![]() |
Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > fsummsndifre | Structured version Visualization version GIF version |
Description: A finite sum with one of its integer summands removed is a real number. (Contributed by Alexander van der Vekens, 31-Aug-2018.) |
Ref | Expression |
---|---|
fsummsndifre | ⊢ ((𝐴 ∈ Fin ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ ℤ) → Σ𝑘 ∈ (𝐴 ∖ {𝑋})𝐵 ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2934 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
2 | nfcsb1v 3767 | . . 3 ⊢ Ⅎ𝑘⦋𝑥 / 𝑘⦌𝐵 | |
3 | csbeq1a 3760 | . . 3 ⊢ (𝑘 = 𝑥 → 𝐵 = ⦋𝑥 / 𝑘⦌𝐵) | |
4 | 1, 2, 3 | cbvsumi 14839 | . 2 ⊢ Σ𝑘 ∈ (𝐴 ∖ {𝑋})𝐵 = Σ𝑥 ∈ (𝐴 ∖ {𝑋})⦋𝑥 / 𝑘⦌𝐵 |
5 | diffi 8482 | . . . 4 ⊢ (𝐴 ∈ Fin → (𝐴 ∖ {𝑋}) ∈ Fin) | |
6 | 5 | adantr 474 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ ℤ) → (𝐴 ∖ {𝑋}) ∈ Fin) |
7 | eldifi 3955 | . . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∖ {𝑋}) → 𝑥 ∈ 𝐴) | |
8 | rspcsbela 4232 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ ℤ) → ⦋𝑥 / 𝑘⦌𝐵 ∈ ℤ) | |
9 | 7, 8 | sylan 575 | . . . . . . 7 ⊢ ((𝑥 ∈ (𝐴 ∖ {𝑋}) ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ ℤ) → ⦋𝑥 / 𝑘⦌𝐵 ∈ ℤ) |
10 | 9 | zred 11838 | . . . . . 6 ⊢ ((𝑥 ∈ (𝐴 ∖ {𝑋}) ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ ℤ) → ⦋𝑥 / 𝑘⦌𝐵 ∈ ℝ) |
11 | 10 | expcom 404 | . . . . 5 ⊢ (∀𝑘 ∈ 𝐴 𝐵 ∈ ℤ → (𝑥 ∈ (𝐴 ∖ {𝑋}) → ⦋𝑥 / 𝑘⦌𝐵 ∈ ℝ)) |
12 | 11 | adantl 475 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ ℤ) → (𝑥 ∈ (𝐴 ∖ {𝑋}) → ⦋𝑥 / 𝑘⦌𝐵 ∈ ℝ)) |
13 | 12 | imp 397 | . . 3 ⊢ (((𝐴 ∈ Fin ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ ℤ) ∧ 𝑥 ∈ (𝐴 ∖ {𝑋})) → ⦋𝑥 / 𝑘⦌𝐵 ∈ ℝ) |
14 | 6, 13 | fsumrecl 14876 | . 2 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ ℤ) → Σ𝑥 ∈ (𝐴 ∖ {𝑋})⦋𝑥 / 𝑘⦌𝐵 ∈ ℝ) |
15 | 4, 14 | syl5eqel 2863 | 1 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ ℤ) → Σ𝑘 ∈ (𝐴 ∖ {𝑋})𝐵 ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2107 ∀wral 3090 ⦋csb 3751 ∖ cdif 3789 {csn 4398 Fincfn 8243 ℝcr 10273 ℤcz 11732 Σcsu 14828 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-inf2 8837 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-se 5317 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-isom 6146 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-sup 8638 df-oi 8706 df-card 9100 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11035 df-nn 11379 df-2 11442 df-3 11443 df-n0 11647 df-z 11733 df-uz 11997 df-rp 12142 df-fz 12648 df-fzo 12789 df-seq 13124 df-exp 13183 df-hash 13440 df-cj 14250 df-re 14251 df-im 14252 df-sqrt 14386 df-abs 14387 df-clim 14631 df-sum 14829 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |