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Mirrors > Home > MPE Home > Th. List > Mathboxes > esum0 | Structured version Visualization version GIF version |
Description: Extended sum of zero. (Contributed by Thierry Arnoux, 3-Mar-2017.) |
Ref | Expression |
---|---|
esum0.k | ⊢ Ⅎ𝑘𝐴 |
Ref | Expression |
---|---|
esum0 | ⊢ (𝐴 ∈ 𝑉 → Σ*𝑘 ∈ 𝐴0 = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | esum0.k | . . . 4 ⊢ Ⅎ𝑘𝐴 | |
2 | 1 | nfel1 2765 | . . 3 ⊢ Ⅎ𝑘 𝐴 ∈ 𝑉 |
3 | id 22 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉) | |
4 | 0e0iccpnf 12154 | . . . 4 ⊢ 0 ∈ (0[,]+∞) | |
5 | 4 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑘 ∈ 𝐴) → 0 ∈ (0[,]+∞)) |
6 | xrge0cmn 19607 | . . . . . 6 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd | |
7 | cmnmnd 18031 | . . . . . 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 3176 | . . . . 5 ⊢ 𝑥 ∈ V | |
10 | xrge00 29017 | . . . . . 6 ⊢ 0 = (0g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
11 | 10 | gsumz 17197 | . . . . 5 ⊢ (((ℝ*𝑠 ↾s (0[,]+∞)) ∈ Mnd ∧ 𝑥 ∈ V) → ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝑥 ↦ 0)) = 0) |
12 | 8, 9, 11 | mp2an 704 | . . . 4 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝑥 ↦ 0)) = 0 |
13 | 12 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝑥 ↦ 0)) = 0) |
14 | 2, 1, 3, 5, 13 | esumval 29435 | . 2 ⊢ (𝐴 ∈ 𝑉 → Σ*𝑘 ∈ 𝐴0 = sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0), ℝ*, < )) |
15 | fconstmpt 5085 | . . . . . . 7 ⊢ ((𝒫 𝐴 ∩ Fin) × {0}) = (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) | |
16 | 15 | eqcomi 2619 | . . . . . 6 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = ((𝒫 𝐴 ∩ Fin) × {0}) |
17 | 0xr 9965 | . . . . . . . . 9 ⊢ 0 ∈ ℝ* | |
18 | 17 | rgenw 2908 | . . . . . . . 8 ⊢ ∀𝑥 ∈ (𝒫 𝐴 ∩ Fin)0 ∈ ℝ* |
19 | eqid 2610 | . . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) | |
20 | 19 | fnmpt 5933 | . . . . . . . 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 4760 | . . . . . . . . 9 ⊢ ∅ ∈ 𝒫 𝐴 | |
23 | 0fin 8073 | . . . . . . . . 9 ⊢ ∅ ∈ Fin | |
24 | elin 3758 | . . . . . . . . 9 ⊢ (∅ ∈ (𝒫 𝐴 ∩ Fin) ↔ (∅ ∈ 𝒫 𝐴 ∧ ∅ ∈ Fin)) | |
25 | 22, 23, 24 | mpbir2an 957 | . . . . . . . 8 ⊢ ∅ ∈ (𝒫 𝐴 ∩ Fin) |
26 | 25 | ne0ii 3882 | . . . . . . 7 ⊢ (𝒫 𝐴 ∩ Fin) ≠ ∅ |
27 | fconst5 6376 | . . . . . . 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 704 | . . . . . 6 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = ((𝒫 𝐴 ∩ Fin) × {0}) ↔ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = {0}) |
29 | 16, 28 | mpbi 219 | . . . . 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 8235 | . . 3 ⊢ (𝐴 ∈ 𝑉 → sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0), ℝ*, < ) = sup({0}, ℝ*, < )) |
32 | xrltso 11850 | . . . 4 ⊢ < Or ℝ* | |
33 | supsn 8261 | . . . 4 ⊢ (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0) | |
34 | 32, 17, 33 | mp2an 704 | . . 3 ⊢ sup({0}, ℝ*, < ) = 0 |
35 | 31, 34 | syl6eq 2660 | . 2 ⊢ (𝐴 ∈ 𝑉 → sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0), ℝ*, < ) = 0) |
36 | 14, 35 | eqtrd 2644 | 1 ⊢ (𝐴 ∈ 𝑉 → Σ*𝑘 ∈ 𝐴0 = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Ⅎwnfc 2738 ≠ wne 2780 ∀wral 2896 Vcvv 3173 ∩ cin 3539 ∅c0 3874 𝒫 cpw 4108 {csn 4125 ↦ cmpt 4643 Or wor 4958 × cxp 5036 ran crn 5039 Fn wfn 5799 (class class class)co 6549 Fincfn 7841 supcsup 8229 0cc0 9815 +∞cpnf 9950 ℝ*cxr 9952 < clt 9953 [,]cicc 12049 ↾s cress 15696 Σg cgsu 15924 ℝ*𝑠cxrs 15983 Mndcmnd 17117 CMndccmn 18016 Σ*cesum 29416 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-xadd 11823 df-ioo 12050 df-ioc 12051 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-tset 15787 df-ple 15788 df-ds 15791 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-ordt 15984 df-xrs 15985 df-mre 16069 df-mrc 16070 df-acs 16072 df-ps 17023 df-tsr 17024 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-cntz 17573 df-cmn 18018 df-fbas 19564 df-fg 19565 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-ntr 20634 df-nei 20712 df-cn 20841 df-haus 20929 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-tsms 21740 df-esum 29417 |
This theorem is referenced by: esumpad 29444 esumrnmpt2 29457 measvunilem0 29603 ddemeas 29626 |
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