Theorem mulgnngsum 13704

Description: Group multiple (exponentiation) operation at a positive integer expressed by a group sum. (Contributed by AV, 28-Dec-2023.)

Hypotheses

Ref	Expression
mulgnngsum.b	|- B = ( Base ` G )
mulgnngsum.t	|- .x. = (.g ` G )
mulgnngsum.f	|- F = ( x e. ( 1 ... N ) |-> X )

Assertion

Ref	Expression
mulgnngsum	|- ( ( N e. NN /\ X e. B ) -> ( N .x. X ) = ( G gsumg F ) )

Distinct variable groups:   x, B   x, N   x, X

Allowed substitution hints:   .x. ( x)   F( x)   G( x)

Proof of Theorem mulgnngsum

Dummy variables a b i are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elnnuz 9783	. . . . 5 |- ( N e. NN <-> N e. ( ZZ>= ` 1 ) )
2	1	biimpi 120	. . . 4 |- ( N e. NN -> N e. ( ZZ>= ` 1 ) )
3	2	adantr 276	. . 3 |- ( ( N e. NN /\ X e. B ) -> N e. ( ZZ>= ` 1 ) )
4		mulgnngsum.f	. . . . . 6 |- F = ( x e. ( 1 ... N ) |-> X )
5	4	a1i 9	. . . . 5 |- ( ( ( N e. NN /\ X e. B ) /\ i e. ( 1 ... N ) ) -> F = ( x e. ( 1 ... N ) |-> X ) )
6		eqidd 2230	. . . . 5 |- ( ( ( ( N e. NN /\ X e. B ) /\ i e. ( 1 ... N ) ) /\ x = i ) -> X = X )
7		simpr 110	. . . . 5 |- ( ( ( N e. NN /\ X e. B ) /\ i e. ( 1 ... N ) ) -> i e. ( 1 ... N ) )
8		simpr 110	. . . . . 6 |- ( ( N e. NN /\ X e. B ) -> X e. B )
9	8	adantr 276	. . . . 5 |- ( ( ( N e. NN /\ X e. B ) /\ i e. ( 1 ... N ) ) -> X e. B )
10	5, 6, 7, 9	fvmptd 5723	. . . 4 |- ( ( ( N e. NN /\ X e. B ) /\ i e. ( 1 ... N ) ) -> ( F ` i ) = X )
11		elfznn 10279	. . . . 5 |- ( i e. ( 1 ... N ) -> i e. NN )
12		fvconst2g 5863	. . . . 5 |- ( ( X e. B /\ i e. NN ) -> ( ( NN X. { X } ) ` i ) = X )
13	8, 11, 12	syl2an 289	. . . 4 |- ( ( ( N e. NN /\ X e. B ) /\ i e. ( 1 ... N ) ) -> ( ( NN X. { X } ) ` i ) = X )
14	10, 13	eqtr4d 2265	. . 3 |- ( ( ( N e. NN /\ X e. B ) /\ i e. ( 1 ... N ) ) -> ( F ` i ) = ( ( NN X. { X } ) ` i ) )
15		1zzd 9496	. . . . . . 7 |- ( ( N e. NN /\ X e. B ) -> 1 e. ZZ )
16		nnz 9488	. . . . . . . 8 |- ( N e. NN -> N e. ZZ )
17	16	adantr 276	. . . . . . 7 |- ( ( N e. NN /\ X e. B ) -> N e. ZZ )
18	15, 17	fzfigd 10683	. . . . . 6 |- ( ( N e. NN /\ X e. B ) -> ( 1 ... N ) e. Fin )
19		mptexg 5874	. . . . . . 7 |- ( ( 1 ... N ) e. Fin -> ( x e. ( 1 ... N ) |-> X ) e. _V )
20	4, 19	eqeltrid 2316	. . . . . 6 |- ( ( 1 ... N ) e. Fin -> F e. _V )
21	18, 20	syl 14	. . . . 5 |- ( ( N e. NN /\ X e. B ) -> F e. _V )
22	21	adantr 276	. . . 4 |- ( ( ( N e. NN /\ X e. B ) /\ a e. ( ZZ>= ` 1 ) ) -> F e. _V )
23		vex 2803	. . . 4 |- a e. _V
24		fvexg 5654	. . . 4 |- ( ( F e. _V /\ a e. _V ) -> ( F ` a ) e. _V )
25	22, 23, 24	sylancl 413	. . 3 |- ( ( ( N e. NN /\ X e. B ) /\ a e. ( ZZ>= ` 1 ) ) -> ( F ` a ) e. _V )
26		nnex 9139	. . . . 5 |- NN e. _V
27	8	adantr 276	. . . . . 6 |- ( ( ( N e. NN /\ X e. B ) /\ a e. ( ZZ>= ` 1 ) ) -> X e. B )
28		snexg 4272	. . . . . 6 |- ( X e. B -> { X } e. _V )
29	27, 28	syl 14	. . . . 5 |- ( ( ( N e. NN /\ X e. B ) /\ a e. ( ZZ>= ` 1 ) ) -> { X } e. _V )
30		xpexg 4838	. . . . 5 |- ( ( NN e. _V /\ { X } e. _V ) -> ( NN X. { X } ) e. _V )
31	26, 29, 30	sylancr 414	. . . 4 |- ( ( ( N e. NN /\ X e. B ) /\ a e. ( ZZ>= ` 1 ) ) -> ( NN X. { X } ) e. _V )
32		fvexg 5654	. . . 4 |- ( ( ( NN X. { X } ) e. _V /\ a e. _V ) -> ( ( NN X. { X } ) ` a ) e. _V )
33	31, 23, 32	sylancl 413	. . 3 |- ( ( ( N e. NN /\ X e. B ) /\ a e. ( ZZ>= ` 1 ) ) -> ( ( NN X. { X } ) ` a ) e. _V )
34		mulgnngsum.b	. . . . . . 7 |- B = ( Base ` G )
35	34	basmex 13132	. . . . . 6 |- ( X e. B -> G e. _V )
36	35	adantl 277	. . . . 5 |- ( ( N e. NN /\ X e. B ) -> G e. _V )
37		plusgslid 13185	. . . . . 6 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
38	37	slotex 13099	. . . . 5 |- ( G e. _V -> ( +g ` G ) e. _V )
39	36, 38	syl 14	. . . 4 |- ( ( N e. NN /\ X e. B ) -> ( +g ` G ) e. _V )
40		simprr 531	. . . 4 |- ( ( ( N e. NN /\ X e. B ) /\ ( a e. _V /\ b e. _V ) ) -> b e. _V )
41		ovexg 6047	. . . 4 |- ( ( a e. _V /\ ( +g ` G ) e. _V /\ b e. _V ) -> ( a ( +g ` G ) b ) e. _V )
42	23, 39, 40, 41	mp3an2ani 1378	. . 3 |- ( ( ( N e. NN /\ X e. B ) /\ ( a e. _V /\ b e. _V ) ) -> ( a ( +g ` G ) b ) e. _V )
43	3, 14, 25, 33, 42	seq3fveq 10731	. 2 |- ( ( N e. NN /\ X e. B ) -> ( seq 1 ( ( +g ` G ) , F ) ` N ) = ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) )
44		eqid 2229	. . 3 |- ( +g ` G ) = ( +g ` G )
45	8	adantr 276	. . . 4 |- ( ( ( N e. NN /\ X e. B ) /\ x e. ( 1 ... N ) ) -> X e. B )
46	45, 4	fmptd 5797	. . 3 |- ( ( N e. NN /\ X e. B ) -> F : ( 1 ... N ) --> B )
47	34, 44, 36, 3, 46	gsumval2 13470	. 2 |- ( ( N e. NN /\ X e. B ) -> ( G gsumg F ) = ( seq 1 ( ( +g ` G ) , F ) ` N ) )
48		mulgnngsum.t	. . 3 |- .x. = (.g ` G )
49		eqid 2229	. . 3 |- seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) = seq 1 ( ( +g ` G ) , ( NN X. { X } ) )
50	34, 44, 48, 49	mulgnn 13703	. 2 |- ( ( N e. NN /\ X e. B ) -> ( N .x. X ) = ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) )
51	43, 47, 50	3eqtr4rd 2273	1 |- ( ( N e. NN /\ X e. B ) -> ( N .x. X ) = ( G gsumg F ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   _Vcvv 2800   {csn 3667   |-> cmpt 4148   X. cxp 4721   ` cfv 5324  (class class class)co 6013   Fincfn 6904   1c1 8023   NNcn 9133   ZZcz 9469   ZZ>=cuz 9745   ...cfz 10233   seqcseq 10699   Basecbs 13072   +g cplusg 13150   gsum_g cgsu 13330  ._gcmg 13696

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-0id 8130  ax-rnegex 8131  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-1o 6577  df-er 6697  df-en 6905  df-fin 6907  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-inn 9134  df-2 9192  df-n0 9393  df-z 9470  df-uz 9746  df-fz 10234  df-seqfrec 10700  df-ndx 13075  df-slot 13076  df-base 13078  df-plusg 13163  df-0g 13331  df-igsum 13332  df-minusg 13577  df-mulg 13697

This theorem is referenced by:  mulgnn0gsum  13705