Theorem mulgnn0gsum 13464

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 9297	. . 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 13463	. . . . 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 12891	. . . . . . . 8 |- ( X e. B -> G e. _V )
8	7	adantl 277	. . . . . . 7 |- ( ( N = 0 /\ X e. B ) -> G e. _V )
9		eqid 2205	. . . . . . . 8 |- ( 0g ` G ) = ( 0g ` G )
10	9	gsum0g 13228	. . . . . . 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 5952	. . . . . . . . . . . 12 |- ( N = 0 -> ( 1 ... N ) = ( 1 ... 0 ) )
13		fz10 10168	. . . . . . . . . . . 12 |- ( 1 ... 0 ) = (/)
14	12, 13	eqtrdi 2254	. . . . . . . . . . 11 |- ( N = 0 -> ( 1 ... N ) = (/) )
15	14	mpteq1d 4129	. . . . . . . . . 10 |- ( N = 0 -> ( x e. ( 1 ... N ) |-> X ) = ( x e. (/) |-> X ) )
16		mpt0 5403	. . . . . . . . . 10 |- ( x e. (/) |-> X ) = (/)
17	15, 16	eqtrdi 2254	. . . . . . . . 9 |- ( N = 0 -> ( x e. ( 1 ... N ) |-> X ) = (/) )
18	4, 17	eqtrid 2250	. . . . . . . 8 |- ( N = 0 -> F = (/) )
19	18	adantr 276	. . . . . . 7 |- ( ( N = 0 /\ X e. B ) -> F = (/) )
20	19	oveq2d 5960	. . . . . 6 |- ( ( N = 0 /\ X e. B ) -> ( G gsumg F ) = ( G gsumg (/) ) )
21		oveq1 5951	. . . . . . 7 |- ( N = 0 -> ( N .x. X ) = ( 0 .x. X ) )
22	2, 9, 3	mulg0 13461	. . . . . . 7 |- ( X e. B -> ( 0 .x. X ) = ( 0g ` G ) )
23	21, 22	sylan9eq 2258	. . . . . 6 |- ( ( N = 0 /\ X e. B ) -> ( N .x. X ) = ( 0g ` G ) )
24	11, 20, 23	3eqtr4rd 2249	. . . . 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 718	. . 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 710   = wceq 1373   e. wcel 2176   _Vcvv 2772   (/)c0 3460   |-> cmpt 4105   ` cfv 5271  (class class class)co 5944   0cc0 7925   1c1 7926   NNcn 9036   NN0cn0 9295   ...cfz 10130   Basecbs 12832   0gc0g 13088   gsum_g cgsu 13089  ._gcmg 13455

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-0id 8033  ax-rnegex 8034  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-iord 4413  df-on 4415  df-ilim 4416  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-frec 6477  df-1o 6502  df-er 6620  df-en 6828  df-fin 6830  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-inn 9037  df-2 9095  df-n0 9296  df-z 9373  df-uz 9649  df-fz 10131  df-seqfrec 10593  df-ndx 12835  df-slot 12836  df-base 12838  df-plusg 12922  df-0g 13090  df-igsum 13091  df-minusg 13336  df-mulg 13456

This theorem is referenced by: (None)