Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumss | Structured version Visualization version GIF version |
Description: Change the index set to a subset by adding zeroes. (Contributed by Thierry Arnoux, 19-Jun-2017.) |
Ref | Expression |
---|---|
esumss.p | ⊢ Ⅎ𝑘𝜑 |
esumss.a | ⊢ Ⅎ𝑘𝐴 |
esumss.b | ⊢ Ⅎ𝑘𝐵 |
esumss.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
esumss.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
esumss.3 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ (0[,]+∞)) |
esumss.4 | ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 0) |
Ref | Expression |
---|---|
esumss | ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐶 = Σ*𝑘 ∈ 𝐵𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | esumss.1 | . . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
2 | esumss.b | . . . . . . 7 ⊢ Ⅎ𝑘𝐵 | |
3 | esumss.a | . . . . . . 7 ⊢ Ⅎ𝑘𝐴 | |
4 | 2, 3 | resmptf 5907 | . . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → ((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴) = (𝑘 ∈ 𝐴 ↦ 𝐶)) |
5 | 1, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → ((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴) = (𝑘 ∈ 𝐴 ↦ 𝐶)) |
6 | 5 | oveq2d 7172 | . . . 4 ⊢ (𝜑 → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums ((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴)) = ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶))) |
7 | xrge0base 30672 | . . . . 5 ⊢ (0[,]+∞) = (Base‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
8 | xrge00 30673 | . . . . 5 ⊢ 0 = (0g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
9 | xrge0cmn 20587 | . . . . . 6 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd | |
10 | 9 | a1i 11 | . . . . 5 ⊢ (𝜑 → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd) |
11 | xrge0tps 31185 | . . . . . 6 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopSp | |
12 | 11 | a1i 11 | . . . . 5 ⊢ (𝜑 → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopSp) |
13 | esumss.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
14 | esumss.p | . . . . . 6 ⊢ Ⅎ𝑘𝜑 | |
15 | nfcv 2977 | . . . . . 6 ⊢ Ⅎ𝑘(0[,]+∞) | |
16 | esumss.3 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ (0[,]+∞)) | |
17 | eqid 2821 | . . . . . 6 ⊢ (𝑘 ∈ 𝐵 ↦ 𝐶) = (𝑘 ∈ 𝐵 ↦ 𝐶) | |
18 | 14, 2, 15, 16, 17 | fmptdF 30401 | . . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐵 ↦ 𝐶):𝐵⟶(0[,]+∞)) |
19 | esumss.4 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 0) | |
20 | 14, 2, 3, 19, 13 | suppss2f 30384 | . . . . 5 ⊢ (𝜑 → ((𝑘 ∈ 𝐵 ↦ 𝐶) supp 0) ⊆ 𝐴) |
21 | 7, 8, 10, 12, 13, 18, 20 | tsmsres 22752 | . . . 4 ⊢ (𝜑 → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums ((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴)) = ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐶))) |
22 | 6, 21 | eqtr3d 2858 | . . 3 ⊢ (𝜑 → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)) = ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐶))) |
23 | 22 | unieqd 4852 | . 2 ⊢ (𝜑 → ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)) = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐶))) |
24 | df-esum 31287 | . 2 ⊢ Σ*𝑘 ∈ 𝐴𝐶 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)) | |
25 | df-esum 31287 | . 2 ⊢ Σ*𝑘 ∈ 𝐵𝐶 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐶)) | |
26 | 23, 24, 25 | 3eqtr4g 2881 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐶 = Σ*𝑘 ∈ 𝐵𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 Ⅎwnf 1784 ∈ wcel 2114 Ⅎwnfc 2961 ∖ cdif 3933 ⊆ wss 3936 ∪ cuni 4838 ↦ cmpt 5146 ↾ cres 5557 (class class class)co 7156 0cc0 10537 +∞cpnf 10672 [,]cicc 12742 ↾s cress 16484 ℝ*𝑠cxrs 16773 CMndccmn 18906 TopSpctps 21540 tsums ctsu 22734 Σ*cesum 31286 |
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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-xadd 12509 df-icc 12746 df-fz 12894 df-fzo 13035 df-seq 13371 df-hash 13692 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-tset 16584 df-ple 16585 df-ds 16587 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-ordt 16774 df-xrs 16775 df-ps 17810 df-tsr 17811 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-cntz 18447 df-cmn 18908 df-fbas 20542 df-fg 20543 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-ntr 21628 df-nei 21706 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-tsms 22735 df-esum 31287 |
This theorem is referenced by: esumpinfval 31332 |
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