Theorem mulgnngsum 13333

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 9655	. . . . 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 2197	. . . . 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 5645	. . . 4 |- ( ( ( N e. NN /\ X e. B ) /\ i e. ( 1 ... N ) ) -> ( F ` i ) = X )
11		elfznn 10146	. . . . 5 |- ( i e. ( 1 ... N ) -> i e. NN )
12		fvconst2g 5779	. . . . 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 2232	. . 3 |- ( ( ( N e. NN /\ X e. B ) /\ i e. ( 1 ... N ) ) -> ( F ` i ) = ( ( NN X. { X } ) ` i ) )
15		1zzd 9370	. . . . . . 7 |- ( ( N e. NN /\ X e. B ) -> 1 e. ZZ )
16		nnz 9362	. . . . . . . 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 10540	. . . . . 6 |- ( ( N e. NN /\ X e. B ) -> ( 1 ... N ) e. Fin )
19		mptexg 5790	. . . . . . 7 |- ( ( 1 ... N ) e. Fin -> ( x e. ( 1 ... N ) |-> X ) e. _V )
20	4, 19	eqeltrid 2283	. . . . . 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 2766	. . . 4 |- a e. _V
24		fvexg 5580	. . . 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 9013	. . . . 5 |- NN e. _V
27	8	adantr 276	. . . . . 6 |- ( ( ( N e. NN /\ X e. B ) /\ a e. ( ZZ>= ` 1 ) ) -> X e. B )
28		snexg 4218	. . . . . 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 4778	. . . . 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 5580	. . . 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 12762	. . . . . 6 |- ( X e. B -> G e. _V )
36	35	adantl 277	. . . . 5 |- ( ( N e. NN /\ X e. B ) -> G e. _V )
37		plusgslid 12815	. . . . . 6 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
38	37	slotex 12730	. . . . 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 5959	. . . 4 |- ( ( a e. _V /\ ( +g ` G ) e. _V /\ b e. _V ) -> ( a ( +g ` G ) b ) e. _V )
42	23, 39, 40, 41	mp3an2ani 1355	. . 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 10588	. 2 |- ( ( N e. NN /\ X e. B ) -> ( seq 1 ( ( +g ` G ) , F ) ` N ) = ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) )
44		eqid 2196	. . 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 5719	. . 3 |- ( ( N e. NN /\ X e. B ) -> F : ( 1 ... N ) --> B )
47	34, 44, 36, 3, 46	gsumval2 13099	. 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 2196	. . 3 |- seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) = seq 1 ( ( +g ` G ) , ( NN X. { X } ) )
50	34, 44, 48, 49	mulgnn 13332	. 2 |- ( ( N e. NN /\ X e. B ) -> ( N .x. X ) = ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) )
51	43, 47, 50	3eqtr4rd 2240	1 |- ( ( N e. NN /\ X e. B ) -> ( N .x. X ) = ( G gsumg F ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   _Vcvv 2763   {csn 3623   |-> cmpt 4095   X. cxp 4662   ` cfv 5259  (class class class)co 5925   Fincfn 6808   1c1 7897   NNcn 9007   ZZcz 9343   ZZ>=cuz 9618   ...cfz 10100   seqcseq 10556   Basecbs 12703   +g cplusg 12780   gsum_g cgsu 12959  ._gcmg 13325

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-1o 6483  df-er 6601  df-en 6809  df-fin 6811  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-inn 9008  df-2 9066  df-n0 9267  df-z 9344  df-uz 9619  df-fz 10101  df-seqfrec 10557  df-ndx 12706  df-slot 12707  df-base 12709  df-plusg 12793  df-0g 12960  df-igsum 12961  df-minusg 13206  df-mulg 13326

This theorem is referenced by:  mulgnn0gsum  13334