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Mirrors > Home > MPE Home > Th. List > fsumconst | Structured version Visualization version GIF version |
Description: The sum of constant terms (𝑘 is not free in 𝐵). (Contributed by NM, 24-Dec-2005.) (Revised by Mario Carneiro, 24-Apr-2014.) |
Ref | Expression |
---|---|
fsumconst | ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ 𝐴 𝐵 = ((♯‘𝐴) · 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mul02 10818 | . . . . 5 ⊢ (𝐵 ∈ ℂ → (0 · 𝐵) = 0) | |
2 | 1 | adantl 484 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) → (0 · 𝐵) = 0) |
3 | 2 | eqcomd 2827 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) → 0 = (0 · 𝐵)) |
4 | sumeq1 15045 | . . . . 5 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ ∅ 𝐵) | |
5 | sum0 15078 | . . . . 5 ⊢ Σ𝑘 ∈ ∅ 𝐵 = 0 | |
6 | 4, 5 | syl6eq 2872 | . . . 4 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 𝐵 = 0) |
7 | fveq2 6670 | . . . . . 6 ⊢ (𝐴 = ∅ → (♯‘𝐴) = (♯‘∅)) | |
8 | hash0 13729 | . . . . . 6 ⊢ (♯‘∅) = 0 | |
9 | 7, 8 | syl6eq 2872 | . . . . 5 ⊢ (𝐴 = ∅ → (♯‘𝐴) = 0) |
10 | 9 | oveq1d 7171 | . . . 4 ⊢ (𝐴 = ∅ → ((♯‘𝐴) · 𝐵) = (0 · 𝐵)) |
11 | 6, 10 | eqeq12d 2837 | . . 3 ⊢ (𝐴 = ∅ → (Σ𝑘 ∈ 𝐴 𝐵 = ((♯‘𝐴) · 𝐵) ↔ 0 = (0 · 𝐵))) |
12 | 3, 11 | syl5ibrcom 249 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) → (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 𝐵 = ((♯‘𝐴) · 𝐵))) |
13 | eqidd 2822 | . . . . . . 7 ⊢ (𝑘 = (𝑓‘𝑛) → 𝐵 = 𝐵) | |
14 | simprl 769 | . . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (♯‘𝐴) ∈ ℕ) | |
15 | simprr 771 | . . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) | |
16 | simpllr 774 | . . . . . . 7 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) | |
17 | simplr 767 | . . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝐵 ∈ ℂ) | |
18 | elfznn 12937 | . . . . . . . 8 ⊢ (𝑛 ∈ (1...(♯‘𝐴)) → 𝑛 ∈ ℕ) | |
19 | fvconst2g 6964 | . . . . . . . 8 ⊢ ((𝐵 ∈ ℂ ∧ 𝑛 ∈ ℕ) → ((ℕ × {𝐵})‘𝑛) = 𝐵) | |
20 | 17, 18, 19 | syl2an 597 | . . . . . . 7 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → ((ℕ × {𝐵})‘𝑛) = 𝐵) |
21 | 13, 14, 15, 16, 20 | fsum 15077 | . . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → Σ𝑘 ∈ 𝐴 𝐵 = (seq1( + , (ℕ × {𝐵}))‘(♯‘𝐴))) |
22 | ser1const 13427 | . . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ (♯‘𝐴) ∈ ℕ) → (seq1( + , (ℕ × {𝐵}))‘(♯‘𝐴)) = ((♯‘𝐴) · 𝐵)) | |
23 | 22 | ad2ant2lr 746 | . . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (seq1( + , (ℕ × {𝐵}))‘(♯‘𝐴)) = ((♯‘𝐴) · 𝐵)) |
24 | 21, 23 | eqtrd 2856 | . . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → Σ𝑘 ∈ 𝐴 𝐵 = ((♯‘𝐴) · 𝐵)) |
25 | 24 | expr 459 | . . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) ∧ (♯‘𝐴) ∈ ℕ) → (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → Σ𝑘 ∈ 𝐴 𝐵 = ((♯‘𝐴) · 𝐵))) |
26 | 25 | exlimdv 1934 | . . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) ∧ (♯‘𝐴) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → Σ𝑘 ∈ 𝐴 𝐵 = ((♯‘𝐴) · 𝐵))) |
27 | 26 | expimpd 456 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) → (((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → Σ𝑘 ∈ 𝐴 𝐵 = ((♯‘𝐴) · 𝐵))) |
28 | fz1f1o 15067 | . . 3 ⊢ (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴))) | |
29 | 28 | adantr 483 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴))) |
30 | 12, 27, 29 | mpjaod 856 | 1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ 𝐴 𝐵 = ((♯‘𝐴) · 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ∅c0 4291 {csn 4567 × cxp 5553 –1-1-onto→wf1o 6354 ‘cfv 6355 (class class class)co 7156 Fincfn 8509 ℂcc 10535 0cc0 10537 1c1 10538 + caddc 10540 · cmul 10542 ℕcn 11638 ...cfz 12893 seqcseq 13370 ♯chash 13691 Σcsu 15042 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-fz 12894 df-fzo 13035 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-sum 15043 |
This theorem is referenced by: fsumdifsnconst 15146 o1fsum 15168 hashiun 15177 hash2iun1dif1 15179 climcndslem1 15204 climcndslem2 15205 harmonic 15214 mertenslem1 15240 sumhash 16232 cshwshashnsame 16437 lagsubg2 18341 sylow2a 18744 lebnumlem3 23567 uniioombllem4 24187 birthdaylem2 25530 basellem8 25665 0sgm 25721 musum 25768 chtleppi 25786 vmasum 25792 logfac2 25793 chpval2 25794 chpchtsum 25795 chpub 25796 logfaclbnd 25798 dchrsum2 25844 sumdchr2 25846 lgsquadlem1 25956 chebbnd1lem1 26045 chtppilimlem1 26049 dchrmusum2 26070 dchrisum0flblem1 26084 rpvmasum2 26088 dchrisum0lem2a 26093 mudivsum 26106 mulogsumlem 26107 selberglem2 26122 pntlemj 26179 rusgrnumwwlks 27753 fusgrhashclwwlkn 27858 fusgreghash2wsp 28117 numclwwlk6 28169 reprlt 31890 hashreprin 31891 reprgt 31892 hgt750lema 31928 rrndstprj2 35124 fltnltalem 39294 stoweidlem11 42316 stoweidlem26 42331 stoweidlem38 42343 dirkertrigeq 42406 fourierdlem73 42484 etransclem32 42571 rrndistlt 42595 sge0rpcpnf 42723 hoiqssbllem2 42925 nn0mulfsum 44704 amgmlemALT 44924 |
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