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Mirrors > Home > ILE Home > Th. List > fsump1i | GIF version |
Description: Optimized version of fsump1 11446 for making sums of a concrete number of terms. (Contributed by Mario Carneiro, 23-Apr-2014.) |
Ref | Expression |
---|---|
fsump1i.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
fsump1i.2 | ⊢ 𝑁 = (𝐾 + 1) |
fsump1i.3 | ⊢ (𝑘 = 𝑁 → 𝐴 = 𝐵) |
fsump1i.4 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) |
fsump1i.5 | ⊢ (𝜑 → (𝐾 ∈ 𝑍 ∧ Σ𝑘 ∈ (𝑀...𝐾)𝐴 = 𝑆)) |
fsump1i.6 | ⊢ (𝜑 → (𝑆 + 𝐵) = 𝑇) |
Ref | Expression |
---|---|
fsump1i | ⊢ (𝜑 → (𝑁 ∈ 𝑍 ∧ Σ𝑘 ∈ (𝑀...𝑁)𝐴 = 𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsump1i.2 | . . 3 ⊢ 𝑁 = (𝐾 + 1) | |
2 | fsump1i.5 | . . . . . 6 ⊢ (𝜑 → (𝐾 ∈ 𝑍 ∧ Σ𝑘 ∈ (𝑀...𝐾)𝐴 = 𝑆)) | |
3 | 2 | simpld 112 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ 𝑍) |
4 | fsump1i.1 | . . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
5 | 3, 4 | eleqtrdi 2282 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑀)) |
6 | peano2uz 9601 | . . . . 5 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → (𝐾 + 1) ∈ (ℤ≥‘𝑀)) | |
7 | 6, 4 | eleqtrrdi 2283 | . . . 4 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → (𝐾 + 1) ∈ 𝑍) |
8 | 5, 7 | syl 14 | . . 3 ⊢ (𝜑 → (𝐾 + 1) ∈ 𝑍) |
9 | 1, 8 | eqeltrid 2276 | . 2 ⊢ (𝜑 → 𝑁 ∈ 𝑍) |
10 | 1 | oveq2i 5902 | . . . . 5 ⊢ (𝑀...𝑁) = (𝑀...(𝐾 + 1)) |
11 | 10 | sumeq1i 11389 | . . . 4 ⊢ Σ𝑘 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ (𝑀...(𝐾 + 1))𝐴 |
12 | elfzuz 10039 | . . . . . . 7 ⊢ (𝑘 ∈ (𝑀...(𝐾 + 1)) → 𝑘 ∈ (ℤ≥‘𝑀)) | |
13 | 12, 4 | eleqtrrdi 2283 | . . . . . 6 ⊢ (𝑘 ∈ (𝑀...(𝐾 + 1)) → 𝑘 ∈ 𝑍) |
14 | fsump1i.4 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) | |
15 | 13, 14 | sylan2 286 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝐾 + 1))) → 𝐴 ∈ ℂ) |
16 | 1 | eqeq2i 2200 | . . . . . 6 ⊢ (𝑘 = 𝑁 ↔ 𝑘 = (𝐾 + 1)) |
17 | fsump1i.3 | . . . . . 6 ⊢ (𝑘 = 𝑁 → 𝐴 = 𝐵) | |
18 | 16, 17 | sylbir 135 | . . . . 5 ⊢ (𝑘 = (𝐾 + 1) → 𝐴 = 𝐵) |
19 | 5, 15, 18 | fsump1 11446 | . . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...(𝐾 + 1))𝐴 = (Σ𝑘 ∈ (𝑀...𝐾)𝐴 + 𝐵)) |
20 | 11, 19 | eqtrid 2234 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (Σ𝑘 ∈ (𝑀...𝐾)𝐴 + 𝐵)) |
21 | 2 | simprd 114 | . . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝐾)𝐴 = 𝑆) |
22 | 21 | oveq1d 5906 | . . 3 ⊢ (𝜑 → (Σ𝑘 ∈ (𝑀...𝐾)𝐴 + 𝐵) = (𝑆 + 𝐵)) |
23 | fsump1i.6 | . . 3 ⊢ (𝜑 → (𝑆 + 𝐵) = 𝑇) | |
24 | 20, 22, 23 | 3eqtrd 2226 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = 𝑇) |
25 | 9, 24 | jca 306 | 1 ⊢ (𝜑 → (𝑁 ∈ 𝑍 ∧ Σ𝑘 ∈ (𝑀...𝑁)𝐴 = 𝑇)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2160 ‘cfv 5231 (class class class)co 5891 ℂcc 7827 1c1 7830 + caddc 7832 ℤ≥cuz 9546 ...cfz 10026 Σcsu 11379 |
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 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-iinf 4602 ax-cnex 7920 ax-resscn 7921 ax-1cn 7922 ax-1re 7923 ax-icn 7924 ax-addcl 7925 ax-addrcl 7926 ax-mulcl 7927 ax-mulrcl 7928 ax-addcom 7929 ax-mulcom 7930 ax-addass 7931 ax-mulass 7932 ax-distr 7933 ax-i2m1 7934 ax-0lt1 7935 ax-1rid 7936 ax-0id 7937 ax-rnegex 7938 ax-precex 7939 ax-cnre 7940 ax-pre-ltirr 7941 ax-pre-ltwlin 7942 ax-pre-lttrn 7943 ax-pre-apti 7944 ax-pre-ltadd 7945 ax-pre-mulgt0 7946 ax-pre-mulext 7947 ax-arch 7948 ax-caucvg 7949 |
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 4308 df-po 4311 df-iso 4312 df-iord 4381 df-on 4383 df-ilim 4384 df-suc 4386 df-iom 4605 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-f1 5236 df-fo 5237 df-f1o 5238 df-fv 5239 df-isom 5240 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-1st 6159 df-2nd 6160 df-recs 6324 df-irdg 6389 df-frec 6410 df-1o 6435 df-oadd 6439 df-er 6553 df-en 6759 df-dom 6760 df-fin 6761 df-pnf 8012 df-mnf 8013 df-xr 8014 df-ltxr 8015 df-le 8016 df-sub 8148 df-neg 8149 df-reap 8550 df-ap 8557 df-div 8648 df-inn 8938 df-2 8996 df-3 8997 df-4 8998 df-n0 9195 df-z 9272 df-uz 9547 df-q 9638 df-rp 9672 df-fz 10027 df-fzo 10161 df-seqfrec 10464 df-exp 10538 df-ihash 10774 df-cj 10869 df-re 10870 df-im 10871 df-rsqrt 11025 df-abs 11026 df-clim 11305 df-sumdc 11380 |
This theorem is referenced by: (None) |
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