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 33798

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 1798	. . . 4 ⊢ Ⅎ𝑥⊤
2		nfcv 2891	. . . 4 ⊢ Ⅎ𝑥∅
3		0ex 5308	. . . . 5 ⊢ ∅ ∈ V
4	3	a1i 11	. . . 4 ⊢ (⊤ → ∅ ∈ V)
5		ral0 4514	. . . . . 6 ⊢ ∀𝑥 ∈ ∅ 𝐴 ∈ (0[,]+∞)
6	5	a1i 11	. . . . 5 ⊢ (⊤ → ∀𝑥 ∈ ∅ 𝐴 ∈ (0[,]+∞))
7	6	r19.21bi 3238	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ∅) → 𝐴 ∈ (0[,]+∞))
8		pw0 4817	. . . . . . . . . . . . 13 ⊢ 𝒫 ∅ = {∅}
9	8	ineq1i 4206	. . . . . . . . . . . 12 ⊢ (𝒫 ∅ ∩ Fin) = ({∅} ∩ Fin)
10		0fin 9196	. . . . . . . . . . . . 13 ⊢ ∅ ∈ Fin
11		snssi 4813	. . . . . . . . . . . . . 14 ⊢ (∅ ∈ Fin → {∅} ⊆ Fin)
12		dfss2 3962	. . . . . . . . . . . . . 14 ⊢ ({∅} ⊆ Fin ↔ ({∅} ∩ Fin) = {∅})
13	11, 12	sylib 217	. . . . . . . . . . . . 13 ⊢ (∅ ∈ Fin → ({∅} ∩ Fin) = {∅})
14	10, 13	ax-mp 5	. . . . . . . . . . . 12 ⊢ ({∅} ∩ Fin) = {∅}
15	9, 14	eqtri 2753	. . . . . . . . . . 11 ⊢ (𝒫 ∅ ∩ Fin) = {∅}
16	15	eleq2i 2817	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↔ 𝑦 ∈ {∅})
17		velsn 4646	. . . . . . . . . 10 ⊢ (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
18	16, 17	sylbb 218	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → 𝑦 = ∅)
19	18	mpteq1d 5244	. . . . . . . 8 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → (𝑥 ∈ 𝑦 ↦ 𝐴) = (𝑥 ∈ ∅ ↦ 𝐴))
20		mpt0 6698	. . . . . . . 8 ⊢ (𝑥 ∈ ∅ ↦ 𝐴) = ∅
21	19, 20	eqtrdi 2781	. . . . . . 7 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → (𝑥 ∈ 𝑦 ↦ 𝐴) = ∅)
22	21	oveq2d 7435	. . . . . 6 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑥 ∈ 𝑦 ↦ 𝐴)) = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g ∅))
23		xrge00 32831	. . . . . . 7 ⊢ 0 = (0_g‘(ℝ^*_𝑠 ↾_s (0[,]+∞)))
24	23	gsum0 18647	. . . . . 6 ⊢ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g ∅) = 0
25	22, 24	eqtrdi 2781	. . . . 5 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑥 ∈ 𝑦 ↦ 𝐴)) = 0)
26	25	adantl 480	. . . 4 ⊢ ((⊤ ∧ 𝑦 ∈ (𝒫 ∅ ∩ Fin)) → ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑥 ∈ 𝑦 ↦ 𝐴)) = 0)
27	1, 2, 4, 7, 26	esumval 33796	. . 3 ⊢ (⊤ → Σ^𝑥 ∈ ∅𝐴 = sup(ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0), ℝ^, < ))
28	27	mptru 1540	. 2 ⊢ Σ^𝑥 ∈ ∅𝐴 = sup(ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0), ℝ^, < )
29		fconstmpt 5740	. . . . 5 ⊢ ((𝒫 ∅ ∩ Fin) × {0}) = (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0)
30	29	eqcomi 2734	. . . 4 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = ((𝒫 ∅ ∩ Fin) × {0})
31		0xr 11293	. . . . . . 7 ⊢ 0 ∈ ℝ^*
32	31	rgenw 3054	. . . . . 6 ⊢ ∀𝑦 ∈ (𝒫 ∅ ∩ Fin)0 ∈ ℝ^*
33		eqid 2725	. . . . . . 7 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0)
34	33	fnmpt 6696	. . . . . 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 4782	. . . . . 6 ⊢ {∅} ≠ ∅
37	15, 36	eqnetri 3000	. . . . 5 ⊢ (𝒫 ∅ ∩ Fin) ≠ ∅
38		fconst5 7218	. . . . 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 690	. . . 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 9472	. 2 ⊢ sup(ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0), ℝ^, < ) = sup({0}, ℝ^, < )
42		xrltso 13155	. . 3 ⊢ < Or ℝ^*
43		supsn 9497	. . 3 ⊢ (( < Or ℝ^* ∧ 0 ∈ ℝ^) → sup({0}, ℝ^, < ) = 0)
44	42, 31, 43	mp2an 690	. 2 ⊢ sup({0}, ℝ^*, < ) = 0
45	28, 41, 44	3eqtri 2757	1 ⊢ Σ^*𝑥 ∈ ∅𝐴 = 0

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 = wceq 1533 ⊤wtru 1534 ∈ wcel 2098 ≠ wne 2929 ∀wral 3050 Vcvv 3461 ∩ cin 3943 ⊆ wss 3944 ∅c0 4322 𝒫 cpw 4604 {csn 4630 ↦ cmpt 5232 Or wor 5589 × cxp 5676 ran crn 5679 Fn wfn 6544 (class class class)co 7419 Fincfn 8964 supcsup 9465 0cc0 11140 +∞cpnf 11277 ℝ^*cxr 11279 < clt 11280 [,]cicc 13362 ↾_s cress 17212 Σ_g cgsu 17425 ℝ^*_𝑠cxrs 17485 Σ^*cesum 33777

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9388 df-fi 9436 df-sup 9467 df-inf 9468 df-oi 9535 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-q 12966 df-xadd 13128 df-ioo 13363 df-ioc 13364 df-ico 13365 df-icc 13366 df-fz 13520 df-fzo 13663 df-seq 14003 df-hash 14326 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-tset 17255 df-ple 17256 df-ds 17258 df-rest 17407 df-topn 17408 df-0g 17426 df-gsum 17427 df-topgen 17428 df-ordt 17486 df-xrs 17487 df-mre 17569 df-mrc 17570 df-acs 17572 df-ps 18561 df-tsr 18562 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18744 df-cntz 19280 df-cmn 19749 df-fbas 21293 df-fg 21294 df-top 22840 df-topon 22857 df-topsp 22879 df-bases 22893 df-ntr 22968 df-nei 23046 df-cn 23175 df-haus 23263 df-fil 23794 df-fm 23886 df-flim 23887 df-flf 23888 df-tsms 24075 df-esum 33778

This theorem is referenced by: esumrnmpt2 33818 esum2dlem 33842 ddemeas 33986 carsgclctunlem1 34068

W3C validator