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Mirrors > Home > ILE Home > Th. List > fisumrev2 | GIF version |
Description: Reversal of a finite sum. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 13-Apr-2016.) |
Ref | Expression |
---|---|
fisumrev2.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
fisumrev2.n | ⊢ (𝜑 → 𝑁 ∈ ℤ) |
fsumrev2.1 | ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ) |
fsumrev2.2 | ⊢ (𝑗 = ((𝑀 + 𝑁) − 𝑘) → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
fisumrev2 | ⊢ (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ (𝑀...𝑁)𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fisumrev2.m | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
2 | 1 | adantr 276 | . . . 4 ⊢ ((𝜑 ∧ 𝑀 ≤ 𝑁) → 𝑀 ∈ ℤ) |
3 | fisumrev2.n | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
4 | 3 | adantr 276 | . . . 4 ⊢ ((𝜑 ∧ 𝑀 ≤ 𝑁) → 𝑁 ∈ ℤ) |
5 | simpr 110 | . . . 4 ⊢ ((𝜑 ∧ 𝑀 ≤ 𝑁) → 𝑀 ≤ 𝑁) | |
6 | eluz2 9565 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁)) | |
7 | 2, 4, 5, 6 | syl3anbrc 1183 | . . 3 ⊢ ((𝜑 ∧ 𝑀 ≤ 𝑁) → 𝑁 ∈ (ℤ≥‘𝑀)) |
8 | 1 | adantr 276 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → 𝑀 ∈ ℤ) |
9 | 3 | adantr 276 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → 𝑁 ∈ ℤ) |
10 | 8, 9 | zaddcld 9410 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (𝑀 + 𝑁) ∈ ℤ) |
11 | fsumrev2.1 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ) | |
12 | 11 | adantlr 477 | . . . . 5 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ) |
13 | fsumrev2.2 | . . . . 5 ⊢ (𝑗 = ((𝑀 + 𝑁) − 𝑘) → 𝐴 = 𝐵) | |
14 | 10, 8, 9, 12, 13 | fsumrev 11486 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ (((𝑀 + 𝑁) − 𝑁)...((𝑀 + 𝑁) − 𝑀))𝐵) |
15 | 8 | zcnd 9407 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → 𝑀 ∈ ℂ) |
16 | 9 | zcnd 9407 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → 𝑁 ∈ ℂ) |
17 | 15, 16 | pncand 8300 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → ((𝑀 + 𝑁) − 𝑁) = 𝑀) |
18 | 15, 16 | pncan2d 8301 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → ((𝑀 + 𝑁) − 𝑀) = 𝑁) |
19 | 17, 18 | oveq12d 5915 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (((𝑀 + 𝑁) − 𝑁)...((𝑀 + 𝑁) − 𝑀)) = (𝑀...𝑁)) |
20 | 19 | sumeq1d 11409 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → Σ𝑘 ∈ (((𝑀 + 𝑁) − 𝑁)...((𝑀 + 𝑁) − 𝑀))𝐵 = Σ𝑘 ∈ (𝑀...𝑁)𝐵) |
21 | 14, 20 | eqtrd 2222 | . . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ (𝑀...𝑁)𝐵) |
22 | 7, 21 | syldan 282 | . 2 ⊢ ((𝜑 ∧ 𝑀 ≤ 𝑁) → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ (𝑀...𝑁)𝐵) |
23 | fzn 10074 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < 𝑀 ↔ (𝑀...𝑁) = ∅)) | |
24 | 1, 3, 23 | syl2anc 411 | . . . 4 ⊢ (𝜑 → (𝑁 < 𝑀 ↔ (𝑀...𝑁) = ∅)) |
25 | 24 | biimpa 296 | . . 3 ⊢ ((𝜑 ∧ 𝑁 < 𝑀) → (𝑀...𝑁) = ∅) |
26 | sum0 11431 | . . . . 5 ⊢ Σ𝑗 ∈ ∅ 𝐴 = 0 | |
27 | sum0 11431 | . . . . 5 ⊢ Σ𝑘 ∈ ∅ 𝐵 = 0 | |
28 | 26, 27 | eqtr4i 2213 | . . . 4 ⊢ Σ𝑗 ∈ ∅ 𝐴 = Σ𝑘 ∈ ∅ 𝐵 |
29 | sumeq1 11398 | . . . 4 ⊢ ((𝑀...𝑁) = ∅ → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑗 ∈ ∅ 𝐴) | |
30 | sumeq1 11398 | . . . 4 ⊢ ((𝑀...𝑁) = ∅ → Σ𝑘 ∈ (𝑀...𝑁)𝐵 = Σ𝑘 ∈ ∅ 𝐵) | |
31 | 28, 29, 30 | 3eqtr4a 2248 | . . 3 ⊢ ((𝑀...𝑁) = ∅ → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ (𝑀...𝑁)𝐵) |
32 | 25, 31 | syl 14 | . 2 ⊢ ((𝜑 ∧ 𝑁 < 𝑀) → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ (𝑀...𝑁)𝐵) |
33 | zlelttric 9329 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ∨ 𝑁 < 𝑀)) | |
34 | 1, 3, 33 | syl2anc 411 | . 2 ⊢ (𝜑 → (𝑀 ≤ 𝑁 ∨ 𝑁 < 𝑀)) |
35 | 22, 32, 34 | mpjaodan 799 | 1 ⊢ (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ (𝑀...𝑁)𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 709 = wceq 1364 ∈ wcel 2160 ∅c0 3437 class class class wbr 4018 ‘cfv 5235 (class class class)co 5897 ℂcc 7840 0cc0 7842 + caddc 7845 < clt 8023 ≤ cle 8024 − cmin 8159 ℤcz 9284 ℤ≥cuz 9559 ...cfz 10040 Σcsu 11396 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 ax-cnex 7933 ax-resscn 7934 ax-1cn 7935 ax-1re 7936 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-mulrcl 7941 ax-addcom 7942 ax-mulcom 7943 ax-addass 7944 ax-mulass 7945 ax-distr 7946 ax-i2m1 7947 ax-0lt1 7948 ax-1rid 7949 ax-0id 7950 ax-rnegex 7951 ax-precex 7952 ax-cnre 7953 ax-pre-ltirr 7954 ax-pre-ltwlin 7955 ax-pre-lttrn 7956 ax-pre-apti 7957 ax-pre-ltadd 7958 ax-pre-mulgt0 7959 ax-pre-mulext 7960 ax-arch 7961 ax-caucvg 7962 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4311 df-po 4314 df-iso 4315 df-iord 4384 df-on 4386 df-ilim 4387 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-isom 5244 df-riota 5852 df-ov 5900 df-oprab 5901 df-mpo 5902 df-1st 6166 df-2nd 6167 df-recs 6331 df-irdg 6396 df-frec 6417 df-1o 6442 df-oadd 6446 df-er 6560 df-en 6768 df-dom 6769 df-fin 6770 df-pnf 8025 df-mnf 8026 df-xr 8027 df-ltxr 8028 df-le 8029 df-sub 8161 df-neg 8162 df-reap 8563 df-ap 8570 df-div 8661 df-inn 8951 df-2 9009 df-3 9010 df-4 9011 df-n0 9208 df-z 9285 df-uz 9560 df-q 9652 df-rp 9686 df-fz 10041 df-fzo 10175 df-seqfrec 10479 df-exp 10554 df-ihash 10791 df-cj 10886 df-re 10887 df-im 10888 df-rsqrt 11042 df-abs 11043 df-clim 11322 df-sumdc 11397 |
This theorem is referenced by: fisum0diag2 11490 efaddlem 11717 |
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