Theorem mulgnn0gzsum 13990

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

Assertion

Ref	Expression
mulgnn0gzsum	|- ( ( N e. NN0 /\ 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 mulgnn0gzsum

Step	Hyp	Ref	Expression
1		elnn0 9576	. . 3 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
2		mulgnngzsum.b	. . . . . 6 |- B = ( Base ` G )
3		mulgnngzsum.t	. . . . . 6 |- .x. = (.g ` G )
4		mulgnngzsum.f	. . . . . 6 |- F = ( x e. ( 1 ... N ) |-> X )
5	2, 3, 4	mulgnngzsum 13989	. . . . 5 |- ( ( N e. NN /\ X e. B ) -> ( N .x. X ) = ( G gzsumgz F ) )
6	5	ex 115	. . . 4 |- ( N e. NN -> ( X e. B -> ( N .x. X ) = ( G gzsumgz F ) ) )
7	2	basmex 13470	. . . . . . . 8 |- ( X e. B -> G e. _V )
8	7	adantl 277	. . . . . . 7 |- ( ( N = 0 /\ X e. B ) -> G e. _V )
9		eqid 2238	. . . . . . . 8 |- ( 0g ` G ) = ( 0g ` G )
10	9	gzsum0 13772	. . . . . . 7 |- ( G e. _V -> ( G gzsumgz (/) ) = ( 0g ` G ) )
11	8, 10	syl 14	. . . . . 6 |- ( ( N = 0 /\ X e. B ) -> ( G gzsumgz (/) ) = ( 0g ` G ) )
12		oveq2 6096	. . . . . . . . . . . 12 |- ( N = 0 -> ( 1 ... N ) = ( 1 ... 0 ) )
13		fz10 10467	. . . . . . . . . . . 12 |- ( 1 ... 0 ) = (/)
14	12, 13	eqtrdi 2287	. . . . . . . . . . 11 |- ( N = 0 -> ( 1 ... N ) = (/) )
15	14	mpteq1d 4217	. . . . . . . . . 10 |- ( N = 0 -> ( x e. ( 1 ... N ) |-> X ) = ( x e. (/) |-> X ) )
16		mpt0 5514	. . . . . . . . . 10 |- ( x e. (/) |-> X ) = (/)
17	15, 16	eqtrdi 2287	. . . . . . . . 9 |- ( N = 0 -> ( x e. ( 1 ... N ) |-> X ) = (/) )
18	4, 17	eqtrid 2283	. . . . . . . 8 |- ( N = 0 -> F = (/) )
19	18	adantr 276	. . . . . . 7 |- ( ( N = 0 /\ X e. B ) -> F = (/) )
20	19	oveq2d 6104	. . . . . 6 |- ( ( N = 0 /\ X e. B ) -> ( G gzsumgz F ) = ( G gzsumgz (/) ) )
21		oveq1 6095	. . . . . . 7 |- ( N = 0 -> ( N .x. X ) = ( 0 .x. X ) )
22	2, 9, 3	mulg0 13987	. . . . . . 7 |- ( X e. B -> ( 0 .x. X ) = ( 0g ` G ) )
23	21, 22	sylan9eq 2291	. . . . . 6 |- ( ( N = 0 /\ X e. B ) -> ( N .x. X ) = ( 0g ` G ) )
24	11, 20, 23	3eqtr4rd 2282	. . . . 5 |- ( ( N = 0 /\ X e. B ) -> ( N .x. X ) = ( G gzsumgz F ) )
25	24	ex 115	. . . 4 |- ( N = 0 -> ( X e. B -> ( N .x. X ) = ( G gzsumgz F ) ) )
26	6, 25	jaoi 728	. . 3 |- ( ( N e. NN \/ N = 0 ) -> ( X e. B -> ( N .x. X ) = ( G gzsumgz F ) ) )
27	1, 26	sylbi 121	. 2 |- ( N e. NN0 -> ( X e. B -> ( N .x. X ) = ( G gzsumgz F ) ) )
28	27	imp 124	1 |- ( ( N e. NN0 /\ X e. B ) -> ( N .x. X ) = ( G gzsumgz F ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   \/ wo 720   = wceq 1402   e. wcel 2209   _Vcvv 2821   (/)c0 3520   |-> cmpt 4193   ` cfv 5380  (class class class)co 6088   0cc0 8185   1c1 8186   NNcn 9313   NN0cn0 9574   ...cfz 10428   Basecbs 13410   0gc0g 13669   gzsum_gz cgzsu 13670  ._gcmg 13981

This proof depends on 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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-addass 8287  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-0id 8293  ax-rnegex 8294  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301

This proof 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 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-1o 6691  df-er 6811  df-en 7027  df-fin 7029  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-inn 9314  df-2 9372  df-n0 9575  df-z 9656  df-uz 9938  df-fz 10429  df-seqfrec 10906  df-ndx 13413  df-slot 13414  df-base 13416  df-plusg 13503  df-0g 13671  df-gzsum 13672  df-minusg 13868  df-mulg 13982

This theorem is used by: (None)