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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumpr | Structured version Visualization version GIF version |
Description: Extended sum over a pair. (Contributed by Thierry Arnoux, 2-Jan-2017.) |
Ref | Expression |
---|---|
esumpr.1 | ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝐶 = 𝐷) |
esumpr.2 | ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐶 = 𝐸) |
esumpr.3 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
esumpr.4 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
esumpr.5 | ⊢ (𝜑 → 𝐷 ∈ (0[,]+∞)) |
esumpr.6 | ⊢ (𝜑 → 𝐸 ∈ (0[,]+∞)) |
esumpr.7 | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
Ref | Expression |
---|---|
esumpr | ⊢ (𝜑 → Σ*𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 +𝑒 𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr 4594 | . . 3 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
2 | esumeq1 32673 | . . 3 ⊢ ({𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) → Σ*𝑘 ∈ {𝐴, 𝐵}𝐶 = Σ*𝑘 ∈ ({𝐴} ∪ {𝐵})𝐶) | |
3 | 1, 2 | mp1i 13 | . 2 ⊢ (𝜑 → Σ*𝑘 ∈ {𝐴, 𝐵}𝐶 = Σ*𝑘 ∈ ({𝐴} ∪ {𝐵})𝐶) |
4 | nfv 1918 | . . 3 ⊢ Ⅎ𝑘𝜑 | |
5 | nfcv 2908 | . . 3 ⊢ Ⅎ𝑘{𝐴} | |
6 | nfcv 2908 | . . 3 ⊢ Ⅎ𝑘{𝐵} | |
7 | snex 5393 | . . . 4 ⊢ {𝐴} ∈ V | |
8 | 7 | a1i 11 | . . 3 ⊢ (𝜑 → {𝐴} ∈ V) |
9 | snex 5393 | . . . 4 ⊢ {𝐵} ∈ V | |
10 | 9 | a1i 11 | . . 3 ⊢ (𝜑 → {𝐵} ∈ V) |
11 | esumpr.7 | . . . 4 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | |
12 | disjsn2 4678 | . . . 4 ⊢ (𝐴 ≠ 𝐵 → ({𝐴} ∩ {𝐵}) = ∅) | |
13 | 11, 12 | syl 17 | . . 3 ⊢ (𝜑 → ({𝐴} ∩ {𝐵}) = ∅) |
14 | elsni 4608 | . . . . 5 ⊢ (𝑘 ∈ {𝐴} → 𝑘 = 𝐴) | |
15 | esumpr.1 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝐶 = 𝐷) | |
16 | 14, 15 | sylan2 594 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝐴}) → 𝐶 = 𝐷) |
17 | esumpr.5 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ (0[,]+∞)) | |
18 | 17 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝐴}) → 𝐷 ∈ (0[,]+∞)) |
19 | 16, 18 | eqeltrd 2838 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝐴}) → 𝐶 ∈ (0[,]+∞)) |
20 | elsni 4608 | . . . . 5 ⊢ (𝑘 ∈ {𝐵} → 𝑘 = 𝐵) | |
21 | esumpr.2 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐶 = 𝐸) | |
22 | 20, 21 | sylan2 594 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝐵}) → 𝐶 = 𝐸) |
23 | esumpr.6 | . . . . 5 ⊢ (𝜑 → 𝐸 ∈ (0[,]+∞)) | |
24 | 23 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝐵}) → 𝐸 ∈ (0[,]+∞)) |
25 | 22, 24 | eqeltrd 2838 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝐵}) → 𝐶 ∈ (0[,]+∞)) |
26 | 4, 5, 6, 8, 10, 13, 19, 25 | esumsplit 32692 | . 2 ⊢ (𝜑 → Σ*𝑘 ∈ ({𝐴} ∪ {𝐵})𝐶 = (Σ*𝑘 ∈ {𝐴}𝐶 +𝑒 Σ*𝑘 ∈ {𝐵}𝐶)) |
27 | esumpr.3 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
28 | 15, 27, 17 | esumsn 32704 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ {𝐴}𝐶 = 𝐷) |
29 | esumpr.4 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
30 | 21, 29, 23 | esumsn 32704 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ {𝐵}𝐶 = 𝐸) |
31 | 28, 30 | oveq12d 7380 | . 2 ⊢ (𝜑 → (Σ*𝑘 ∈ {𝐴}𝐶 +𝑒 Σ*𝑘 ∈ {𝐵}𝐶) = (𝐷 +𝑒 𝐸)) |
32 | 3, 26, 31 | 3eqtrd 2781 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 +𝑒 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2944 Vcvv 3448 ∪ cun 3913 ∩ cin 3914 ∅c0 4287 {csn 4591 {cpr 4593 (class class class)co 7362 0cc0 11058 +∞cpnf 11193 +𝑒 cxad 13038 [,]cicc 13274 Σ*cesum 32666 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 ax-addf 11137 ax-mulf 11138 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-iin 4962 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7622 df-om 7808 df-1st 7926 df-2nd 7927 df-supp 8098 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-2o 8418 df-er 8655 df-map 8774 df-pm 8775 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9313 df-fi 9354 df-sup 9385 df-inf 9386 df-oi 9453 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-z 12507 df-dec 12626 df-uz 12771 df-q 12881 df-rp 12923 df-xneg 13040 df-xadd 13041 df-xmul 13042 df-ioo 13275 df-ioc 13276 df-ico 13277 df-icc 13278 df-fz 13432 df-fzo 13575 df-fl 13704 df-mod 13782 df-seq 13914 df-exp 13975 df-fac 14181 df-bc 14210 df-hash 14238 df-shft 14959 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-limsup 15360 df-clim 15377 df-rlim 15378 df-sum 15578 df-ef 15957 df-sin 15959 df-cos 15960 df-pi 15962 df-struct 17026 df-sets 17043 df-slot 17061 df-ndx 17073 df-base 17091 df-ress 17120 df-plusg 17153 df-mulr 17154 df-starv 17155 df-sca 17156 df-vsca 17157 df-ip 17158 df-tset 17159 df-ple 17160 df-ds 17162 df-unif 17163 df-hom 17164 df-cco 17165 df-rest 17311 df-topn 17312 df-0g 17330 df-gsum 17331 df-topgen 17332 df-pt 17333 df-prds 17336 df-ordt 17390 df-xrs 17391 df-qtop 17396 df-imas 17397 df-xps 17399 df-mre 17473 df-mrc 17474 df-acs 17476 df-ps 18462 df-tsr 18463 df-plusf 18503 df-mgm 18504 df-sgrp 18553 df-mnd 18564 df-mhm 18608 df-submnd 18609 df-grp 18758 df-minusg 18759 df-sbg 18760 df-mulg 18880 df-subg 18932 df-cntz 19104 df-cmn 19571 df-abl 19572 df-mgp 19904 df-ur 19921 df-ring 19973 df-cring 19974 df-subrg 20236 df-abv 20292 df-lmod 20340 df-scaf 20341 df-sra 20649 df-rgmod 20650 df-psmet 20804 df-xmet 20805 df-met 20806 df-bl 20807 df-mopn 20808 df-fbas 20809 df-fg 20810 df-cnfld 20813 df-top 22259 df-topon 22276 df-topsp 22298 df-bases 22312 df-cld 22386 df-ntr 22387 df-cls 22388 df-nei 22465 df-lp 22503 df-perf 22504 df-cn 22594 df-cnp 22595 df-haus 22682 df-tx 22929 df-hmeo 23122 df-fil 23213 df-fm 23305 df-flim 23306 df-flf 23307 df-tmd 23439 df-tgp 23440 df-tsms 23494 df-trg 23527 df-xms 23689 df-ms 23690 df-tms 23691 df-nm 23954 df-ngp 23955 df-nrg 23957 df-nlm 23958 df-ii 24256 df-cncf 24257 df-limc 25246 df-dv 25247 df-log 25928 df-esum 32667 |
This theorem is referenced by: esumpr2 32706 carsgsigalem 32955 pmeasmono 32964 probun 33059 |
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