esumnul - Mathbox for Thierry Arnoux

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Theorem esumnul 31916

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 1808	. . . 4 ⊢ Ⅎ𝑥⊤
2		nfcv 2906	. . . 4 ⊢ Ⅎ𝑥∅
3		0ex 5226	. . . . 5 ⊢ ∅ ∈ V
4	3	a1i 11	. . . 4 ⊢ (⊤ → ∅ ∈ V)
5		ral0 4440	. . . . . 6 ⊢ ∀𝑥 ∈ ∅ 𝐴 ∈ (0[,]+∞)
6	5	a1i 11	. . . . 5 ⊢ (⊤ → ∀𝑥 ∈ ∅ 𝐴 ∈ (0[,]+∞))
7	6	r19.21bi 3132	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ∅) → 𝐴 ∈ (0[,]+∞))
8		pw0 4742	. . . . . . . . . . . . 13 ⊢ 𝒫 ∅ = {∅}
9	8	ineq1i 4139	. . . . . . . . . . . 12 ⊢ (𝒫 ∅ ∩ Fin) = ({∅} ∩ Fin)
10		0fin 8916	. . . . . . . . . . . . 13 ⊢ ∅ ∈ Fin
11		snssi 4738	. . . . . . . . . . . . . 14 ⊢ (∅ ∈ Fin → {∅} ⊆ Fin)
12		df-ss 3900	. . . . . . . . . . . . . 14 ⊢ ({∅} ⊆ Fin ↔ ({∅} ∩ Fin) = {∅})
13	11, 12	sylib 217	. . . . . . . . . . . . 13 ⊢ (∅ ∈ Fin → ({∅} ∩ Fin) = {∅})
14	10, 13	ax-mp 5	. . . . . . . . . . . 12 ⊢ ({∅} ∩ Fin) = {∅}
15	9, 14	eqtri 2766	. . . . . . . . . . 11 ⊢ (𝒫 ∅ ∩ Fin) = {∅}
16	15	eleq2i 2830	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↔ 𝑦 ∈ {∅})
17		velsn 4574	. . . . . . . . . 10 ⊢ (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
18	16, 17	sylbb 218	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → 𝑦 = ∅)
19	18	mpteq1d 5165	. . . . . . . 8 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → (𝑥 ∈ 𝑦 ↦ 𝐴) = (𝑥 ∈ ∅ ↦ 𝐴))
20		mpt0 6559	. . . . . . . 8 ⊢ (𝑥 ∈ ∅ ↦ 𝐴) = ∅
21	19, 20	eqtrdi 2795	. . . . . . 7 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → (𝑥 ∈ 𝑦 ↦ 𝐴) = ∅)
22	21	oveq2d 7271	. . . . . 6 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑥 ∈ 𝑦 ↦ 𝐴)) = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g ∅))
23		xrge00 31197	. . . . . . 7 ⊢ 0 = (0_g‘(ℝ^*_𝑠 ↾_s (0[,]+∞)))
24	23	gsum0 18283	. . . . . 6 ⊢ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g ∅) = 0
25	22, 24	eqtrdi 2795	. . . . 5 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑥 ∈ 𝑦 ↦ 𝐴)) = 0)
26	25	adantl 481	. . . 4 ⊢ ((⊤ ∧ 𝑦 ∈ (𝒫 ∅ ∩ Fin)) → ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑥 ∈ 𝑦 ↦ 𝐴)) = 0)
27	1, 2, 4, 7, 26	esumval 31914	. . 3 ⊢ (⊤ → Σ^𝑥 ∈ ∅𝐴 = sup(ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0), ℝ^, < ))
28	27	mptru 1546	. 2 ⊢ Σ^𝑥 ∈ ∅𝐴 = sup(ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0), ℝ^, < )
29		fconstmpt 5640	. . . . 5 ⊢ ((𝒫 ∅ ∩ Fin) × {0}) = (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0)
30	29	eqcomi 2747	. . . 4 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = ((𝒫 ∅ ∩ Fin) × {0})
31		0xr 10953	. . . . . . 7 ⊢ 0 ∈ ℝ^*
32	31	rgenw 3075	. . . . . 6 ⊢ ∀𝑦 ∈ (𝒫 ∅ ∩ Fin)0 ∈ ℝ^*
33		eqid 2738	. . . . . . 7 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0)
34	33	fnmpt 6557	. . . . . 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 4709	. . . . . 6 ⊢ {∅} ≠ ∅
37	15, 36	eqnetri 3013	. . . . 5 ⊢ (𝒫 ∅ ∩ Fin) ≠ ∅
38		fconst5 7063	. . . . 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 688	. . . 4 ⊢ ((𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = ((𝒫 ∅ ∩ Fin) × {0}) ↔ ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = {0})
40	30, 39	mpbi 229	. . 3 ⊢ ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = {0}
41	40	supeq1i 9136	. 2 ⊢ sup(ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0), ℝ^, < ) = sup({0}, ℝ^, < )
42		xrltso 12804	. . 3 ⊢ < Or ℝ^*
43		supsn 9161	. . 3 ⊢ (( < Or ℝ^* ∧ 0 ∈ ℝ^) → sup({0}, ℝ^, < ) = 0)
44	42, 31, 43	mp2an 688	. 2 ⊢ sup({0}, ℝ^*, < ) = 0
45	28, 41, 44	3eqtri 2770	1 ⊢ Σ^*𝑥 ∈ ∅𝐴 = 0

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 = wceq 1539 ⊤wtru 1540 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 Vcvv 3422 ∩ cin 3882 ⊆ wss 3883 ∅c0 4253 𝒫 cpw 4530 {csn 4558 ↦ cmpt 5153 Or wor 5493 × cxp 5578 ran crn 5581 Fn wfn 6413 (class class class)co 7255 Fincfn 8691 supcsup 9129 0cc0 10802 +∞cpnf 10937 ℝ^*cxr 10939 < clt 10940 [,]cicc 13011 ↾_s cress 16867 Σ_g cgsu 17068 ℝ^*_𝑠cxrs 17128 Σ^*cesum 31895

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-xadd 12778 df-ioo 13012 df-ioc 13013 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-seq 13650 df-hash 13973 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-tset 16907 df-ple 16908 df-ds 16910 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-ordt 17129 df-xrs 17130 df-mre 17212 df-mrc 17213 df-acs 17215 df-ps 18199 df-tsr 18200 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-cntz 18838 df-cmn 19303 df-fbas 20507 df-fg 20508 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-ntr 22079 df-nei 22157 df-cn 22286 df-haus 22374 df-fil 22905 df-fm 22997 df-flim 22998 df-flf 22999 df-tsms 23186 df-esum 31896

This theorem is referenced by: esumrnmpt2 31936 esum2dlem 31960 ddemeas 32104 carsgclctunlem1 32184

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