esum0 - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > esum0		Structured version Visualization version GIF version

Theorem esum0 34155

Description: Extended sum of zero. (Contributed by Thierry Arnoux, 3-Mar-2017.)

**Hypothesis**
Ref	Expression
esum0.k	⊢ Ⅎ𝑘𝐴

**Assertion**
Ref	Expression
esum0	⊢ (𝐴 ∈ 𝑉 → Σ^*𝑘 ∈ 𝐴0 = 0)

Distinct variable group: 𝑘,𝑉 Allowed substitution hint: 𝐴(𝑘)

Proof of Theorem esum0 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		esum0.k	. . . 4 ⊢ Ⅎ𝑘𝐴
2	1	nfel1 2913	. . 3 ⊢ Ⅎ𝑘 𝐴 ∈ 𝑉
3		id 22	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉)
4		0e0iccpnf 13373	. . . 4 ⊢ 0 ∈ (0[,]+∞)
5	4	a1i 11	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑘 ∈ 𝐴) → 0 ∈ (0[,]+∞))
6		xrge0cmn 21397	. . . . . 6 ⊢ (ℝ^*_𝑠 ↾_s (0[,]+∞)) ∈ CMnd
7		cmnmnd 19724	. . . . . 6 ⊢ ((ℝ^_𝑠 ↾_s (0[,]+∞)) ∈ CMnd → (ℝ^_𝑠 ↾_s (0[,]+∞)) ∈ Mnd)
8	6, 7	ax-mp 5	. . . . 5 ⊢ (ℝ^*_𝑠 ↾_s (0[,]+∞)) ∈ Mnd
9		vex 3442	. . . . 5 ⊢ 𝑥 ∈ V
10		xrge00 33045	. . . . . 6 ⊢ 0 = (0_g‘(ℝ^*_𝑠 ↾_s (0[,]+∞)))
11	10	gsumz 18759	. . . . 5 ⊢ (((ℝ^_𝑠 ↾_s (0[,]+∞)) ∈ Mnd ∧ 𝑥 ∈ V) → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝑥 ↦ 0)) = 0)
12	8, 9, 11	mp2an 692	. . . 4 ⊢ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝑥 ↦ 0)) = 0
13	12	a1i 11	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝑥 ↦ 0)) = 0)
14	2, 1, 3, 5, 13	esumval 34152	. 2 ⊢ (𝐴 ∈ 𝑉 → Σ^𝑘 ∈ 𝐴0 = sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0), ℝ^, < ))
15		fconstmpt 5684	. . . . . . 7 ⊢ ((𝒫 𝐴 ∩ Fin) × {0}) = (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0)
16	15	eqcomi 2743	. . . . . 6 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = ((𝒫 𝐴 ∩ Fin) × {0})
17		0xr 11177	. . . . . . . . 9 ⊢ 0 ∈ ℝ^*
18	17	rgenw 3053	. . . . . . . 8 ⊢ ∀𝑥 ∈ (𝒫 𝐴 ∩ Fin)0 ∈ ℝ^*
19		eqid 2734	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0)
20	19	fnmpt 6630	. . . . . . . 8 ⊢ (∀𝑥 ∈ (𝒫 𝐴 ∩ Fin)0 ∈ ℝ^* → (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) Fn (𝒫 𝐴 ∩ Fin))
21	18, 20	ax-mp 5	. . . . . . 7 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) Fn (𝒫 𝐴 ∩ Fin)
22		0elpw 5299	. . . . . . . . 9 ⊢ ∅ ∈ 𝒫 𝐴
23		0fi 8977	. . . . . . . . 9 ⊢ ∅ ∈ Fin
24		elin 3915	. . . . . . . . 9 ⊢ (∅ ∈ (𝒫 𝐴 ∩ Fin) ↔ (∅ ∈ 𝒫 𝐴 ∧ ∅ ∈ Fin))
25	22, 23, 24	mpbir2an 711	. . . . . . . 8 ⊢ ∅ ∈ (𝒫 𝐴 ∩ Fin)
26	25	ne0ii 4294	. . . . . . 7 ⊢ (𝒫 𝐴 ∩ Fin) ≠ ∅
27		fconst5 7150	. . . . . . 7 ⊢ (((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) Fn (𝒫 𝐴 ∩ Fin) ∧ (𝒫 𝐴 ∩ Fin) ≠ ∅) → ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = ((𝒫 𝐴 ∩ Fin) × {0}) ↔ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = {0}))
28	21, 26, 27	mp2an 692	. . . . . 6 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = ((𝒫 𝐴 ∩ Fin) × {0}) ↔ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = {0})
29	16, 28	mpbi 230	. . . . 5 ⊢ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = {0}
30	29	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = {0})
31	30	supeq1d 9347	. . 3 ⊢ (𝐴 ∈ 𝑉 → sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0), ℝ^, < ) = sup({0}, ℝ^, < ))
32		xrltso 13053	. . . 4 ⊢ < Or ℝ^*
33		supsn 9374	. . . 4 ⊢ (( < Or ℝ^* ∧ 0 ∈ ℝ^) → sup({0}, ℝ^, < ) = 0)
34	32, 17, 33	mp2an 692	. . 3 ⊢ sup({0}, ℝ^*, < ) = 0
35	31, 34	eqtrdi 2785	. 2 ⊢ (𝐴 ∈ 𝑉 → sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0), ℝ^*, < ) = 0)
36	14, 35	eqtrd 2769	1 ⊢ (𝐴 ∈ 𝑉 → Σ^*𝑘 ∈ 𝐴0 = 0)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 Ⅎwnfc 2881 ≠ wne 2930 ∀wral 3049 Vcvv 3438 ∩ cin 3898 ∅c0 4283 𝒫 cpw 4552 {csn 4578 ↦ cmpt 5177 Or wor 5529 × cxp 5620 ran crn 5623 Fn wfn 6485 (class class class)co 7356 Fincfn 8881 supcsup 9341 0cc0 11024 +∞cpnf 11161 ℝ^*cxr 11163 < clt 11164 [,]cicc 13262 ↾_s cress 17155 Σ_g cgsu 17358 ℝ^*_𝑠cxrs 17419 Mndcmnd 18657 CMndccmn 19707 Σ^*cesum 34133

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-pre-sup 11102

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-iin 4947 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 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 8763 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-fsupp 9263 df-fi 9312 df-sup 9343 df-inf 9344 df-oi 9413 df-card 9849 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-div 11793 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-5 12209 df-6 12210 df-7 12211 df-8 12212 df-9 12213 df-n0 12400 df-z 12487 df-dec 12606 df-uz 12750 df-q 12860 df-xadd 13025 df-ioo 13263 df-ioc 13264 df-ico 13265 df-icc 13266 df-fz 13422 df-fzo 13569 df-seq 13923 df-hash 14252 df-struct 17072 df-sets 17089 df-slot 17107 df-ndx 17119 df-base 17135 df-ress 17156 df-plusg 17188 df-mulr 17189 df-tset 17194 df-ple 17195 df-ds 17197 df-rest 17340 df-topn 17341 df-0g 17359 df-gsum 17360 df-topgen 17361 df-ordt 17420 df-xrs 17421 df-mre 17503 df-mrc 17504 df-acs 17506 df-ps 18487 df-tsr 18488 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-submnd 18707 df-cntz 19244 df-cmn 19709 df-fbas 21304 df-fg 21305 df-top 22836 df-topon 22853 df-topsp 22875 df-bases 22888 df-ntr 22962 df-nei 23040 df-cn 23169 df-haus 23257 df-fil 23788 df-fm 23880 df-flim 23881 df-flf 23882 df-tsms 24069 df-esum 34134

This theorem is referenced by: esumpad 34161 esumrnmpt2 34174 measvunilem0 34319 ddemeas 34342

W3C validator