Theorem gsumfzsnfd 13848

Description: Group sum of a singleton, deduction form, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Thierry Arnoux, 28-Mar-2018.) (Revised by AV, 11-Dec-2019.)

Hypotheses

Ref	Expression
gsumsnd.b	|- B = ( Base ` G )
gsumsnd.g	|- ( ph -> G e. Mnd )
gsumfzsnd.m	|- ( ph -> M e. ZZ )
gsumsnd.c	|- ( ph -> C e. B )
gsumsnd.s	|- ( ( ph /\ k = M ) -> A = C )
gsumsnfd.p	|- F/ k ph
gsumsnfd.c	|- F/_ k C

Assertion

Ref	Expression
gsumfzsnfd	|- ( ph -> ( G gsumg ( k e. { M } |-> A ) ) = C )

Distinct variable group:   k, M

Allowed substitution hints:   ph( k)   A( k)   B( k)   C( k)   G( k)

Proof of Theorem gsumfzsnfd

Step	Hyp	Ref	Expression
1		gsumsnfd.p	. . . . 5 |- F/ k ph
2		elsni 3664	. . . . . 6 |- ( k e. { M } -> k = M )
3		gsumsnd.s	. . . . . 6 |- ( ( ph /\ k = M ) -> A = C )
4	2, 3	sylan2 286	. . . . 5 |- ( ( ph /\ k e. { M } ) -> A = C )
5	1, 4	mpteq2da 4152	. . . 4 |- ( ph -> ( k e. { M } |-> A ) = ( k e. { M } |-> C ) )
6	5	oveq2d 5990	. . 3 |- ( ph -> ( G gsumg ( k e. { M } |-> A ) ) = ( G gsumg ( k e. { M } |-> C ) ) )
7		gsumfzsnd.m	. . . . . 6 |- ( ph -> M e. ZZ )
8		fzsn 10230	. . . . . 6 |- ( M e. ZZ -> ( M ... M ) = { M } )
9	7, 8	syl 14	. . . . 5 |- ( ph -> ( M ... M ) = { M } )
10	9	mpteq1d 4148	. . . 4 |- ( ph -> ( k e. ( M ... M ) |-> C ) = ( k e. { M } |-> C ) )
11	10	oveq2d 5990	. . 3 |- ( ph -> ( G gsumg ( k e. ( M ... M ) |-> C ) ) = ( G gsumg ( k e. { M } |-> C ) ) )
12		gsumsnd.g	. . . 4 |- ( ph -> G e. Mnd )
13	7	uzidd 9705	. . . 4 |- ( ph -> M e. ( ZZ>= ` M ) )
14		gsumsnd.c	. . . 4 |- ( ph -> C e. B )
15		gsumsnfd.c	. . . . 5 |- F/_ k C
16		gsumsnd.b	. . . . 5 |- B = ( Base ` G )
17		eqid 2209	. . . . 5 |- (.g ` G ) = (.g ` G )
18	15, 16, 17	gsumfzconstf 13845	. . . 4 |- ( ( G e. Mnd /\ M e. ( ZZ>= ` M ) /\ C e. B ) -> ( G gsumg ( k e. ( M ... M ) |-> C ) ) = ( ( ( M - M ) + 1 ) (.g ` G ) C ) )
19	12, 13, 14, 18	syl3anc 1252	. . 3 |- ( ph -> ( G gsumg ( k e. ( M ... M ) |-> C ) ) = ( ( ( M - M ) + 1 ) (.g ` G ) C ) )
20	6, 11, 19	3eqtr2d 2248	. 2 |- ( ph -> ( G gsumg ( k e. { M } |-> A ) ) = ( ( ( M - M ) + 1 ) (.g ` G ) C ) )
21	7	zcnd 9538	. . . . . 6 |- ( ph -> M e. CC )
22	21	subidd 8413	. . . . 5 |- ( ph -> ( M - M ) = 0 )
23	22	oveq1d 5989	. . . 4 |- ( ph -> ( ( M - M ) + 1 ) = ( 0 + 1 ) )
24		0p1e1 9192	. . . 4 |- ( 0 + 1 ) = 1
25	23, 24	eqtrdi 2258	. . 3 |- ( ph -> ( ( M - M ) + 1 ) = 1 )
26	25	oveq1d 5989	. 2 |- ( ph -> ( ( ( M - M ) + 1 ) (.g ` G ) C ) = ( 1 (.g ` G ) C ) )
27	16, 17	mulg1 13632	. . 3 |- ( C e. B -> ( 1 (.g ` G ) C ) = C )
28	14, 27	syl 14	. 2 |- ( ph -> ( 1 (.g ` G ) C ) = C )
29	20, 26, 28	3eqtrd 2246	1 |- ( ph -> ( G gsumg ( k e. { M } |-> A ) ) = C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1375   F/wnf 1486   e. wcel 2180   F/_wnfc 2339   {csn 3646   |-> cmpt 4124   ` cfv 5294  (class class class)co 5974   0cc0 7967   1c1 7968   + caddc 7970   - cmin 8285   ZZcz 9414   ZZ>=cuz 9690   ...cfz 10172   Basecbs 12998   gsum_g cgsu 13256   Mndcmnd 13415  ._gcmg 13622

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-coll 4178  ax-sep 4181  ax-nul 4189  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-iinf 4657  ax-cnex 8058  ax-resscn 8059  ax-1cn 8060  ax-1re 8061  ax-icn 8062  ax-addcl 8063  ax-addrcl 8064  ax-mulcl 8065  ax-addcom 8067  ax-addass 8069  ax-distr 8071  ax-i2m1 8072  ax-0lt1 8073  ax-0id 8075  ax-rnegex 8076  ax-cnre 8078  ax-pre-ltirr 8079  ax-pre-ltwlin 8080  ax-pre-lttrn 8081  ax-pre-apti 8082  ax-pre-ltadd 8083

This theorem depends on definitions:  df-bi 117  df-dc 839  df-3or 984  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-nel 2476  df-ral 2493  df-rex 2494  df-reu 2495  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-if 3583  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-tr 4162  df-id 4361  df-iord 4434  df-on 4436  df-ilim 4437  df-suc 4439  df-iom 4660  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-riota 5927  df-ov 5977  df-oprab 5978  df-mpo 5979  df-1st 6256  df-2nd 6257  df-recs 6421  df-frec 6507  df-1o 6532  df-er 6650  df-en 6858  df-fin 6860  df-pnf 8151  df-mnf 8152  df-xr 8153  df-ltxr 8154  df-le 8155  df-sub 8287  df-neg 8288  df-inn 9079  df-2 9137  df-n0 9338  df-z 9415  df-uz 9691  df-fz 10173  df-seqfrec 10637  df-ndx 13001  df-slot 13002  df-base 13004  df-plusg 13089  df-0g 13257  df-igsum 13258  df-minusg 13503  df-mulg 13623

This theorem is referenced by:  gsumfzfsumlemm  14516