![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
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 6032 | . . . . . 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 7420 | . . . 4 ⊢ (𝜑 → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums ((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴)) = ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶))) |
7 | xrge0base 32687 | . . . . 5 ⊢ (0[,]+∞) = (Base‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
8 | xrge00 32688 | . . . . 5 ⊢ 0 = (0g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
9 | xrge0cmn 21298 | . . . . . 6 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd | |
10 | 9 | a1i 11 | . . . . 5 ⊢ (𝜑 → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd) |
11 | xrge0tps 33452 | . . . . . 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 2897 | . . . . . 6 ⊢ Ⅎ𝑘(0[,]+∞) | |
16 | esumss.3 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ (0[,]+∞)) | |
17 | eqid 2726 | . . . . . 6 ⊢ (𝑘 ∈ 𝐵 ↦ 𝐶) = (𝑘 ∈ 𝐵 ↦ 𝐶) | |
18 | 14, 2, 15, 16, 17 | fmptdF 32386 | . . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐵 ↦ 𝐶):𝐵⟶(0[,]+∞)) |
19 | esumss.4 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 0) | |
20 | 14, 2, 3, 19, 13 | suppss2f 32368 | . . . . 5 ⊢ (𝜑 → ((𝑘 ∈ 𝐵 ↦ 𝐶) supp 0) ⊆ 𝐴) |
21 | 7, 8, 10, 12, 13, 18, 20 | tsmsres 23999 | . . . 4 ⊢ (𝜑 → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums ((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴)) = ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐶))) |
22 | 6, 21 | eqtr3d 2768 | . . 3 ⊢ (𝜑 → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)) = ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐶))) |
23 | 22 | unieqd 4915 | . 2 ⊢ (𝜑 → ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)) = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐶))) |
24 | df-esum 33556 | . 2 ⊢ Σ*𝑘 ∈ 𝐴𝐶 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐶)) | |
25 | df-esum 33556 | . 2 ⊢ Σ*𝑘 ∈ 𝐵𝐶 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐵 ↦ 𝐶)) | |
26 | 23, 24, 25 | 3eqtr4g 2791 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐶 = Σ*𝑘 ∈ 𝐵𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 Ⅎwnf 1777 ∈ wcel 2098 Ⅎwnfc 2877 ∖ cdif 3940 ⊆ wss 3943 ∪ cuni 4902 ↦ cmpt 5224 ↾ cres 5671 (class class class)co 7404 0cc0 11109 +∞cpnf 11246 [,]cicc 13330 ↾s cress 17180 ℝ*𝑠cxrs 17453 CMndccmn 19698 TopSpctps 22785 tsums ctsu 23981 Σ*cesum 33555 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8144 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-fi 9405 df-oi 9504 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-xadd 13096 df-icc 13334 df-fz 13488 df-fzo 13631 df-seq 13970 df-hash 14294 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-tset 17223 df-ple 17224 df-ds 17226 df-rest 17375 df-topn 17376 df-0g 17394 df-gsum 17395 df-topgen 17396 df-ordt 17454 df-xrs 17455 df-ps 18529 df-tsr 18530 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-cntz 19231 df-cmn 19700 df-fbas 21233 df-fg 21234 df-top 22747 df-topon 22764 df-topsp 22786 df-bases 22800 df-ntr 22875 df-nei 22953 df-fil 23701 df-fm 23793 df-flim 23794 df-flf 23795 df-tsms 23982 df-esum 33556 |
This theorem is referenced by: esumpinfval 33601 |
Copyright terms: Public domain | W3C validator |