Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
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 31327. (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 487 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵) | |
2 | dfsn2 4582 | . . . . . 6 ⊢ {𝐴} = {𝐴, 𝐴} | |
3 | preq2 4672 | . . . . . 6 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐴} = {𝐴, 𝐵}) | |
4 | 2, 3 | syl5req 2871 | . . . . 5 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴}) |
5 | esumeq1 31295 | . . . . 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 31326 | . . . . 5 ⊢ (𝜑 → Σ*𝑘 ∈ {𝐴}𝐶 = 𝐷) |
11 | 10 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → Σ*𝑘 ∈ {𝐴}𝐶 = 𝐷) |
12 | 6, 11 | eqtrd 2858 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → Σ*𝑘 ∈ {𝐴, 𝐵}𝐶 = 𝐷) |
13 | esumpr2.1 | . . . . 5 ⊢ (𝜑 → (𝐴 = 𝐵 → (𝐷 = 0 ∨ 𝐷 = +∞))) | |
14 | oveq2 7166 | . . . . . . 7 ⊢ (𝐷 = 0 → (𝐷 +𝑒 𝐷) = (𝐷 +𝑒 0)) | |
15 | 0xr 10690 | . . . . . . . . 9 ⊢ 0 ∈ ℝ* | |
16 | eleq1 2902 | . . . . . . . . 9 ⊢ (𝐷 = 0 → (𝐷 ∈ ℝ* ↔ 0 ∈ ℝ*)) | |
17 | 15, 16 | mpbiri 260 | . . . . . . . 8 ⊢ (𝐷 = 0 → 𝐷 ∈ ℝ*) |
18 | xaddid1 12637 | . . . . . . . 8 ⊢ (𝐷 ∈ ℝ* → (𝐷 +𝑒 0) = 𝐷) | |
19 | 17, 18 | syl 17 | . . . . . . 7 ⊢ (𝐷 = 0 → (𝐷 +𝑒 0) = 𝐷) |
20 | 14, 19 | eqtrd 2858 | . . . . . 6 ⊢ (𝐷 = 0 → (𝐷 +𝑒 𝐷) = 𝐷) |
21 | pnfxr 10697 | . . . . . . . . 9 ⊢ +∞ ∈ ℝ* | |
22 | eleq1 2902 | . . . . . . . . 9 ⊢ (𝐷 = +∞ → (𝐷 ∈ ℝ* ↔ +∞ ∈ ℝ*)) | |
23 | 21, 22 | mpbiri 260 | . . . . . . . 8 ⊢ (𝐷 = +∞ → 𝐷 ∈ ℝ*) |
24 | pnfnemnf 10698 | . . . . . . . . 9 ⊢ +∞ ≠ -∞ | |
25 | neeq1 3080 | . . . . . . . . 9 ⊢ (𝐷 = +∞ → (𝐷 ≠ -∞ ↔ +∞ ≠ -∞)) | |
26 | 24, 25 | mpbiri 260 | . . . . . . . 8 ⊢ (𝐷 = +∞ → 𝐷 ≠ -∞) |
27 | xaddpnf1 12622 | . . . . . . . 8 ⊢ ((𝐷 ∈ ℝ* ∧ 𝐷 ≠ -∞) → (𝐷 +𝑒 +∞) = +∞) | |
28 | 23, 26, 27 | syl2anc 586 | . . . . . . 7 ⊢ (𝐷 = +∞ → (𝐷 +𝑒 +∞) = +∞) |
29 | oveq2 7166 | . . . . . . 7 ⊢ (𝐷 = +∞ → (𝐷 +𝑒 𝐷) = (𝐷 +𝑒 +∞)) | |
30 | id 22 | . . . . . . 7 ⊢ (𝐷 = +∞ → 𝐷 = +∞) | |
31 | 28, 29, 30 | 3eqtr4d 2868 | . . . . . 6 ⊢ (𝐷 = +∞ → (𝐷 +𝑒 𝐷) = 𝐷) |
32 | 20, 31 | jaoi 853 | . . . . 5 ⊢ ((𝐷 = 0 ∨ 𝐷 = +∞) → (𝐷 +𝑒 𝐷) = 𝐷) |
33 | 13, 32 | syl6 35 | . . . 4 ⊢ (𝜑 → (𝐴 = 𝐵 → (𝐷 +𝑒 𝐷) = 𝐷)) |
34 | 33 | imp 409 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐷 +𝑒 𝐷) = 𝐷) |
35 | simpll 765 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑘 = 𝐵) → 𝜑) | |
36 | eqeq2 2835 | . . . . . . . . . 10 ⊢ (𝐴 = 𝐵 → (𝑘 = 𝐴 ↔ 𝑘 = 𝐵)) | |
37 | 36 | biimprd 250 | . . . . . . . . 9 ⊢ (𝐴 = 𝐵 → (𝑘 = 𝐵 → 𝑘 = 𝐴)) |
38 | 1, 37 | syl 17 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝑘 = 𝐵 → 𝑘 = 𝐴)) |
39 | 38 | imp 409 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑘 = 𝐵) → 𝑘 = 𝐴) |
40 | 35, 39, 7 | syl2anc 586 | . . . . . 6 ⊢ (((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑘 = 𝐵) → 𝐶 = 𝐷) |
41 | esumpr.4 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
42 | 41 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐵 ∈ 𝑊) |
43 | 9 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐷 ∈ (0[,]+∞)) |
44 | 40, 42, 43 | esumsn 31326 | . . . . 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 31326 | . . . . . 6 ⊢ (𝜑 → Σ*𝑘 ∈ {𝐵}𝐶 = 𝐸) |
48 | 47 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → Σ*𝑘 ∈ {𝐵}𝐶 = 𝐸) |
49 | 44, 48 | eqtr3d 2860 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐷 = 𝐸) |
50 | 49 | oveq2d 7174 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐷 +𝑒 𝐷) = (𝐷 +𝑒 𝐸)) |
51 | 12, 34, 50 | 3eqtr2d 2864 | . 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → Σ*𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 +𝑒 𝐸)) |
52 | 7 | adantlr 713 | . . 3 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ 𝑘 = 𝐴) → 𝐶 = 𝐷) |
53 | 45 | adantlr 713 | . . 3 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ 𝑘 = 𝐵) → 𝐶 = 𝐸) |
54 | 8 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ 𝑉) |
55 | 41 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ 𝑊) |
56 | 9 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐷 ∈ (0[,]+∞)) |
57 | 46 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐸 ∈ (0[,]+∞)) |
58 | simpr 487 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐴 ≠ 𝐵) | |
59 | 52, 53, 54, 55, 56, 57, 58 | esumpr 31327 | . 2 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → Σ*𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 +𝑒 𝐸)) |
60 | 51, 59 | pm2.61dane 3106 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 +𝑒 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 {csn 4569 {cpr 4571 (class class class)co 7158 0cc0 10539 +∞cpnf 10674 -∞cmnf 10675 ℝ*cxr 10676 +𝑒 cxad 12508 [,]cicc 12744 Σ*cesum 31288 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ioc 12746 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-fac 13637 df-bc 13666 df-hash 13694 df-shft 14428 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-limsup 14830 df-clim 14847 df-rlim 14848 df-sum 15045 df-ef 15423 df-sin 15425 df-cos 15426 df-pi 15428 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-hom 16591 df-cco 16592 df-rest 16698 df-topn 16699 df-0g 16717 df-gsum 16718 df-topgen 16719 df-pt 16720 df-prds 16723 df-ordt 16776 df-xrs 16777 df-qtop 16782 df-imas 16783 df-xps 16785 df-mre 16859 df-mrc 16860 df-acs 16862 df-ps 17812 df-tsr 17813 df-plusf 17853 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-submnd 17959 df-grp 18108 df-minusg 18109 df-sbg 18110 df-mulg 18227 df-subg 18278 df-cntz 18449 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-cring 19302 df-subrg 19535 df-abv 19590 df-lmod 19638 df-scaf 19639 df-sra 19946 df-rgmod 19947 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-fbas 20544 df-fg 20545 df-cnfld 20548 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cld 21629 df-ntr 21630 df-cls 21631 df-nei 21708 df-lp 21746 df-perf 21747 df-cn 21837 df-cnp 21838 df-haus 21925 df-tx 22172 df-hmeo 22365 df-fil 22456 df-fm 22548 df-flim 22549 df-flf 22550 df-tmd 22682 df-tgp 22683 df-tsms 22737 df-trg 22770 df-xms 22932 df-ms 22933 df-tms 22934 df-nm 23194 df-ngp 23195 df-nrg 23197 df-nlm 23198 df-ii 23487 df-cncf 23488 df-limc 24466 df-dv 24467 df-log 25142 df-esum 31289 |
This theorem is referenced by: measxun2 31471 measssd 31476 carsgclctun 31581 |
Copyright terms: Public domain | W3C validator |