Theorem mulgnngsum 13434

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 9684	. . . . 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 2205	. . . . 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 5659	. . . 4 |- ( ( ( N e. NN /\ X e. B ) /\ i e. ( 1 ... N ) ) -> ( F ` i ) = X )
11		elfznn 10175	. . . . 5 |- ( i e. ( 1 ... N ) -> i e. NN )
12		fvconst2g 5797	. . . . 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 2240	. . 3 |- ( ( ( N e. NN /\ X e. B ) /\ i e. ( 1 ... N ) ) -> ( F ` i ) = ( ( NN X. { X } ) ` i ) )
15		1zzd 9398	. . . . . . 7 |- ( ( N e. NN /\ X e. B ) -> 1 e. ZZ )
16		nnz 9390	. . . . . . . 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 10574	. . . . . 6 |- ( ( N e. NN /\ X e. B ) -> ( 1 ... N ) e. Fin )
19		mptexg 5808	. . . . . . 7 |- ( ( 1 ... N ) e. Fin -> ( x e. ( 1 ... N ) |-> X ) e. _V )
20	4, 19	eqeltrid 2291	. . . . . 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 2774	. . . 4 |- a e. _V
24		fvexg 5594	. . . 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 9041	. . . . 5 |- NN e. _V
27	8	adantr 276	. . . . . 6 |- ( ( ( N e. NN /\ X e. B ) /\ a e. ( ZZ>= ` 1 ) ) -> X e. B )
28		snexg 4227	. . . . . 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 4788	. . . . 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 5594	. . . 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 12862	. . . . . 6 |- ( X e. B -> G e. _V )
36	35	adantl 277	. . . . 5 |- ( ( N e. NN /\ X e. B ) -> G e. _V )
37		plusgslid 12915	. . . . . 6 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
38	37	slotex 12830	. . . . 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 5977	. . . 4 |- ( ( a e. _V /\ ( +g ` G ) e. _V /\ b e. _V ) -> ( a ( +g ` G ) b ) e. _V )
42	23, 39, 40, 41	mp3an2ani 1356	. . 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 10622	. 2 |- ( ( N e. NN /\ X e. B ) -> ( seq 1 ( ( +g ` G ) , F ) ` N ) = ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) )
44		eqid 2204	. . 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 5733	. . 3 |- ( ( N e. NN /\ X e. B ) -> F : ( 1 ... N ) --> B )
47	34, 44, 36, 3, 46	gsumval2 13200	. 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 2204	. . 3 |- seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) = seq 1 ( ( +g ` G ) , ( NN X. { X } ) )
50	34, 44, 48, 49	mulgnn 13433	. 2 |- ( ( N e. NN /\ X e. B ) -> ( N .x. X ) = ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) )
51	43, 47, 50	3eqtr4rd 2248	1 |- ( ( N e. NN /\ X e. B ) -> ( N .x. X ) = ( G gsumg F ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1372   e. wcel 2175   _Vcvv 2771   {csn 3632   |-> cmpt 4104   X. cxp 4672   ` cfv 5270  (class class class)co 5943   Fincfn 6826   1c1 7925   NNcn 9035   ZZcz 9371   ZZ>=cuz 9647   ...cfz 10129   seqcseq 10590   Basecbs 12803   +g cplusg 12880   gsum_g cgsu 13060  ._gcmg 13426

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-addcom 8024  ax-addass 8026  ax-distr 8028  ax-i2m1 8029  ax-0lt1 8030  ax-0id 8032  ax-rnegex 8033  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-ltwlin 8037  ax-pre-lttrn 8038  ax-pre-apti 8039  ax-pre-ltadd 8040

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-iord 4412  df-on 4414  df-ilim 4415  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-recs 6390  df-frec 6476  df-1o 6501  df-er 6619  df-en 6827  df-fin 6829  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-sub 8244  df-neg 8245  df-inn 9036  df-2 9094  df-n0 9295  df-z 9372  df-uz 9648  df-fz 10130  df-seqfrec 10591  df-ndx 12806  df-slot 12807  df-base 12809  df-plusg 12893  df-0g 13061  df-igsum 13062  df-minusg 13307  df-mulg 13427

This theorem is referenced by:  mulgnn0gsum  13435