Theorem mulgnngsum 13197

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 9629	. . . . 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 2194	. . . . 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 5638	. . . 4 |- ( ( ( N e. NN /\ X e. B ) /\ i e. ( 1 ... N ) ) -> ( F ` i ) = X )
11		elfznn 10120	. . . . 5 |- ( i e. ( 1 ... N ) -> i e. NN )
12		fvconst2g 5772	. . . . 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 2229	. . 3 |- ( ( ( N e. NN /\ X e. B ) /\ i e. ( 1 ... N ) ) -> ( F ` i ) = ( ( NN X. { X } ) ` i ) )
15		1zzd 9344	. . . . . . 7 |- ( ( N e. NN /\ X e. B ) -> 1 e. ZZ )
16		nnz 9336	. . . . . . . 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 10502	. . . . . 6 |- ( ( N e. NN /\ X e. B ) -> ( 1 ... N ) e. Fin )
19		mptexg 5783	. . . . . . 7 |- ( ( 1 ... N ) e. Fin -> ( x e. ( 1 ... N ) |-> X ) e. _V )
20	4, 19	eqeltrid 2280	. . . . . 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 2763	. . . 4 |- a e. _V
24		fvexg 5573	. . . 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 8988	. . . . 5 |- NN e. _V
27	8	adantr 276	. . . . . 6 |- ( ( ( N e. NN /\ X e. B ) /\ a e. ( ZZ>= ` 1 ) ) -> X e. B )
28		snexg 4213	. . . . . 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 4773	. . . . 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 5573	. . . 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 12677	. . . . . 6 |- ( X e. B -> G e. _V )
36	35	adantl 277	. . . . 5 |- ( ( N e. NN /\ X e. B ) -> G e. _V )
37		plusgslid 12730	. . . . . 6 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
38	37	slotex 12645	. . . . 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 5952	. . . 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 10550	. 2 |- ( ( N e. NN /\ X e. B ) -> ( seq 1 ( ( +g ` G ) , F ) ` N ) = ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) )
44		eqid 2193	. . 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 5712	. . 3 |- ( ( N e. NN /\ X e. B ) -> F : ( 1 ... N ) --> B )
47	34, 44, 36, 3, 46	gsumval2 12980	. 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 2193	. . 3 |- seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) = seq 1 ( ( +g ` G ) , ( NN X. { X } ) )
50	34, 44, 48, 49	mulgnn 13196	. 2 |- ( ( N e. NN /\ X e. B ) -> ( N .x. X ) = ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) )
51	43, 47, 50	3eqtr4rd 2237	1 |- ( ( N e. NN /\ X e. B ) -> ( N .x. X ) = ( G gsumg F ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   _Vcvv 2760   {csn 3618   |-> cmpt 4090   X. cxp 4657   ` cfv 5254  (class class class)co 5918   Fincfn 6794   1c1 7873   NNcn 8982   ZZcz 9317   ZZ>=cuz 9592   ...cfz 10074   seqcseq 10518   Basecbs 12618   +g cplusg 12695   gsum_g cgsu 12868  ._gcmg 13189

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988

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 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-1o 6469  df-er 6587  df-en 6795  df-fin 6797  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-inn 8983  df-2 9041  df-n0 9241  df-z 9318  df-uz 9593  df-fz 10075  df-seqfrec 10519  df-ndx 12621  df-slot 12622  df-base 12624  df-plusg 12708  df-0g 12869  df-igsum 12870  df-minusg 13076  df-mulg 13190

This theorem is referenced by:  mulgnn0gsum  13198