esumnul - Mathbox for Thierry Arnoux

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Theorem esumnul 34232

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 1811	. . . 4 ⊢ Ⅎ𝑥⊤
2		nfcv 2901	. . . 4 ⊢ Ⅎ𝑥∅
3		0ex 5229	. . . . 5 ⊢ ∅ ∈ V
4	3	a1i 11	. . . 4 ⊢ (⊤ → ∅ ∈ V)
5		ral0 4426	. . . . . 6 ⊢ ∀𝑥 ∈ ∅ 𝐴 ∈ (0[,]+∞)
6	5	a1i 11	. . . . 5 ⊢ (⊤ → ∀𝑥 ∈ ∅ 𝐴 ∈ (0[,]+∞))
7	6	r19.21bi 3231	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ∅) → 𝐴 ∈ (0[,]+∞))
8		pw0 4743	. . . . . . . . . . . . 13 ⊢ 𝒫 ∅ = {∅}
9	8	ineq1i 4145	. . . . . . . . . . . 12 ⊢ (𝒫 ∅ ∩ Fin) = ({∅} ∩ Fin)
10		0fi 8979	. . . . . . . . . . . . 13 ⊢ ∅ ∈ Fin
11		snssi 4717	. . . . . . . . . . . . . 14 ⊢ (∅ ∈ Fin → {∅} ⊆ Fin)
12		dfss2 3901	. . . . . . . . . . . . . 14 ⊢ ({∅} ⊆ Fin ↔ ({∅} ∩ Fin) = {∅})
13	11, 12	sylib 219	. . . . . . . . . . . . 13 ⊢ (∅ ∈ Fin → ({∅} ∩ Fin) = {∅})
14	10, 13	ax-mp 5	. . . . . . . . . . . 12 ⊢ ({∅} ∩ Fin) = {∅}
15	9, 14	eqtri 2762	. . . . . . . . . . 11 ⊢ (𝒫 ∅ ∩ Fin) = {∅}
16	15	eleq2i 2831	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↔ 𝑦 ∈ {∅})
17		velsn 4571	. . . . . . . . . 10 ⊢ (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
18	16, 17	sylbb 220	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → 𝑦 = ∅)
19	18	mpteq1d 5162	. . . . . . . 8 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → (𝑥 ∈ 𝑦 ↦ 𝐴) = (𝑥 ∈ ∅ ↦ 𝐴))
20		mpt0 6627	. . . . . . . 8 ⊢ (𝑥 ∈ ∅ ↦ 𝐴) = ∅
21	19, 20	eqtrdi 2790	. . . . . . 7 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → (𝑥 ∈ 𝑦 ↦ 𝐴) = ∅)
22	21	oveq2d 7372	. . . . . 6 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑥 ∈ 𝑦 ↦ 𝐴)) = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g ∅))
23		xrge00 33093	. . . . . . 7 ⊢ 0 = (0_g‘(ℝ^*_𝑠 ↾_s (0[,]+∞)))
24	23	gsum0 18643	. . . . . 6 ⊢ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g ∅) = 0
25	22, 24	eqtrdi 2790	. . . . 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 34230	. . 3 ⊢ (⊤ → Σ^𝑥 ∈ ∅𝐴 = sup(ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0), ℝ^, < ))
28	27	mptru 1554	. 2 ⊢ Σ^𝑥 ∈ ∅𝐴 = sup(ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0), ℝ^, < )
29		fconstmpt 5680	. . . . 5 ⊢ ((𝒫 ∅ ∩ Fin) × {0}) = (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0)
30	29	eqcomi 2748	. . . 4 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = ((𝒫 ∅ ∩ Fin) × {0})
31		0xr 11183	. . . . . . 7 ⊢ 0 ∈ ℝ^*
32	31	rgenw 3057	. . . . . 6 ⊢ ∀𝑦 ∈ (𝒫 ∅ ∩ Fin)0 ∈ ℝ^*
33		eqid 2739	. . . . . . 7 ⊢ (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0)
34	33	fnmpt 6625	. . . . . 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 4708	. . . . . 6 ⊢ {∅} ≠ ∅
37	15, 36	eqnetri 3004	. . . . 5 ⊢ (𝒫 ∅ ∩ Fin) ≠ ∅
38		fconst5 7150	. . . . 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 698	. . . 4 ⊢ ((𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = ((𝒫 ∅ ∩ Fin) × {0}) ↔ ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = {0})
40	30, 39	mpbi 231	. . 3 ⊢ ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0) = {0}
41	40	supeq1i 9350	. 2 ⊢ sup(ran (𝑦 ∈ (𝒫 ∅ ∩ Fin) ↦ 0), ℝ^, < ) = sup({0}, ℝ^, < )
42		xrltso 13083	. . 3 ⊢ < Or ℝ^*
43		supsn 9376	. . 3 ⊢ (( < Or ℝ^* ∧ 0 ∈ ℝ^) → sup({0}, ℝ^, < ) = 0)
44	42, 31, 43	mp2an 698	. 2 ⊢ sup({0}, ℝ^*, < ) = 0
45	28, 41, 44	3eqtri 2766	1 ⊢ Σ^*𝑥 ∈ ∅𝐴 = 0

Colors of variables: wff setvar class

Syntax hints: ↔ wb 207 = wceq 1547 ⊤wtru 1548 ∈ wcel 2119 ≠ wne 2934 ∀wral 3053 Vcvv 3431 ∩ cin 3882 ⊆ wss 3883 ∅c0 4261 𝒫 cpw 4529 {csn 4555 ↦ cmpt 5153 Or wor 5525 × cxp 5616 ran crn 5619 Fn wfn 6480 (class class class)co 7356 Fincfn 8883 supcsup 9343 0cc0 11029 +∞cpnf 11167 ℝ^*cxr 11169 < clt 11170 [,]cicc 13292 ↾_s cress 17191 Σ_g cgsu 17394 ℝ^*_𝑠cxrs 17455 Σ^*cesum 34211

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-iin 4924 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-se 5572 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-isom 6494 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8633 df-map 8765 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-xadd 13055 df-ioo 13293 df-ioc 13294 df-ico 13295 df-icc 13296 df-fz 13453 df-fzo 13600 df-seq 13955 df-hash 14284 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-tset 17230 df-ple 17231 df-ds 17233 df-rest 17376 df-topn 17377 df-0g 17395 df-gsum 17396 df-topgen 17397 df-ordt 17456 df-xrs 17457 df-mre 17539 df-mrc 17540 df-acs 17542 df-ps 18523 df-tsr 18524 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-cntz 19283 df-cmn 19748 df-fbas 21344 df-fg 21345 df-top 22877 df-topon 22894 df-topsp 22916 df-bases 22929 df-ntr 23003 df-nei 23081 df-cn 23210 df-haus 23298 df-fil 23829 df-fm 23921 df-flim 23922 df-flf 23923 df-tsms 24110 df-esum 34212

This theorem is referenced by: esumrnmpt2 34252 esum2dlem 34276 ddemeas 34420 carsgclctunlem1 34501

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