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Theorem fprodconst 14708
Description: The product of constant terms ( k is not free in B.) (Contributed by Scott Fenton, 12-Jan-2018.)
Assertion
Ref Expression
fprodconst |- ( ( A e. Fin /\ B e. CC ) -> prod_ k e. A B = ( B ^ ( # ` A ) ) )
Distinct variable groups:   A, k   B, k
Proof of Theorem fprodconst
Dummy variables f n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 exp0 12864 . . . . 5 |- ( B e. CC -> ( B ^ 0 ) = 1 )
21eqcomd 2628 . . . 4 |- ( B e. CC -> 1 = ( B ^ 0 ) )
3 prodeq1 14639 . . . . . 6 |- ( A = (/) -> prod_ k e. A B = prod_ k e. (/) B )
4 prod0 14673 . . . . . 6 |- prod_ k e. (/) B = 1
53, 4syl6eq 2672 . . . . 5 |- ( A = (/) -> prod_ k e. A B = 1 )
6 fveq2 6191 . . . . . . 7 |- ( A = (/) -> ( # ` A ) = ( # ` (/) ) )
7 hash0 13158 . . . . . . 7 |- ( # ` (/) ) = 0
86, 7syl6eq 2672 . . . . . 6 |- ( A = (/) -> ( # ` A ) = 0 )
98oveq2d 6666 . . . . 5 |- ( A = (/) -> ( B ^ ( # ` A ) ) = ( B ^ 0 ) )
105, 9eqeq12d 2637 . . . 4 |- ( A = (/) -> ( prod_ k e. A B = ( B ^ ( # ` A ) ) <-> 1 = ( B ^ 0 ) ) )
112, 10syl5ibrcom 237 . . 3 |- ( B e. CC -> ( A = (/) -> prod_ k e. A B = ( B ^ ( # ` A ) ) ) )
1211adantl 482 . 2 |- ( ( A e. Fin /\ B e. CC ) -> ( A = (/) -> prod_ k e. A B = ( B ^ ( # ` A ) ) ) )
13 eqidd 2623 . . . . . . 7 |- ( k = ( f ` n ) -> B = B )
14 simprl 794 . . . . . . 7 |- ( ( ( A e. Fin /\ B e. CC ) /\ ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) -> ( # ` A ) e. NN )
15 simprr 796 . . . . . . 7 |- ( ( ( A e. Fin /\ B e. CC ) /\ ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) -> f : ( 1 ... ( # ` A ) ) -1-1-onto-> A )
16 simpllr 799 . . . . . . 7 |- ( ( ( ( A e. Fin /\ B e. CC ) /\ ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) /\ k e. A ) -> B e. CC )
17 simpllr 799 . . . . . . . 8 |- ( ( ( ( A e. Fin /\ B e. CC ) /\ ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) /\ n e. ( 1 ... ( # ` A ) ) ) -> B e. CC )
18 elfznn 12370 . . . . . . . . 9 |- ( n e. ( 1 ... ( # ` A ) ) -> n e. NN )
1918adantl 482 . . . . . . . 8 |- ( ( ( ( A e. Fin /\ B e. CC ) /\ ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) /\ n e. ( 1 ... ( # ` A ) ) ) -> n e. NN )
20 fvconst2g 6467 . . . . . . . 8 |- ( ( B e. CC /\ n e. NN ) -> ( ( NN X. { B } ) ` n ) = B )
2117, 19, 20syl2anc 693 . . . . . . 7 |- ( ( ( ( A e. Fin /\ B e. CC ) /\ ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) /\ n e. ( 1 ... ( # ` A ) ) ) -> ( ( NN X. { B } ) ` n ) = B )
2213, 14, 15, 16, 21fprod 14671 . . . . . 6 |- ( ( ( A e. Fin /\ B e. CC ) /\ ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) -> prod_ k e. A B = ( seq 1 ( x. , ( NN X. { B } ) ) ` ( # ` A ) ) )
23 expnnval 12863 . . . . . . 7 |- ( ( B e. CC /\ ( # ` A ) e. NN ) -> ( B ^ ( # ` A ) ) = ( seq 1 ( x. , ( NN X. { B } ) ) ` ( # ` A ) ) )
2423ad2ant2lr 784 . . . . . 6 |- ( ( ( A e. Fin /\ B e. CC ) /\ ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) -> ( B ^ ( # ` A ) ) = ( seq 1 ( x. , ( NN X. { B } ) ) ` ( # ` A ) ) )
2522, 24eqtr4d 2659 . . . . 5 |- ( ( ( A e. Fin /\ B e. CC ) /\ ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) -> prod_ k e. A B = ( B ^ ( # ` A ) ) )
2625expr 643 . . . 4 |- ( ( ( A e. Fin /\ B e. CC ) /\ ( # ` A ) e. NN ) -> ( f : ( 1 ... ( # ` A ) ) -1-1-onto-> A -> prod_ k e. A B = ( B ^ ( # ` A ) ) ) )
2726exlimdv 1861 . . 3 |- ( ( ( A e. Fin /\ B e. CC ) /\ ( # ` A ) e. NN ) -> ( E. f f : ( 1 ... ( # ` A ) ) -1-1-onto-> A -> prod_ k e. A B = ( B ^ ( # ` A ) ) ) )
2827expimpd 629 . 2 |- ( ( A e. Fin /\ B e. CC ) -> ( ( ( # ` A ) e. NN /\ E. f f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) -> prod_ k e. A B = ( B ^ ( # ` A ) ) ) )
29 fz1f1o 14441 . . 3 |- ( A e. Fin -> ( A = (/) \/ ( ( # ` A ) e. NN /\ E. f f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) )
3029adantr 481 . 2 |- ( ( A e. Fin /\ B e. CC ) -> ( A = (/) \/ ( ( # ` A ) e. NN /\ E. f f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) )
3112, 28, 30mpjaod 396 1 |- ( ( A e. Fin /\ B e. CC ) -> prod_ k e. A B = ( B ^ ( # ` A ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   (/)c0 3915   {csn 4177   X. cxp 5112   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   Fincfn 7955   CCcc 9934   0cc0 9936   1c1 9937   x. cmul 9941   NNcn 11020   ...cfz 12326   seqcseq 12801   ^cexp 12860   #chash 13117   prod_cprod 14635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-prod 14636
This theorem is referenced by:  risefallfac  14755  gausslemma2dlem5  25096  gausslemma2dlem6  25097  breprexpnat  30712  circlemethnat  30719  circlevma  30720  circlemethhgt  30721  bcprod  31624  etransclem23  40474  hoicvrrex  40770  ovnhoilem1  40815  vonsn  40905
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