnoux

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Theorem esumnul 29883

Description: Extended sum over the empty set. (Contributed by Thierry Arnoux, 19-Feb-2017.)

**Assertion**
Ref	Expression
esumnul	⊢ Σ^*𝑥 ∈ ∅𝐴 = 0

Proof of Theorem esumnul Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nftru 1727	. . . 4 ⊢ Ⅎ𝑥⊤
2		nfcv 2767	. . . 4 ⊢ Ⅎ𝑥∅
3		0ex 4755	. . . . 5 ⊢ ∅ ∈ V
4	3	a1i 11	. . . 4 ⊢ (⊤ → ∅ ∈ V)
5		ral0 4053	. . . . . 6 ⊢ ∀𝑥 ∈ ∅ 𝐴 ∈ (0[,]+∞)
6	5	a1i 11	. . . . 5 ⊢ (⊤ → ∀𝑥 ∈ ∅ 𝐴 ∈ (0[,]+∞))
7	6	r19.21bi 2932	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ∅) → 𝐴 ∈ (0[,]+∞))
8		pw0 4316	. . . . . . . . . . . . 13 ⊢ 𝒫 ∅ = {∅}
9	8	ineq1i 3793	. . . . . . . . . . . 12 ⊢ (𝒫 ∅ ∩ Fin) = ({∅} ∩ Fin)
10		0fin 8133	. . . . . . . . . . . . 13 ⊢ ∅ ∈ Fin
11		snssi 4313	. . . . . . . . . . . . . 14 ⊢ (∅ ∈ Fin → {∅} ⊆ Fin)
12		df-ss 3574	. . . . . . . . . . . . . 14 ⊢ ({∅} ⊆ Fin ↔ ({∅} ∩ Fin) = {∅})
13	11, 12	sylib 208	. . . . . . . . . . . . 13 ⊢ (∅ ∈ Fin → ({∅} ∩ Fin) = {∅})
14	10, 13	ax-mp 5	. . . . . . . . . . . 12 ⊢ ({∅} ∩ Fin) = {∅}
15	9, 14	eqtri 2648	. . . . . . . . . . 11 ⊢ (𝒫 ∅ ∩ Fin) = {∅}
16	15	eleq2i 2696	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↔ 𝑦 ∈ {∅})
17		velsn 4169	. . . . . . . . . 10 ⊢ (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
18	16, 17	sylbb 209	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → 𝑦 = ∅)
19	18	mpteq1d 4703	. . . . . . . 8 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → (𝑥 ∈ 𝑦 ↦ 𝐴) = (𝑥 ∈ ∅ ↦ 𝐴))
20		mpt0 5980	. . . . . . . 8 ⊢ (𝑥 ∈ ∅ ↦ 𝐴) = ∅
21	19, 20	syl6eq 2676	. . . . . . 7 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → (𝑥 ∈ 𝑦 ↦ 𝐴) = ∅)
22	21	oveq2d 6621	. . . . . 6 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑥 ∈ 𝑦 ↦ 𝐴)) = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g ∅))
23		xrge00 29463	. . . . . . 7 ⊢ 0 = (0_g‘(ℝ^*_𝑠 ↾_s (0[,]+∞)))
24	23	gsum0 17194	. . . . . 6 ⊢ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g ∅) = 0
25	22, 24	syl6eq 2676	. . . . 5 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑥 ∈ 𝑦 ↦ 𝐴)) = 0)
26	25	adantl 482	. . . 4 ⊢ ((⊤ ∧ 𝑦 ∈ (𝒫 ∅ ∩ Fin)) → ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑥 ∈ 𝑦 ↦ 𝐴)) = 0)
27	1, 2, 4, 7, 26	esumval 29881	. . 3 ⊢ (⊤ → Σ^𝑥 ∈ ∅𝐴 = sup(ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0), ℝ^, < ))
28	27	trud 1490	. 2 ⊢ Σ^𝑥 ∈ ∅𝐴 = sup(ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0), ℝ^, < )
29		fconstmpt 5128	. . . . 5 ⊢ ((𝒫 ∅ ∩ Fin) × {0}) = (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0)
30	29	eqcomi 2635	. . . 4 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = ((𝒫 ∅ ∩ Fin) × {0})
31		0xr 10031	. . . . . . 7 ⊢ 0 ∈ ℝ^*
32	31	rgenw 2924	. . . . . 6 ⊢ ∀𝑦 ∈ (𝒫 ∅ ∩ Fin)0 ∈ ℝ^*
33		eqid 2626	. . . . . . 7 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0)
34	33	fnmpt 5979	. . . . . 6 ⊢ (∀𝑦 ∈ (𝒫 ∅ ∩ Fin)0 ∈ ℝ^* → (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) Fn (𝒫 ∅ ∩ Fin))
35	32, 34	ax-mp 5	. . . . 5 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) Fn (𝒫 ∅ ∩ Fin)
36	3	snnz 4284	. . . . . 6 ⊢ {∅} ≠ ∅
37	15, 36	eqnetri 2866	. . . . 5 ⊢ (𝒫 ∅ ∩ Fin) ≠ ∅
38		fconst5 6426	. . . . 5 ⊢ (((𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) Fn (𝒫 ∅ ∩ Fin) ∧ (𝒫 ∅ ∩ Fin) ≠ ∅) → ((𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = ((𝒫 ∅ ∩ Fin) × {0}) ↔ ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = {0}))
39	35, 37, 38	mp2an 707	. . . 4 ⊢ ((𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = ((𝒫 ∅ ∩ Fin) × {0}) ↔ ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = {0})
40	30, 39	mpbi 220	. . 3 ⊢ ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = {0}
41	40	supeq1i 8298	. 2 ⊢ sup(ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0), ℝ^, < ) = sup({0}, ℝ^, < )
42		xrltso 11918	. . 3 ⊢ < Or ℝ^*
43		supsn 8323	. . 3 ⊢ (( < Or ℝ^* ∧ 0 ∈ ℝ^) → sup({0}, ℝ^, < ) = 0)
44	42, 31, 43	mp2an 707	. 2 ⊢ sup({0}, ℝ^*, < ) = 0
45	28, 41, 44	3eqtri 2652	1 ⊢ Σ^*𝑥 ∈ ∅𝐴 = 0

Colors of variables: wff setvar class

Syntax hints: ↔ wb 196 = wceq 1480 ⊤wtru 1481 ∈ wcel 1992 ≠ wne 2796 ∀wral 2912 Vcvv 3191 ∩ cin 3559 ⊆ wss 3560 ∅c0 3896 𝒫 cpw 4135 {csn 4153 ↦ cmpt 4678 Or wor 4999 × cxp 5077 ran crn 5080 Fn wfn 5845 (class class class)co 6605 Fincfn 7900 supcsup 8291 0cc0 9881 +∞cpnf 10016 ℝ^*cxr 10018 < clt 10019 [,]cicc 12117 ↾_s cress 15777 Σ_g cgsu 16017 ℝ^*_𝑠cxrs 16076 Σ^*cesum 29862

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1719 ax-4 1734 ax-5 1841 ax-6 1890 ax-7 1937 ax-8 1994 ax-9 2001 ax-10 2021 ax-11 2036 ax-12 2049 ax-13 2250 ax-ext 2606 ax-rep 4736 ax-sep 4746 ax-nul 4754 ax-pow 4808 ax-pr 4872 ax-un 6903 ax-cnex 9937 ax-resscn 9938 ax-1cn 9939 ax-icn 9940 ax-addcl 9941 ax-addrcl 9942 ax-mulcl 9943 ax-mulrcl 9944 ax-mulcom 9945 ax-addass 9946 ax-mulass 9947 ax-distr 9948 ax-i2m1 9949 ax-1ne0 9950 ax-1rid 9951 ax-rnegex 9952 ax-rrecex 9953 ax-cnre 9954 ax-pre-lttri 9955 ax-pre-lttrn 9956 ax-pre-ltadd 9957 ax-pre-mulgt0 9958 ax-pre-sup 9959

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1037 df-3an 1038 df-tru 1483 df-fal 1486 df-ex 1702 df-nf 1707 df-sb 1883 df-eu 2478 df-mo 2479 df-clab 2613 df-cleq 2619 df-clel 2622 df-nfc 2756 df-ne 2797 df-nel 2900 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3193 df-sbc 3423 df-csb 3520 df-dif 3563 df-un 3565 df-in 3567 df-ss 3574 df-pss 3576 df-nul 3897 df-if 4064 df-pw 4137 df-sn 4154 df-pr 4156 df-tp 4158 df-op 4160 df-uni 4408 df-int 4446 df-iun 4492 df-iin 4493 df-br 4619 df-opab 4679 df-mpt 4680 df-tr 4718 df-eprel 4990 df-id 4994 df-po 5000 df-so 5001 df-fr 5038 df-se 5039 df-we 5040 df-xp 5085 df-rel 5086 df-cnv 5087 df-co 5088 df-dm 5089 df-rn 5090 df-res 5091 df-ima 5092 df-pred 5642 df-ord 5688 df-on 5689 df-lim 5690 df-suc 5691 df-iota 5813 df-fun 5852 df-fn 5853 df-f 5854 df-f1 5855 df-fo 5856 df-f1o 5857 df-fv 5858 df-isom 5859 df-riota 6566 df-ov 6608 df-oprab 6609 df-mpt2 6610 df-of 6851 df-om 7014 df-1st 7116 df-2nd 7117 df-supp 7242 df-wrecs 7353 df-recs 7414 df-rdg 7452 df-1o 7506 df-oadd 7510 df-er 7688 df-map 7805 df-en 7901 df-dom 7902 df-sdom 7903 df-fin 7904 df-fsupp 8221 df-fi 8262 df-sup 8293 df-inf 8294 df-oi 8360 df-card 8710 df-pnf 10021 df-mnf 10022 df-xr 10023 df-ltxr 10024 df-le 10025 df-sub 10213 df-neg 10214 df-div 10630 df-nn 10966 df-2 11024 df-3 11025 df-4 11026 df-5 11027 df-6 11028 df-7 11029 df-8 11030 df-9 11031 df-n0 11238 df-z 11323 df-dec 11438 df-uz 11632 df-q 11733 df-xadd 11891 df-ioo 12118 df-ioc 12119 df-ico 12120 df-icc 12121 df-fz 12266 df-fzo 12404 df-seq 12739 df-hash 13055 df-struct 15778 df-ndx 15779 df-slot 15780 df-base 15781 df-sets 15782 df-ress 15783 df-plusg 15870 df-mulr 15871 df-tset 15876 df-ple 15877 df-ds 15880 df-rest 15999 df-topn 16000 df-0g 16018 df-gsum 16019 df-topgen 16020 df-ordt 16077 df-xrs 16078 df-mre 16162 df-mrc 16163 df-acs 16165 df-ps 17116 df-tsr 17117 df-mgm 17158 df-sgrp 17200 df-mnd 17211 df-submnd 17252 df-cntz 17666 df-cmn 18111 df-fbas 19657 df-fg 19658 df-top 20616 df-bases 20617 df-topon 20618 df-topsp 20619 df-ntr 20729 df-nei 20807 df-cn 20936 df-haus 21024 df-fil 21555 df-fm 21647 df-flim 21648 df-flf 21649 df-tsms 21835 df-esum 29863

This theorem is referenced by: esumrnmpt2 29903 esum2dlem 29927 ddemeas 30072 carsgclctunlem1 30152

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