fsumconst - Metamath Proof Explorer

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Theorem fsumconst 15714

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 11312	. . . . 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 15613	. . . . 5 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ ∅ 𝐵)
5		sum0 15645	. . . . 5 ⊢ Σ𝑘 ∈ ∅ 𝐵 = 0
6	4, 5	eqtrdi 2788	. . . 4 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 𝐵 = 0)
7		fveq2 6832	. . . . . 6 ⊢ (𝐴 = ∅ → (♯‘𝐴) = (♯‘∅))
8		hash0 14291	. . . . . 6 ⊢ (♯‘∅) = 0
9	7, 8	eqtrdi 2788	. . . . 5 ⊢ (𝐴 = ∅ → (♯‘𝐴) = 0)
10	9	oveq1d 7373	. . . 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 13470	. . . . . . . 8 ⊢ (𝑛 ∈ (1...(♯‘𝐴)) → 𝑛 ∈ ℕ)
19		fvconst2g 7148	. . . . . . . 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 15644	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → Σ𝑘 ∈ 𝐴 𝐵 = (seq1( + , (ℕ × {𝐵}))‘(♯‘𝐴)))
22		ser1const 13982	. . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ (♯‘𝐴) ∈ ℕ) → (seq1( + , (ℕ × {𝐵}))‘(♯‘𝐴)) = ((♯‘𝐴) · 𝐵))
23	22	ad2ant2lr 749	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (seq1( + , (ℕ × {𝐵}))‘(♯‘𝐴)) = ((♯‘𝐴) · 𝐵))
24	21, 23	eqtrd 2772	. . . . 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 1935	. . 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 15634	. . 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 1542 ∃wex 1781 ∈ wcel 2114 ∅c0 4274 {csn 4568 × cxp 5620 –1-1-onto→wf1o 6489 ‘cfv 6490 (class class class)co 7358 Fincfn 8884 ℂcc 11025 0cc0 11027 1c1 11028 + caddc 11030 · cmul 11032 ℕcn 12146 ...cfz 13424 seqcseq 13925 ♯chash 14254 Σcsu 15610

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680 ax-inf2 9551 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104 ax-pre-sup 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-er 8634 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-sup 9346 df-oi 9416 df-card 9852 df-pnf 11169 df-mnf 11170 df-xr 11171 df-ltxr 11172 df-le 11173 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-3 12210 df-n0 12403 df-z 12490 df-uz 12753 df-rp 12907 df-fz 13425 df-fzo 13572 df-seq 13926 df-exp 13986 df-hash 14255 df-cj 15023 df-re 15024 df-im 15025 df-sqrt 15159 df-abs 15160 df-clim 15412 df-sum 15611

This theorem is referenced by: fsumdifsnconst 15715 o1fsum 15737 hashiun 15746 hash2iun1dif1 15748 climcndslem1 15773 climcndslem2 15774 harmonic 15783 mertenslem1 15808 sumhash 16825 cshwshashnsame 17032 lagsubg2 19127 sylow2a 19552 lebnumlem3 24908 uniioombllem4 25531 birthdaylem2 26902 basellem8 27038 0sgm 27094 musum 27141 chtleppi 27161 vmasum 27167 logfac2 27168 chpval2 27169 chpchtsum 27170 chpub 27171 logfaclbnd 27173 dchrsum2 27219 sumdchr2 27221 lgsquadlem1 27331 chebbnd1lem1 27420 chtppilimlem1 27424 dchrmusum2 27445 dchrisum0flblem1 27459 rpvmasum2 27463 dchrisum0lem2a 27468 mudivsum 27481 mulogsumlem 27482 selberglem2 27497 pntlemj 27554 rusgrnumwwlks 30034 fusgrhashclwwlkn 30138 fusgreghash2wsp 30397 numclwwlk6 30449 vietadeg1 33727 reprlt 34769 hashreprin 34770 reprgt 34771 hgt750lema 34807 rrndstprj2 38143 lcmineqlem17 42476 sticksstones10 42586 sticksstones12a 42588 fz1sumconst 42740 fltnltalem 43094 stoweidlem11 46443 stoweidlem26 46458 stoweidlem38 46470 dirkertrigeq 46533 fourierdlem73 46611 etransclem32 46698 rrndistlt 46722 sge0rpcpnf 46853 hoiqssbllem2 47055 nn0mulfsum 49058 amgmlemALT 50236

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