Theorem mulgnngzsum 13907

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

Hypotheses

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

Assertion

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

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

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

Proof of Theorem mulgnngzsum

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

Step	Hyp	Ref	Expression
1		elnnuz 9938	. . . . 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		mulgnngzsum.f	. . . . 5 |- F = ( x e. ( 1 ... N ) |-> X )
5		eqidd 2239	. . . . 5 |- ( ( ( ( N e. NN /\ X e. B ) /\ i e. ( 1 ... N ) ) /\ x = i ) -> X = X )
6		simpr 110	. . . . 5 |- ( ( ( N e. NN /\ X e. B ) /\ i e. ( 1 ... N ) ) -> i e. ( 1 ... N ) )
7		simpr 110	. . . . . 6 |- ( ( N e. NN /\ X e. B ) -> X e. B )
8	7	adantr 276	. . . . 5 |- ( ( ( N e. NN /\ X e. B ) /\ i e. ( 1 ... N ) ) -> X e. B )
9	4, 5, 6, 8	fvmptd2 5781	. . . 4 |- ( ( ( N e. NN /\ X e. B ) /\ i e. ( 1 ... N ) ) -> ( F ` i ) = X )
10		elfznn 10438	. . . . 5 |- ( i e. ( 1 ... N ) -> i e. NN )
11		fvconst2g 5920	. . . . 5 |- ( ( X e. B /\ i e. NN ) -> ( ( NN X. { X } ) ` i ) = X )
12	7, 10, 11	syl2an 289	. . . 4 |- ( ( ( N e. NN /\ X e. B ) /\ i e. ( 1 ... N ) ) -> ( ( NN X. { X } ) ` i ) = X )
13	9, 12	eqtr4d 2274	. . 3 |- ( ( ( N e. NN /\ X e. B ) /\ i e. ( 1 ... N ) ) -> ( F ` i ) = ( ( NN X. { X } ) ` i ) )
14		1zzd 9650	. . . . . . 7 |- ( ( N e. NN /\ X e. B ) -> 1 e. ZZ )
15		nnz 9642	. . . . . . . 8 |- ( N e. NN -> N e. ZZ )
16	15	adantr 276	. . . . . . 7 |- ( ( N e. NN /\ X e. B ) -> N e. ZZ )
17	14, 16	fzfigd 10846	. . . . . 6 |- ( ( N e. NN /\ X e. B ) -> ( 1 ... N ) e. Fin )
18		mptexg 5933	. . . . . . 7 |- ( ( 1 ... N ) e. Fin -> ( x e. ( 1 ... N ) |-> X ) e. _V )
19	4, 18	eqeltrid 2325	. . . . . 6 |- ( ( 1 ... N ) e. Fin -> F e. _V )
20	17, 19	syl 14	. . . . 5 |- ( ( N e. NN /\ X e. B ) -> F e. _V )
21	20	adantr 276	. . . 4 |- ( ( ( N e. NN /\ X e. B ) /\ a e. ( ZZ>= ` 1 ) ) -> F e. _V )
22		vex 2824	. . . 4 |- a e. _V
23		fvexg 5709	. . . 4 |- ( ( F e. _V /\ a e. _V ) -> ( F ` a ) e. _V )
24	21, 22, 23	sylancl 417	. . 3 |- ( ( ( N e. NN /\ X e. B ) /\ a e. ( ZZ>= ` 1 ) ) -> ( F ` a ) e. _V )
25		nnex 9289	. . . . 5 |- NN e. _V
26	7	adantr 276	. . . . . 6 |- ( ( ( N e. NN /\ X e. B ) /\ a e. ( ZZ>= ` 1 ) ) -> X e. B )
27		snexg 4316	. . . . . 6 |- ( X e. B -> { X } e. _V )
28	26, 27	syl 14	. . . . 5 |- ( ( ( N e. NN /\ X e. B ) /\ a e. ( ZZ>= ` 1 ) ) -> { X } e. _V )
29		xpexg 4884	. . . . 5 |- ( ( NN e. _V /\ { X } e. _V ) -> ( NN X. { X } ) e. _V )
30	25, 28, 29	sylancr 418	. . . 4 |- ( ( ( N e. NN /\ X e. B ) /\ a e. ( ZZ>= ` 1 ) ) -> ( NN X. { X } ) e. _V )
31		fvexg 5709	. . . 4 |- ( ( ( NN X. { X } ) e. _V /\ a e. _V ) -> ( ( NN X. { X } ) ` a ) e. _V )
32	30, 22, 31	sylancl 417	. . 3 |- ( ( ( N e. NN /\ X e. B ) /\ a e. ( ZZ>= ` 1 ) ) -> ( ( NN X. { X } ) ` a ) e. _V )
33		mulgnngzsum.b	. . . . . . 7 |- B = ( Base ` G )
34	33	basmex 13390	. . . . . 6 |- ( X e. B -> G e. _V )
35	34	adantl 277	. . . . 5 |- ( ( N e. NN /\ X e. B ) -> G e. _V )
36		plusgslid 13443	. . . . . 6 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
37	36	slotex 13357	. . . . 5 |- ( G e. _V -> ( +g ` G ) e. _V )
38	35, 37	syl 14	. . . 4 |- ( ( N e. NN /\ X e. B ) -> ( +g ` G ) e. _V )
39		simprr 537	. . . 4 |- ( ( ( N e. NN /\ X e. B ) /\ ( a e. _V /\ b e. _V ) ) -> b e. _V )
40		ovexg 6109	. . . 4 |- ( ( a e. _V /\ ( +g ` G ) e. _V /\ b e. _V ) -> ( a ( +g ` G ) b ) e. _V )
41	22, 38, 39, 40	mp3an2ani 1385	. . 3 |- ( ( ( N e. NN /\ X e. B ) /\ ( a e. _V /\ b e. _V ) ) -> ( a ( +g ` G ) b ) e. _V )
42	3, 13, 24, 32, 41	seq3fveq 10894	. 2 |- ( ( N e. NN /\ X e. B ) -> ( seq 1 ( ( +g ` G ) , F ) ` N ) = ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) )
43		eqid 2238	. . 3 |- ( +g ` G ) = ( +g ` G )
44	7	adantr 276	. . . 4 |- ( ( ( N e. NN /\ X e. B ) /\ x e. ( 1 ... N ) ) -> X e. B )
45	44, 4	fmptd 5853	. . 3 |- ( ( N e. NN /\ X e. B ) -> F : ( 1 ... N ) --> B )
46	33, 43, 35, 3, 45	gzsumval2 13691	. 2 |- ( ( N e. NN /\ X e. B ) -> ( G gzsumgz F ) = ( seq 1 ( ( +g ` G ) , F ) ` N ) )
47		mulgnngzsum.t	. . 3 |- .x. = (.g ` G )
48		eqid 2238	. . 3 |- seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) = seq 1 ( ( +g ` G ) , ( NN X. { X } ) )
49	33, 43, 47, 48	mulgnn 13906	. 2 |- ( ( N e. NN /\ X e. B ) -> ( N .x. X ) = ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) )
50	42, 46, 49	3eqtr4rd 2282	1 |- ( ( N e. NN /\ X e. B ) -> ( N .x. X ) = ( G gzsumgz F ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   _Vcvv 2821   {csn 3705   |-> cmpt 4187   X. cxp 4767   ` cfv 5372  (class class class)co 6075   Fincfn 7012   1c1 8170   NNcn 9283   ZZcz 9623   ZZ>=cuz 9900   ...cfz 10390   seqcseq 10862   Basecbs 13330   +g cplusg 13408   gzsum_gz cgzsu 13588  ._gcmg 13899

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-1o 6677  df-er 6797  df-en 7013  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-2 9342  df-n0 9543  df-z 9624  df-uz 9901  df-fz 10391  df-seqfrec 10863  df-ndx 13333  df-slot 13334  df-base 13336  df-plusg 13421  df-0g 13589  df-gzsum 13590  df-minusg 13786  df-mulg 13900

This theorem is referenced by:  mulgnn0gzsum  13908