Theorem mulgnn0gsum 13702

Description: Group multiple (exponentiation) operation at a nonnegative integer expressed by a group sum. This corresponds to the definition in [Lang] p. 6, second formula. (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
mulgnn0gsum	|- ( ( N e. NN0 /\ 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 mulgnn0gsum

Step	Hyp	Ref	Expression
1		elnn0 9392	. . 3 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
2		mulgnngsum.b	. . . . . 6 |- B = ( Base ` G )
3		mulgnngsum.t	. . . . . 6 |- .x. = (.g ` G )
4		mulgnngsum.f	. . . . . 6 |- F = ( x e. ( 1 ... N ) |-> X )
5	2, 3, 4	mulgnngsum 13701	. . . . 5 |- ( ( N e. NN /\ X e. B ) -> ( N .x. X ) = ( G gsumg F ) )
6	5	ex 115	. . . 4 |- ( N e. NN -> ( X e. B -> ( N .x. X ) = ( G gsumg F ) ) )
7	2	basmex 13129	. . . . . . . 8 |- ( X e. B -> G e. _V )
8	7	adantl 277	. . . . . . 7 |- ( ( N = 0 /\ X e. B ) -> G e. _V )
9		eqid 2229	. . . . . . . 8 |- ( 0g ` G ) = ( 0g ` G )
10	9	gsum0g 13466	. . . . . . 7 |- ( G e. _V -> ( G gsumg (/) ) = ( 0g ` G ) )
11	8, 10	syl 14	. . . . . 6 |- ( ( N = 0 /\ X e. B ) -> ( G gsumg (/) ) = ( 0g ` G ) )
12		oveq2 6019	. . . . . . . . . . . 12 |- ( N = 0 -> ( 1 ... N ) = ( 1 ... 0 ) )
13		fz10 10269	. . . . . . . . . . . 12 |- ( 1 ... 0 ) = (/)
14	12, 13	eqtrdi 2278	. . . . . . . . . . 11 |- ( N = 0 -> ( 1 ... N ) = (/) )
15	14	mpteq1d 4170	. . . . . . . . . 10 |- ( N = 0 -> ( x e. ( 1 ... N ) |-> X ) = ( x e. (/) |-> X ) )
16		mpt0 5455	. . . . . . . . . 10 |- ( x e. (/) |-> X ) = (/)
17	15, 16	eqtrdi 2278	. . . . . . . . 9 |- ( N = 0 -> ( x e. ( 1 ... N ) |-> X ) = (/) )
18	4, 17	eqtrid 2274	. . . . . . . 8 |- ( N = 0 -> F = (/) )
19	18	adantr 276	. . . . . . 7 |- ( ( N = 0 /\ X e. B ) -> F = (/) )
20	19	oveq2d 6027	. . . . . 6 |- ( ( N = 0 /\ X e. B ) -> ( G gsumg F ) = ( G gsumg (/) ) )
21		oveq1 6018	. . . . . . 7 |- ( N = 0 -> ( N .x. X ) = ( 0 .x. X ) )
22	2, 9, 3	mulg0 13699	. . . . . . 7 |- ( X e. B -> ( 0 .x. X ) = ( 0g ` G ) )
23	21, 22	sylan9eq 2282	. . . . . 6 |- ( ( N = 0 /\ X e. B ) -> ( N .x. X ) = ( 0g ` G ) )
24	11, 20, 23	3eqtr4rd 2273	. . . . 5 |- ( ( N = 0 /\ X e. B ) -> ( N .x. X ) = ( G gsumg F ) )
25	24	ex 115	. . . 4 |- ( N = 0 -> ( X e. B -> ( N .x. X ) = ( G gsumg F ) ) )
26	6, 25	jaoi 721	. . 3 |- ( ( N e. NN \/ N = 0 ) -> ( X e. B -> ( N .x. X ) = ( G gsumg F ) ) )
27	1, 26	sylbi 121	. 2 |- ( N e. NN0 -> ( X e. B -> ( N .x. X ) = ( G gsumg F ) ) )
28	27	imp 124	1 |- ( ( N e. NN0 /\ X e. B ) -> ( N .x. X ) = ( G gsumg F ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 713   = wceq 1395   e. wcel 2200   _Vcvv 2800   (/)c0 3492   |-> cmpt 4146   ` cfv 5322  (class class class)co 6011   0cc0 8020   1c1 8021   NNcn 9131   NN0cn0 9390   ...cfz 10231   Basecbs 13069   0gc0g 13326   gsum_g cgsu 13327  ._gcmg 13693

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 4200  ax-sep 4203  ax-nul 4211  ax-pow 4260  ax-pr 4295  ax-un 4526  ax-setind 4631  ax-iinf 4682  ax-cnex 8111  ax-resscn 8112  ax-1cn 8113  ax-1re 8114  ax-icn 8115  ax-addcl 8116  ax-addrcl 8117  ax-mulcl 8118  ax-addcom 8120  ax-addass 8122  ax-distr 8124  ax-i2m1 8125  ax-0lt1 8126  ax-0id 8128  ax-rnegex 8129  ax-cnre 8131  ax-pre-ltirr 8132  ax-pre-ltwlin 8133  ax-pre-lttrn 8134  ax-pre-apti 8135  ax-pre-ltadd 8136

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 3890  df-int 3925  df-iun 3968  df-br 4085  df-opab 4147  df-mpt 4148  df-tr 4184  df-id 4386  df-iord 4459  df-on 4461  df-ilim 4462  df-suc 4464  df-iom 4685  df-xp 4727  df-rel 4728  df-cnv 4729  df-co 4730  df-dm 4731  df-rn 4732  df-res 4733  df-ima 4734  df-iota 5282  df-fun 5324  df-fn 5325  df-f 5326  df-f1 5327  df-fo 5328  df-f1o 5329  df-fv 5330  df-riota 5964  df-ov 6014  df-oprab 6015  df-mpo 6016  df-1st 6296  df-2nd 6297  df-recs 6464  df-frec 6550  df-1o 6575  df-er 6695  df-en 6903  df-fin 6905  df-pnf 8204  df-mnf 8205  df-xr 8206  df-ltxr 8207  df-le 8208  df-sub 8340  df-neg 8341  df-inn 9132  df-2 9190  df-n0 9391  df-z 9468  df-uz 9744  df-fz 10232  df-seqfrec 10698  df-ndx 13072  df-slot 13073  df-base 13075  df-plusg 13160  df-0g 13328  df-igsum 13329  df-minusg 13574  df-mulg 13694

This theorem is referenced by: (None)