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Theorem fsumconst 15752

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 11324	. . . . 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 15651	. . . . 5 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ ∅ 𝐵)
5		sum0 15683	. . . . 5 ⊢ Σ𝑘 ∈ ∅ 𝐵 = 0
6	4, 5	eqtrdi 2788	. . . 4 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 𝐵 = 0)
7		fveq2 6841	. . . . . 6 ⊢ (𝐴 = ∅ → (♯‘𝐴) = (♯‘∅))
8		hash0 14329	. . . . . 6 ⊢ (♯‘∅) = 0
9	7, 8	eqtrdi 2788	. . . . 5 ⊢ (𝐴 = ∅ → (♯‘𝐴) = 0)
10	9	oveq1d 7382	. . . 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 13507	. . . . . . . 8 ⊢ (𝑛 ∈ (1...(♯‘𝐴)) → 𝑛 ∈ ℕ)
19		fvconst2g 7157	. . . . . . . 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 15682	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → Σ𝑘 ∈ 𝐴 𝐵 = (seq1( + , (ℕ × {𝐵}))‘(♯‘𝐴)))
22		ser1const 14020	. . . . . . 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 15672	. . 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 5629 –1-1-onto→wf1o 6498 ‘cfv 6499 (class class class)co 7367 Fincfn 8893 ℂcc 11036 0cc0 11038 1c1 11039 + caddc 11041 · cmul 11043 ℕcn 12174 ...cfz 13461 seqcseq 13963 ♯chash 14292 Σcsu 15648

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 5213 ax-sep 5232 ax-nul 5242 ax-pow 5308 ax-pr 5376 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

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 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6266 df-ord 6327 df-on 6328 df-lim 6329 df-suc 6330 df-iota 6455 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-er 8643 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-sup 9355 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-n0 12438 df-z 12525 df-uz 12789 df-rp 12943 df-fz 13462 df-fzo 13609 df-seq 13964 df-exp 14024 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-clim 15450 df-sum 15649

This theorem is referenced by: fsumconst1 15753 fsumdifsnconst 15754 o1fsum 15776 hashiun 15785 hash2iun1dif1 15787 climcndslem1 15814 climcndslem2 15815 harmonic 15824 mertenslem1 15849 sumhash 16867 cshwshashnsame 17074 lagsubg2 19169 sylow2a 19594 lebnumlem3 24930 uniioombllem4 25553 birthdaylem2 26916 basellem8 27051 0sgm 27107 musum 27154 chtleppi 27173 vmasum 27179 logfac2 27180 chpval2 27181 chpchtsum 27182 chpub 27183 logfaclbnd 27185 dchrsum2 27231 sumdchr2 27233 lgsquadlem1 27343 chebbnd1lem1 27432 chtppilimlem1 27436 dchrmusum2 27457 dchrisum0flblem1 27471 rpvmasum2 27475 dchrisum0lem2a 27480 mudivsum 27493 mulogsumlem 27494 selberglem2 27509 pntlemj 27566 rusgrnumwwlks 30045 fusgrhashclwwlkn 30149 fusgreghash2wsp 30408 numclwwlk6 30460 vietadeg1 33722 reprlt 34763 hashreprin 34764 reprgt 34765 hgt750lema 34801 rrndstprj2 38152 lcmineqlem17 42484 sticksstones10 42594 sticksstones12a 42596 fz1sumconst 42741 fltnltalem 43095 stoweidlem11 46439 stoweidlem26 46454 stoweidlem38 46466 dirkertrigeq 46529 fourierdlem73 46607 etransclem32 46694 rrndistlt 46718 sge0rpcpnf 46849 hoiqssbllem2 47051 nn0mulfsum 49094 amgmlemALT 50272

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