esumnul - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > esumnul		Structured version Visualization version GIF version

Theorem esumnul 34038

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 1804	. . . 4 ⊢ Ⅎ𝑥⊤
2		nfcv 2891	. . . 4 ⊢ Ⅎ𝑥∅
3		0ex 5262	. . . . 5 ⊢ ∅ ∈ V
4	3	a1i 11	. . . 4 ⊢ (⊤ → ∅ ∈ V)
5		ral0 4476	. . . . . 6 ⊢ ∀𝑥 ∈ ∅ 𝐴 ∈ (0[,]+∞)
6	5	a1i 11	. . . . 5 ⊢ (⊤ → ∀𝑥 ∈ ∅ 𝐴 ∈ (0[,]+∞))
7	6	r19.21bi 3229	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ∅) → 𝐴 ∈ (0[,]+∞))
8		pw0 4776	. . . . . . . . . . . . 13 ⊢ 𝒫 ∅ = {∅}
9	8	ineq1i 4179	. . . . . . . . . . . 12 ⊢ (𝒫 ∅ ∩ Fin) = ({∅} ∩ Fin)
10		0fi 9013	. . . . . . . . . . . . 13 ⊢ ∅ ∈ Fin
11		snssi 4772	. . . . . . . . . . . . . 14 ⊢ (∅ ∈ Fin → {∅} ⊆ Fin)
12		dfss2 3932	. . . . . . . . . . . . . 14 ⊢ ({∅} ⊆ Fin ↔ ({∅} ∩ Fin) = {∅})
13	11, 12	sylib 218	. . . . . . . . . . . . 13 ⊢ (∅ ∈ Fin → ({∅} ∩ Fin) = {∅})
14	10, 13	ax-mp 5	. . . . . . . . . . . 12 ⊢ ({∅} ∩ Fin) = {∅}
15	9, 14	eqtri 2752	. . . . . . . . . . 11 ⊢ (𝒫 ∅ ∩ Fin) = {∅}
16	15	eleq2i 2820	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↔ 𝑦 ∈ {∅})
17		velsn 4605	. . . . . . . . . 10 ⊢ (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
18	16, 17	sylbb 219	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → 𝑦 = ∅)
19	18	mpteq1d 5197	. . . . . . . 8 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → (𝑥 ∈ 𝑦 ↦ 𝐴) = (𝑥 ∈ ∅ ↦ 𝐴))
20		mpt0 6660	. . . . . . . 8 ⊢ (𝑥 ∈ ∅ ↦ 𝐴) = ∅
21	19, 20	eqtrdi 2780	. . . . . . 7 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → (𝑥 ∈ 𝑦 ↦ 𝐴) = ∅)
22	21	oveq2d 7403	. . . . . 6 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑥 ∈ 𝑦 ↦ 𝐴)) = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g ∅))
23		xrge00 32953	. . . . . . 7 ⊢ 0 = (0_g‘(ℝ^*_𝑠 ↾_s (0[,]+∞)))
24	23	gsum0 18611	. . . . . 6 ⊢ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g ∅) = 0
25	22, 24	eqtrdi 2780	. . . . 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 34036	. . 3 ⊢ (⊤ → Σ^𝑥 ∈ ∅𝐴 = sup(ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0), ℝ^, < ))
28	27	mptru 1547	. 2 ⊢ Σ^𝑥 ∈ ∅𝐴 = sup(ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0), ℝ^, < )
29		fconstmpt 5700	. . . . 5 ⊢ ((𝒫 ∅ ∩ Fin) × {0}) = (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0)
30	29	eqcomi 2738	. . . 4 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = ((𝒫 ∅ ∩ Fin) × {0})
31		0xr 11221	. . . . . . 7 ⊢ 0 ∈ ℝ^*
32	31	rgenw 3048	. . . . . 6 ⊢ ∀𝑦 ∈ (𝒫 ∅ ∩ Fin)0 ∈ ℝ^*
33		eqid 2729	. . . . . . 7 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0)
34	33	fnmpt 6658	. . . . . 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 4740	. . . . . 6 ⊢ {∅} ≠ ∅
37	15, 36	eqnetri 2995	. . . . 5 ⊢ (𝒫 ∅ ∩ Fin) ≠ ∅
38		fconst5 7180	. . . . 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 692	. . . 4 ⊢ ((𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = ((𝒫 ∅ ∩ Fin) × {0}) ↔ ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = {0})
40	30, 39	mpbi 230	. . 3 ⊢ ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = {0}
41	40	supeq1i 9398	. 2 ⊢ sup(ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0), ℝ^, < ) = sup({0}, ℝ^, < )
42		xrltso 13101	. . 3 ⊢ < Or ℝ^*
43		supsn 9424	. . 3 ⊢ (( < Or ℝ^* ∧ 0 ∈ ℝ^) → sup({0}, ℝ^, < ) = 0)
44	42, 31, 43	mp2an 692	. 2 ⊢ sup({0}, ℝ^*, < ) = 0
45	28, 41, 44	3eqtri 2756	1 ⊢ Σ^*𝑥 ∈ ∅𝐴 = 0

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 = wceq 1540 ⊤wtru 1541 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 Vcvv 3447 ∩ cin 3913 ⊆ wss 3914 ∅c0 4296 𝒫 cpw 4563 {csn 4589 ↦ cmpt 5188 Or wor 5545 × cxp 5636 ran crn 5639 Fn wfn 6506 (class class class)co 7387 Fincfn 8918 supcsup 9391 0cc0 11068 +∞cpnf 11205 ℝ^*cxr 11207 < clt 11208 [,]cicc 13309 ↾_s cress 17200 Σ_g cgsu 17403 ℝ^*_𝑠cxrs 17463 Σ^*cesum 34017

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-map 8801 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-fi 9362 df-sup 9393 df-inf 9394 df-oi 9463 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-q 12908 df-xadd 13073 df-ioo 13310 df-ioc 13311 df-ico 13312 df-icc 13313 df-fz 13469 df-fzo 13616 df-seq 13967 df-hash 14296 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-tset 17239 df-ple 17240 df-ds 17242 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-ordt 17464 df-xrs 17465 df-mre 17547 df-mrc 17548 df-acs 17550 df-ps 18525 df-tsr 18526 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-cntz 19249 df-cmn 19712 df-fbas 21261 df-fg 21262 df-top 22781 df-topon 22798 df-topsp 22820 df-bases 22833 df-ntr 22907 df-nei 22985 df-cn 23114 df-haus 23202 df-fil 23733 df-fm 23825 df-flim 23826 df-flf 23827 df-tsms 24014 df-esum 34018

This theorem is referenced by: esumrnmpt2 34058 esum2dlem 34082 ddemeas 34226 carsgclctunlem1 34308

W3C validator