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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumpad | Structured version Visualization version GIF version |
Description: Extend an extended sum by padding outside with zeroes. (Contributed by Thierry Arnoux, 31-May-2020.) |
Ref | Expression |
---|---|
esumpad.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
esumpad.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
esumpad.3 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (0[,]+∞)) |
esumpad.4 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐶 = 0) |
Ref | Expression |
---|---|
esumpad | ⊢ (𝜑 → Σ*𝑘 ∈ (𝐴 ∪ 𝐵)𝐶 = Σ*𝑘 ∈ 𝐴𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1909 | . . 3 ⊢ Ⅎ𝑘𝜑 | |
2 | nfcv 2891 | . . 3 ⊢ Ⅎ𝑘𝐴 | |
3 | nfcv 2891 | . . 3 ⊢ Ⅎ𝑘(𝐵 ∖ 𝐴) | |
4 | esumpad.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
5 | elex 3481 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝜑 → 𝐴 ∈ V) |
7 | esumpad.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
8 | 7 | difexd 5335 | . . 3 ⊢ (𝜑 → (𝐵 ∖ 𝐴) ∈ V) |
9 | disjdif 4475 | . . . 4 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅ | |
10 | 9 | a1i 11 | . . 3 ⊢ (𝜑 → (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅) |
11 | esumpad.3 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (0[,]+∞)) | |
12 | difssd 4131 | . . . . 5 ⊢ (𝜑 → (𝐵 ∖ 𝐴) ⊆ 𝐵) | |
13 | 12 | sselda 3978 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝑘 ∈ 𝐵) |
14 | esumpad.4 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐶 = 0) | |
15 | 0e0iccpnf 13485 | . . . . 5 ⊢ 0 ∈ (0[,]+∞) | |
16 | 14, 15 | eqeltrdi 2833 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ (0[,]+∞)) |
17 | 13, 16 | syldan 589 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 ∈ (0[,]+∞)) |
18 | 1, 2, 3, 6, 8, 10, 11, 17 | esumsplit 33842 | . 2 ⊢ (𝜑 → Σ*𝑘 ∈ (𝐴 ∪ (𝐵 ∖ 𝐴))𝐶 = (Σ*𝑘 ∈ 𝐴𝐶 +𝑒 Σ*𝑘 ∈ (𝐵 ∖ 𝐴)𝐶)) |
19 | undif2 4480 | . . . 4 ⊢ (𝐴 ∪ (𝐵 ∖ 𝐴)) = (𝐴 ∪ 𝐵) | |
20 | esumeq1 33823 | . . . 4 ⊢ ((𝐴 ∪ (𝐵 ∖ 𝐴)) = (𝐴 ∪ 𝐵) → Σ*𝑘 ∈ (𝐴 ∪ (𝐵 ∖ 𝐴))𝐶 = Σ*𝑘 ∈ (𝐴 ∪ 𝐵)𝐶) | |
21 | 19, 20 | ax-mp 5 | . . 3 ⊢ Σ*𝑘 ∈ (𝐴 ∪ (𝐵 ∖ 𝐴))𝐶 = Σ*𝑘 ∈ (𝐴 ∪ 𝐵)𝐶 |
22 | 21 | a1i 11 | . 2 ⊢ (𝜑 → Σ*𝑘 ∈ (𝐴 ∪ (𝐵 ∖ 𝐴))𝐶 = Σ*𝑘 ∈ (𝐴 ∪ 𝐵)𝐶) |
23 | 13, 14 | syldan 589 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 0) |
24 | 23 | ralrimiva 3135 | . . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ (𝐵 ∖ 𝐴)𝐶 = 0) |
25 | 1, 24 | esumeq2d 33826 | . . . . 5 ⊢ (𝜑 → Σ*𝑘 ∈ (𝐵 ∖ 𝐴)𝐶 = Σ*𝑘 ∈ (𝐵 ∖ 𝐴)0) |
26 | 3 | esum0 33838 | . . . . . 6 ⊢ ((𝐵 ∖ 𝐴) ∈ V → Σ*𝑘 ∈ (𝐵 ∖ 𝐴)0 = 0) |
27 | 8, 26 | syl 17 | . . . . 5 ⊢ (𝜑 → Σ*𝑘 ∈ (𝐵 ∖ 𝐴)0 = 0) |
28 | 25, 27 | eqtrd 2765 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ (𝐵 ∖ 𝐴)𝐶 = 0) |
29 | 28 | oveq2d 7439 | . . 3 ⊢ (𝜑 → (Σ*𝑘 ∈ 𝐴𝐶 +𝑒 Σ*𝑘 ∈ (𝐵 ∖ 𝐴)𝐶) = (Σ*𝑘 ∈ 𝐴𝐶 +𝑒 0)) |
30 | iccssxr 13456 | . . . . 5 ⊢ (0[,]+∞) ⊆ ℝ* | |
31 | 11 | ralrimiva 3135 | . . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 ∈ (0[,]+∞)) |
32 | 2 | esumcl 33819 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐶 ∈ (0[,]+∞)) → Σ*𝑘 ∈ 𝐴𝐶 ∈ (0[,]+∞)) |
33 | 4, 31, 32 | syl2anc 582 | . . . . 5 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐶 ∈ (0[,]+∞)) |
34 | 30, 33 | sselid 3976 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐶 ∈ ℝ*) |
35 | xaddrid 13269 | . . . 4 ⊢ (Σ*𝑘 ∈ 𝐴𝐶 ∈ ℝ* → (Σ*𝑘 ∈ 𝐴𝐶 +𝑒 0) = Σ*𝑘 ∈ 𝐴𝐶) | |
36 | 34, 35 | syl 17 | . . 3 ⊢ (𝜑 → (Σ*𝑘 ∈ 𝐴𝐶 +𝑒 0) = Σ*𝑘 ∈ 𝐴𝐶) |
37 | 29, 36 | eqtrd 2765 | . 2 ⊢ (𝜑 → (Σ*𝑘 ∈ 𝐴𝐶 +𝑒 Σ*𝑘 ∈ (𝐵 ∖ 𝐴)𝐶) = Σ*𝑘 ∈ 𝐴𝐶) |
38 | 18, 22, 37 | 3eqtr3d 2773 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ (𝐴 ∪ 𝐵)𝐶 = Σ*𝑘 ∈ 𝐴𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3050 Vcvv 3461 ∖ cdif 3943 ∪ cun 3944 ∩ cin 3945 ∅c0 4324 (class class class)co 7423 0cc0 11154 +∞cpnf 11291 ℝ*cxr 11293 +𝑒 cxad 13139 [,]cicc 13376 Σ*cesum 33816 |
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 2166 ax-ext 2696 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-inf2 9680 ax-cnex 11210 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 ax-pre-mulgt0 11231 ax-pre-sup 11232 ax-addf 11233 ax-mulf 11234 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-se 5637 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-of 7689 df-om 7876 df-1st 8002 df-2nd 8003 df-supp 8174 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-2o 8496 df-er 8733 df-map 8856 df-pm 8857 df-ixp 8926 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fsupp 9402 df-fi 9450 df-sup 9481 df-inf 9482 df-oi 9549 df-card 9978 df-pnf 11296 df-mnf 11297 df-xr 11298 df-ltxr 11299 df-le 11300 df-sub 11492 df-neg 11493 df-div 11918 df-nn 12260 df-2 12322 df-3 12323 df-4 12324 df-5 12325 df-6 12326 df-7 12327 df-8 12328 df-9 12329 df-n0 12520 df-z 12606 df-dec 12725 df-uz 12870 df-q 12980 df-rp 13024 df-xneg 13141 df-xadd 13142 df-xmul 13143 df-ioo 13377 df-ioc 13378 df-ico 13379 df-icc 13380 df-fz 13534 df-fzo 13677 df-fl 13807 df-mod 13885 df-seq 14017 df-exp 14077 df-fac 14286 df-bc 14315 df-hash 14343 df-shft 15067 df-cj 15099 df-re 15100 df-im 15101 df-sqrt 15235 df-abs 15236 df-limsup 15468 df-clim 15485 df-rlim 15486 df-sum 15686 df-ef 16064 df-sin 16066 df-cos 16067 df-pi 16069 df-struct 17144 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-mulr 17275 df-starv 17276 df-sca 17277 df-vsca 17278 df-ip 17279 df-tset 17280 df-ple 17281 df-ds 17283 df-unif 17284 df-hom 17285 df-cco 17286 df-rest 17432 df-topn 17433 df-0g 17451 df-gsum 17452 df-topgen 17453 df-pt 17454 df-prds 17457 df-ordt 17511 df-xrs 17512 df-qtop 17517 df-imas 17518 df-xps 17520 df-mre 17594 df-mrc 17595 df-acs 17597 df-ps 18586 df-tsr 18587 df-plusf 18627 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-mhm 18768 df-submnd 18769 df-grp 18926 df-minusg 18927 df-sbg 18928 df-mulg 19057 df-subg 19112 df-cntz 19306 df-cmn 19775 df-abl 19776 df-mgp 20113 df-rng 20131 df-ur 20160 df-ring 20213 df-cring 20214 df-subrng 20523 df-subrg 20548 df-abv 20737 df-lmod 20785 df-scaf 20786 df-sra 21098 df-rgmod 21099 df-psmet 21327 df-xmet 21328 df-met 21329 df-bl 21330 df-mopn 21331 df-fbas 21332 df-fg 21333 df-cnfld 21336 df-top 22879 df-topon 22896 df-topsp 22918 df-bases 22932 df-cld 23006 df-ntr 23007 df-cls 23008 df-nei 23085 df-lp 23123 df-perf 23124 df-cn 23214 df-cnp 23215 df-haus 23302 df-tx 23549 df-hmeo 23742 df-fil 23833 df-fm 23925 df-flim 23926 df-flf 23927 df-tmd 24059 df-tgp 24060 df-tsms 24114 df-trg 24147 df-xms 24309 df-ms 24310 df-tms 24311 df-nm 24574 df-ngp 24575 df-nrg 24577 df-nlm 24578 df-ii 24880 df-cncf 24881 df-limc 25878 df-dv 25879 df-log 26575 df-esum 33817 |
This theorem is referenced by: esumpad2 33845 carsggect 34108 |
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