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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 11439 | . . . . 5 ⊢ (𝐵 ∈ ℂ → (0 · 𝐵) = 0) | |
| 2 | 1 | adantl 481 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) → (0 · 𝐵) = 0) |
| 3 | 2 | eqcomd 2743 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) → 0 = (0 · 𝐵)) |
| 4 | sumeq1 15725 | . . . . 5 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ ∅ 𝐵) | |
| 5 | sum0 15757 | . . . . 5 ⊢ Σ𝑘 ∈ ∅ 𝐵 = 0 | |
| 6 | 4, 5 | eqtrdi 2793 | . . . 4 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 𝐵 = 0) |
| 7 | fveq2 6906 | . . . . . 6 ⊢ (𝐴 = ∅ → (♯‘𝐴) = (♯‘∅)) | |
| 8 | hash0 14406 | . . . . . 6 ⊢ (♯‘∅) = 0 | |
| 9 | 7, 8 | eqtrdi 2793 | . . . . 5 ⊢ (𝐴 = ∅ → (♯‘𝐴) = 0) |
| 10 | 9 | oveq1d 7446 | . . . 4 ⊢ (𝐴 = ∅ → ((♯‘𝐴) · 𝐵) = (0 · 𝐵)) |
| 11 | 6, 10 | eqeq12d 2753 | . . 3 ⊢ (𝐴 = ∅ → (Σ𝑘 ∈ 𝐴 𝐵 = ((♯‘𝐴) · 𝐵) ↔ 0 = (0 · 𝐵))) |
| 12 | 3, 11 | syl5ibrcom 247 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) → (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 𝐵 = ((♯‘𝐴) · 𝐵))) |
| 13 | eqidd 2738 | . . . . . . 7 ⊢ (𝑘 = (𝑓‘𝑛) → 𝐵 = 𝐵) | |
| 14 | simprl 771 | . . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (♯‘𝐴) ∈ ℕ) | |
| 15 | simprr 773 | . . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) | |
| 16 | simpllr 776 | . . . . . . 7 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) | |
| 17 | simplr 769 | . . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝐵 ∈ ℂ) | |
| 18 | elfznn 13593 | . . . . . . . 8 ⊢ (𝑛 ∈ (1...(♯‘𝐴)) → 𝑛 ∈ ℕ) | |
| 19 | fvconst2g 7222 | . . . . . . . 8 ⊢ ((𝐵 ∈ ℂ ∧ 𝑛 ∈ ℕ) → ((ℕ × {𝐵})‘𝑛) = 𝐵) | |
| 20 | 17, 18, 19 | syl2an 596 | . . . . . . 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 15756 | . . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → Σ𝑘 ∈ 𝐴 𝐵 = (seq1( + , (ℕ × {𝐵}))‘(♯‘𝐴))) |
| 22 | ser1const 14099 | . . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ (♯‘𝐴) ∈ ℕ) → (seq1( + , (ℕ × {𝐵}))‘(♯‘𝐴)) = ((♯‘𝐴) · 𝐵)) | |
| 23 | 22 | ad2ant2lr 748 | . . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (seq1( + , (ℕ × {𝐵}))‘(♯‘𝐴)) = ((♯‘𝐴) · 𝐵)) |
| 24 | 21, 23 | eqtrd 2777 | . . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → Σ𝑘 ∈ 𝐴 𝐵 = ((♯‘𝐴) · 𝐵)) |
| 25 | 24 | expr 456 | . . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) ∧ (♯‘𝐴) ∈ ℕ) → (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → Σ𝑘 ∈ 𝐴 𝐵 = ((♯‘𝐴) · 𝐵))) |
| 26 | 25 | exlimdv 1933 | . . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) ∧ (♯‘𝐴) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → Σ𝑘 ∈ 𝐴 𝐵 = ((♯‘𝐴) · 𝐵))) |
| 27 | 26 | expimpd 453 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) → (((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → Σ𝑘 ∈ 𝐴 𝐵 = ((♯‘𝐴) · 𝐵))) |
| 28 | fz1f1o 15746 | . . 3 ⊢ (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴))) | |
| 29 | 28 | adantr 480 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴))) |
| 30 | 12, 27, 29 | mpjaod 861 | 1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ 𝐴 𝐵 = ((♯‘𝐴) · 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ∅c0 4333 {csn 4626 × cxp 5683 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 ℂcc 11153 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 ℕcn 12266 ...cfz 13547 seqcseq 14042 ♯chash 14369 Σcsu 15722 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-sum 15723 |
| This theorem is referenced by: fsumdifsnconst 15827 o1fsum 15849 hashiun 15858 hash2iun1dif1 15860 climcndslem1 15885 climcndslem2 15886 harmonic 15895 mertenslem1 15920 sumhash 16934 cshwshashnsame 17141 lagsubg2 19212 sylow2a 19637 lebnumlem3 24995 uniioombllem4 25621 birthdaylem2 26995 basellem8 27131 0sgm 27187 musum 27234 chtleppi 27254 vmasum 27260 logfac2 27261 chpval2 27262 chpchtsum 27263 chpub 27264 logfaclbnd 27266 dchrsum2 27312 sumdchr2 27314 lgsquadlem1 27424 chebbnd1lem1 27513 chtppilimlem1 27517 dchrmusum2 27538 dchrisum0flblem1 27552 rpvmasum2 27556 dchrisum0lem2a 27561 mudivsum 27574 mulogsumlem 27575 selberglem2 27590 pntlemj 27647 rusgrnumwwlks 29994 fusgrhashclwwlkn 30098 fusgreghash2wsp 30357 numclwwlk6 30409 reprlt 34634 hashreprin 34635 reprgt 34636 hgt750lema 34672 rrndstprj2 37838 lcmineqlem17 42046 sticksstones10 42156 sticksstones12a 42158 fz1sumconst 42343 fltnltalem 42672 stoweidlem11 46026 stoweidlem26 46041 stoweidlem38 46053 dirkertrigeq 46116 fourierdlem73 46194 etransclem32 46281 rrndistlt 46305 sge0rpcpnf 46436 hoiqssbllem2 46638 nn0mulfsum 48545 amgmlemALT 49322 |
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