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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumpr2 | Structured version Visualization version GIF version |
Description: Extended sum over a pair, with a relaxed condition compared to esumpr 30669. (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[,]+∞)) |
esumpr2.1 | ⊢ (𝜑 → (𝐴 = 𝐵 → (𝐷 = 0 ∨ 𝐷 = +∞))) |
Ref | Expression |
---|---|
esumpr2 | ⊢ (𝜑 → Σ*𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 +𝑒 𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵) | |
2 | dfsn2 4412 | . . . . . 6 ⊢ {𝐴} = {𝐴, 𝐴} | |
3 | preq2 4489 | . . . . . 6 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐴} = {𝐴, 𝐵}) | |
4 | 2, 3 | syl5req 2874 | . . . . 5 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴}) |
5 | esumeq1 30637 | . . . . 5 ⊢ ({𝐴, 𝐵} = {𝐴} → Σ*𝑘 ∈ {𝐴, 𝐵}𝐶 = Σ*𝑘 ∈ {𝐴}𝐶) | |
6 | 1, 4, 5 | 3syl 18 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → Σ*𝑘 ∈ {𝐴, 𝐵}𝐶 = Σ*𝑘 ∈ {𝐴}𝐶) |
7 | esumpr.1 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝐶 = 𝐷) | |
8 | esumpr.3 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
9 | esumpr.5 | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (0[,]+∞)) | |
10 | 7, 8, 9 | esumsn 30668 | . . . . 5 ⊢ (𝜑 → Σ*𝑘 ∈ {𝐴}𝐶 = 𝐷) |
11 | 10 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → Σ*𝑘 ∈ {𝐴}𝐶 = 𝐷) |
12 | 6, 11 | eqtrd 2861 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → Σ*𝑘 ∈ {𝐴, 𝐵}𝐶 = 𝐷) |
13 | esumpr2.1 | . . . . 5 ⊢ (𝜑 → (𝐴 = 𝐵 → (𝐷 = 0 ∨ 𝐷 = +∞))) | |
14 | oveq2 6918 | . . . . . . 7 ⊢ (𝐷 = 0 → (𝐷 +𝑒 𝐷) = (𝐷 +𝑒 0)) | |
15 | 0xr 10410 | . . . . . . . . 9 ⊢ 0 ∈ ℝ* | |
16 | eleq1 2894 | . . . . . . . . 9 ⊢ (𝐷 = 0 → (𝐷 ∈ ℝ* ↔ 0 ∈ ℝ*)) | |
17 | 15, 16 | mpbiri 250 | . . . . . . . 8 ⊢ (𝐷 = 0 → 𝐷 ∈ ℝ*) |
18 | xaddid1 12367 | . . . . . . . 8 ⊢ (𝐷 ∈ ℝ* → (𝐷 +𝑒 0) = 𝐷) | |
19 | 17, 18 | syl 17 | . . . . . . 7 ⊢ (𝐷 = 0 → (𝐷 +𝑒 0) = 𝐷) |
20 | 14, 19 | eqtrd 2861 | . . . . . 6 ⊢ (𝐷 = 0 → (𝐷 +𝑒 𝐷) = 𝐷) |
21 | pnfxr 10417 | . . . . . . . . 9 ⊢ +∞ ∈ ℝ* | |
22 | eleq1 2894 | . . . . . . . . 9 ⊢ (𝐷 = +∞ → (𝐷 ∈ ℝ* ↔ +∞ ∈ ℝ*)) | |
23 | 21, 22 | mpbiri 250 | . . . . . . . 8 ⊢ (𝐷 = +∞ → 𝐷 ∈ ℝ*) |
24 | pnfnemnf 10419 | . . . . . . . . 9 ⊢ +∞ ≠ -∞ | |
25 | neeq1 3061 | . . . . . . . . 9 ⊢ (𝐷 = +∞ → (𝐷 ≠ -∞ ↔ +∞ ≠ -∞)) | |
26 | 24, 25 | mpbiri 250 | . . . . . . . 8 ⊢ (𝐷 = +∞ → 𝐷 ≠ -∞) |
27 | xaddpnf1 12352 | . . . . . . . 8 ⊢ ((𝐷 ∈ ℝ* ∧ 𝐷 ≠ -∞) → (𝐷 +𝑒 +∞) = +∞) | |
28 | 23, 26, 27 | syl2anc 579 | . . . . . . 7 ⊢ (𝐷 = +∞ → (𝐷 +𝑒 +∞) = +∞) |
29 | oveq2 6918 | . . . . . . 7 ⊢ (𝐷 = +∞ → (𝐷 +𝑒 𝐷) = (𝐷 +𝑒 +∞)) | |
30 | id 22 | . . . . . . 7 ⊢ (𝐷 = +∞ → 𝐷 = +∞) | |
31 | 28, 29, 30 | 3eqtr4d 2871 | . . . . . 6 ⊢ (𝐷 = +∞ → (𝐷 +𝑒 𝐷) = 𝐷) |
32 | 20, 31 | jaoi 888 | . . . . 5 ⊢ ((𝐷 = 0 ∨ 𝐷 = +∞) → (𝐷 +𝑒 𝐷) = 𝐷) |
33 | 13, 32 | syl6 35 | . . . 4 ⊢ (𝜑 → (𝐴 = 𝐵 → (𝐷 +𝑒 𝐷) = 𝐷)) |
34 | 33 | imp 397 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐷 +𝑒 𝐷) = 𝐷) |
35 | simpll 783 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑘 = 𝐵) → 𝜑) | |
36 | eqeq2 2836 | . . . . . . . . . 10 ⊢ (𝐴 = 𝐵 → (𝑘 = 𝐴 ↔ 𝑘 = 𝐵)) | |
37 | 36 | biimprd 240 | . . . . . . . . 9 ⊢ (𝐴 = 𝐵 → (𝑘 = 𝐵 → 𝑘 = 𝐴)) |
38 | 1, 37 | syl 17 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝑘 = 𝐵 → 𝑘 = 𝐴)) |
39 | 38 | imp 397 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑘 = 𝐵) → 𝑘 = 𝐴) |
40 | 35, 39, 7 | syl2anc 579 | . . . . . 6 ⊢ (((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑘 = 𝐵) → 𝐶 = 𝐷) |
41 | esumpr.4 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
42 | 41 | adantr 474 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐵 ∈ 𝑊) |
43 | 9 | adantr 474 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐷 ∈ (0[,]+∞)) |
44 | 40, 42, 43 | esumsn 30668 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → Σ*𝑘 ∈ {𝐵}𝐶 = 𝐷) |
45 | esumpr.2 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐶 = 𝐸) | |
46 | esumpr.6 | . . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ (0[,]+∞)) | |
47 | 45, 41, 46 | esumsn 30668 | . . . . . 6 ⊢ (𝜑 → Σ*𝑘 ∈ {𝐵}𝐶 = 𝐸) |
48 | 47 | adantr 474 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → Σ*𝑘 ∈ {𝐵}𝐶 = 𝐸) |
49 | 44, 48 | eqtr3d 2863 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐷 = 𝐸) |
50 | 49 | oveq2d 6926 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐷 +𝑒 𝐷) = (𝐷 +𝑒 𝐸)) |
51 | 12, 34, 50 | 3eqtr2d 2867 | . 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → Σ*𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 +𝑒 𝐸)) |
52 | 7 | adantlr 706 | . . 3 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ 𝑘 = 𝐴) → 𝐶 = 𝐷) |
53 | 45 | adantlr 706 | . . 3 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ 𝑘 = 𝐵) → 𝐶 = 𝐸) |
54 | 8 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ 𝑉) |
55 | 41 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ 𝑊) |
56 | 9 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐷 ∈ (0[,]+∞)) |
57 | 46 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐸 ∈ (0[,]+∞)) |
58 | simpr 479 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐴 ≠ 𝐵) | |
59 | 52, 53, 54, 55, 56, 57, 58 | esumpr 30669 | . 2 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → Σ*𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 +𝑒 𝐸)) |
60 | 51, 59 | pm2.61dane 3086 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 +𝑒 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∨ wo 878 = wceq 1656 ∈ wcel 2164 ≠ wne 2999 {csn 4399 {cpr 4401 (class class class)co 6910 0cc0 10259 +∞cpnf 10395 -∞cmnf 10396 ℝ*cxr 10397 +𝑒 cxad 12237 [,]cicc 12473 Σ*cesum 30630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-inf2 8822 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-pre-sup 10337 ax-addf 10338 ax-mulf 10339 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-fal 1670 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-iin 4745 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-se 5306 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-isom 6136 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-of 7162 df-om 7332 df-1st 7433 df-2nd 7434 df-supp 7565 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-2o 7832 df-oadd 7835 df-er 8014 df-map 8129 df-pm 8130 df-ixp 8182 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-fsupp 8551 df-fi 8592 df-sup 8623 df-inf 8624 df-oi 8691 df-card 9085 df-cda 9312 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-div 11017 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-9 11428 df-n0 11626 df-z 11712 df-dec 11829 df-uz 11976 df-q 12079 df-rp 12120 df-xneg 12239 df-xadd 12240 df-xmul 12241 df-ioo 12474 df-ioc 12475 df-ico 12476 df-icc 12477 df-fz 12627 df-fzo 12768 df-fl 12895 df-mod 12971 df-seq 13103 df-exp 13162 df-fac 13361 df-bc 13390 df-hash 13418 df-shft 14191 df-cj 14223 df-re 14224 df-im 14225 df-sqrt 14359 df-abs 14360 df-limsup 14586 df-clim 14603 df-rlim 14604 df-sum 14801 df-ef 15177 df-sin 15179 df-cos 15180 df-pi 15182 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-mulr 16326 df-starv 16327 df-sca 16328 df-vsca 16329 df-ip 16330 df-tset 16331 df-ple 16332 df-ds 16334 df-unif 16335 df-hom 16336 df-cco 16337 df-rest 16443 df-topn 16444 df-0g 16462 df-gsum 16463 df-topgen 16464 df-pt 16465 df-prds 16468 df-ordt 16521 df-xrs 16522 df-qtop 16527 df-imas 16528 df-xps 16530 df-mre 16606 df-mrc 16607 df-acs 16609 df-ps 17560 df-tsr 17561 df-plusf 17601 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-mhm 17695 df-submnd 17696 df-grp 17786 df-minusg 17787 df-sbg 17788 df-mulg 17902 df-subg 17949 df-cntz 18107 df-cmn 18555 df-abl 18556 df-mgp 18851 df-ur 18863 df-ring 18910 df-cring 18911 df-subrg 19141 df-abv 19180 df-lmod 19228 df-scaf 19229 df-sra 19540 df-rgmod 19541 df-psmet 20105 df-xmet 20106 df-met 20107 df-bl 20108 df-mopn 20109 df-fbas 20110 df-fg 20111 df-cnfld 20114 df-top 21076 df-topon 21093 df-topsp 21115 df-bases 21128 df-cld 21201 df-ntr 21202 df-cls 21203 df-nei 21280 df-lp 21318 df-perf 21319 df-cn 21409 df-cnp 21410 df-haus 21497 df-tx 21743 df-hmeo 21936 df-fil 22027 df-fm 22119 df-flim 22120 df-flf 22121 df-tmd 22253 df-tgp 22254 df-tsms 22307 df-trg 22340 df-xms 22502 df-ms 22503 df-tms 22504 df-nm 22764 df-ngp 22765 df-nrg 22767 df-nlm 22768 df-ii 23057 df-cncf 23058 df-limc 24036 df-dv 24037 df-log 24709 df-esum 30631 |
This theorem is referenced by: measxun2 30814 measssd 30819 carsgclctun 30924 |
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