fsumconst - Metamath Proof Explorer

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Theorem fsumconst 15815

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.)

**Assertion**
Ref	Expression
fsumconst	⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ 𝐴 𝐵 = ((♯‘𝐴) · 𝐵))

Distinct variable groups: 𝐴,𝑘 𝐵,𝑘

Proof of Theorem fsumconst Dummy variables 𝑓 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mul02 11449	. . . . 5 ⊢ (𝐵 ∈ ℂ → (0 · 𝐵) = 0)
2	1	adantl 480	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) → (0 · 𝐵) = 0)
3	2	eqcomd 2735	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) → 0 = (0 · 𝐵))
4		sumeq1 15714	. . . . 5 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ ∅ 𝐵)
5		sum0 15746	. . . . 5 ⊢ Σ𝑘 ∈ ∅ 𝐵 = 0
6	4, 5	eqtrdi 2785	. . . 4 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 𝐵 = 0)
7		fveq2 6905	. . . . . 6 ⊢ (𝐴 = ∅ → (♯‘𝐴) = (♯‘∅))
8		hash0 14396	. . . . . 6 ⊢ (♯‘∅) = 0
9	7, 8	eqtrdi 2785	. . . . 5 ⊢ (𝐴 = ∅ → (♯‘𝐴) = 0)
10	9	oveq1d 7445	. . . 4 ⊢ (𝐴 = ∅ → ((♯‘𝐴) · 𝐵) = (0 · 𝐵))
11	6, 10	eqeq12d 2745	. . 3 ⊢ (𝐴 = ∅ → (Σ𝑘 ∈ 𝐴 𝐵 = ((♯‘𝐴) · 𝐵) ↔ 0 = (0 · 𝐵)))
12	3, 11	syl5ibrcom 246	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) → (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 𝐵 = ((♯‘𝐴) · 𝐵)))
13		eqidd 2730	. . . . . . 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 13594	. . . . . . . 8 ⊢ (𝑛 ∈ (1...(♯‘𝐴)) → 𝑛 ∈ ℕ)
19		fvconst2g 7224	. . . . . . . 8 ⊢ ((𝐵 ∈ ℂ ∧ 𝑛 ∈ ℕ) → ((ℕ × {𝐵})‘𝑛) = 𝐵)
20	17, 18, 19	syl2an 594	. . . . . . 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 15745	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → Σ𝑘 ∈ 𝐴 𝐵 = (seq1( + , (ℕ × {𝐵}))‘(♯‘𝐴)))
22		ser1const 14089	. . . . . . 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 2769	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → Σ𝑘 ∈ 𝐴 𝐵 = ((♯‘𝐴) · 𝐵))
25	24	expr 455	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) ∧ (♯‘𝐴) ∈ ℕ) → (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → Σ𝑘 ∈ 𝐴 𝐵 = ((♯‘𝐴) · 𝐵)))
26	25	exlimdv 1929	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) ∧ (♯‘𝐴) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → Σ𝑘 ∈ 𝐴 𝐵 = ((♯‘𝐴) · 𝐵)))
27	26	expimpd 452	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) → (((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → Σ𝑘 ∈ 𝐴 𝐵 = ((♯‘𝐴) · 𝐵)))
28		fz1f1o 15735	. . 3 ⊢ (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)))
29	28	adantr 479	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)))
30	12, 27, 29	mpjaod 858	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ 𝐴 𝐵 = ((♯‘𝐴) · 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 ∨ wo 845 = wceq 1534 ∃wex 1774 ∈ wcel 2100 ∅c0 4335 {csn 4636 × cxp 5684 –1-1-onto→wf1o 6557 ‘cfv 6558 (class class class)co 7430 Fincfn 8982 ℂcc 11163 0cc0 11165 1c1 11166 + caddc 11168 · cmul 11170 ℕcn 12274 ...cfz 13548 seqcseq 14032 ♯chash 14359 Σcsu 15711

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2102 ax-9 2110 ax-10 2133 ax-11 2150 ax-12 2170 ax-ext 2700 ax-rep 5293 ax-sep 5307 ax-nul 5314 ax-pow 5373 ax-pr 5437 ax-un 7751 ax-inf2 9691 ax-cnex 11221 ax-resscn 11222 ax-1cn 11223 ax-icn 11224 ax-addcl 11225 ax-addrcl 11226 ax-mulcl 11227 ax-mulrcl 11228 ax-mulcom 11229 ax-addass 11230 ax-mulass 11231 ax-distr 11232 ax-i2m1 11233 ax-1ne0 11234 ax-1rid 11235 ax-rnegex 11236 ax-rrecex 11237 ax-cnre 11238 ax-pre-lttri 11239 ax-pre-lttrn 11240 ax-pre-ltadd 11241 ax-pre-mulgt0 11242 ax-pre-sup 11243

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2062 df-mo 2532 df-eu 2561 df-clab 2707 df-cleq 2721 df-clel 2806 df-nfc 2881 df-ne 2934 df-nel 3040 df-ral 3055 df-rex 3064 df-rmo 3373 df-reu 3374 df-rab 3429 df-v 3474 df-sbc 3789 df-csb 3905 df-dif 3962 df-un 3964 df-in 3966 df-ss 3976 df-pss 3979 df-nul 4336 df-if 4537 df-pw 4612 df-sn 4637 df-pr 4639 df-op 4643 df-uni 4919 df-int 4960 df-iun 5008 df-br 5157 df-opab 5219 df-mpt 5240 df-tr 5274 df-id 5584 df-eprel 5590 df-po 5598 df-so 5599 df-fr 5641 df-se 5642 df-we 5643 df-xp 5692 df-rel 5693 df-cnv 5694 df-co 5695 df-dm 5696 df-rn 5697 df-res 5698 df-ima 5699 df-pred 6317 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6510 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-isom 6567 df-riota 7386 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7885 df-1st 8011 df-2nd 8012 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8742 df-en 8983 df-dom 8984 df-sdom 8985 df-fin 8986 df-sup 9492 df-oi 9560 df-card 9989 df-pnf 11307 df-mnf 11308 df-xr 11309 df-ltxr 11310 df-le 11311 df-sub 11503 df-neg 11504 df-div 11929 df-nn 12275 df-2 12337 df-3 12338 df-n0 12535 df-z 12621 df-uz 12885 df-rp 13039 df-fz 13549 df-fzo 13692 df-seq 14033 df-exp 14093 df-hash 14360 df-cj 15125 df-re 15126 df-im 15127 df-sqrt 15261 df-abs 15262 df-clim 15511 df-sum 15712

This theorem is referenced by: fsumdifsnconst 15816 o1fsum 15838 hashiun 15847 hash2iun1dif1 15849 climcndslem1 15874 climcndslem2 15875 harmonic 15884 mertenslem1 15909 sumhash 16919 cshwshashnsame 17127 lagsubg2 19210 sylow2a 19639 lebnumlem3 25003 uniioombllem4 25629 birthdaylem2 27003 basellem8 27139 0sgm 27195 musum 27242 chtleppi 27262 vmasum 27268 logfac2 27269 chpval2 27270 chpchtsum 27271 chpub 27272 logfaclbnd 27274 dchrsum2 27320 sumdchr2 27322 lgsquadlem1 27432 chebbnd1lem1 27521 chtppilimlem1 27525 dchrmusum2 27546 dchrisum0flblem1 27560 rpvmasum2 27564 dchrisum0lem2a 27569 mudivsum 27582 mulogsumlem 27583 selberglem2 27598 pntlemj 27655 rusgrnumwwlks 29931 fusgrhashclwwlkn 30035 fusgreghash2wsp 30294 numclwwlk6 30346 reprlt 34523 hashreprin 34524 reprgt 34525 hgt750lema 34561 rrndstprj2 37599 lcmineqlem17 41811 sticksstones10 41921 sticksstones12a 41923 fz1sumconst 42090 fltnltalem 42406 stoweidlem11 45722 stoweidlem26 45737 stoweidlem38 45749 dirkertrigeq 45812 fourierdlem73 45890 etransclem32 45977 rrndistlt 46001 sge0rpcpnf 46132 hoiqssbllem2 46334 nn0mulfsum 48113 amgmlemALT 48652

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