Theorem gsummpt1n0 17188

Description: If only one summand in a finite group sum is not zero, the whole sum equals this summand. More general version of gsummptif1n0 17189. (Contributed by AV, 11-Oct-2019.)

Hypotheses

Ref	Expression
gsummpt1n0.0	|- .0. = ( 0g ` G )
gsummpt1n0.g	|- ( ph -> G e. Mnd )
gsummpt1n0.i	|- ( ph -> I e. W )
gsummpt1n0.x	|- ( ph -> X e. I )
gsummpt1n0.f	|- F = ( n e. I |-> if ( n = X , A , .0. ) )
gsummpt1n0.a	|- ( ph -> A. n e. I A e. ( Base ` G ) )

Assertion

Ref	Expression
gsummpt1n0	|- ( ph -> ( G gsumg F ) = [_ X / n ]_ A )

Distinct variable groups:   n, G   n, I   n, X   ph, n   .0. , n

Allowed substitution hints:   A( n)   F( n)   W( n)

Proof of Theorem gsummpt1n0

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2454	. . 3 |- ( Base ` G ) = ( Base ` G )
2		gsummpt1n0.0	. . 3 |- .0. = ( 0g ` G )
3		gsummpt1n0.g	. . 3 |- ( ph -> G e. Mnd )
4		gsummpt1n0.i	. . 3 |- ( ph -> I e. W )
5		gsummpt1n0.x	. . 3 |- ( ph -> X e. I )
6		gsummpt1n0.a	. . . . . 6 |- ( ph -> A. n e. I A e. ( Base ` G ) )
7	6	r19.21bi 2823	. . . . 5 |- ( ( ph /\ n e. I ) -> A e. ( Base ` G ) )
8	1, 2	mndidcl 16137	. . . . . . 7 |- ( G e. Mnd -> .0. e. ( Base ` G ) )
9	3, 8	syl 16	. . . . . 6 |- ( ph -> .0. e. ( Base ` G ) )
10	9	adantr 463	. . . . 5 |- ( ( ph /\ n e. I ) -> .0. e. ( Base ` G ) )
11	7, 10	ifcld 3972	. . . 4 |- ( ( ph /\ n e. I ) -> if ( n = X , A , .0. ) e. ( Base ` G ) )
12		gsummpt1n0.f	. . . 4 |- F = ( n e. I |-> if ( n = X , A , .0. ) )
13	11, 12	fmptd 6031	. . 3 |- ( ph -> F : I --> ( Base ` G ) )
14	12	oveq1i 6280	. . . 4 |- ( F supp .0. ) = ( ( n e. I |-> if ( n = X , A , .0. ) ) supp .0. )
15		eldifsni 4142	. . . . . . 7 |- ( n e. ( I \ { X } ) -> n =/= X )
16	15	adantl 464	. . . . . 6 |- ( ( ph /\ n e. ( I \ { X } ) ) -> n =/= X )
17		ifnefalse 3941	. . . . . 6 |- ( n =/= X -> if ( n = X , A , .0. ) = .0. )
18	16, 17	syl 16	. . . . 5 |- ( ( ph /\ n e. ( I \ { X } ) ) -> if ( n = X , A , .0. ) = .0. )
19	18, 4	suppss2 6926	. . . 4 |- ( ph -> ( ( n e. I |-> if ( n = X , A , .0. ) ) supp .0. ) C_ { X } )
20	14, 19	syl5eqss 3533	. . 3 |- ( ph -> ( F supp .0. ) C_ { X } )
21	1, 2, 3, 4, 5, 13, 20	gsumpt 17184	. 2 |- ( ph -> ( G gsumg F ) = ( F ` X ) )
22		rspcsbela 3845	. . . 4 |- ( ( X e. I /\ A. n e. I A e. ( Base ` G ) ) -> [_ X / n ]_ A e. ( Base ` G ) )
23	5, 6, 22	syl2anc 659	. . 3 |- ( ph -> [_ X / n ]_ A e. ( Base ` G ) )
24		iftrue 3935	. . . . 5 |- ( y = X -> if ( y = X , [_ y / n ]_ A , .0. ) = [_ y / n ]_ A )
25		csbeq1 3423	. . . . 5 |- ( y = X -> [_ y / n ]_ A = [_ X / n ]_ A )
26	24, 25	eqtrd 2495	. . . 4 |- ( y = X -> if ( y = X , [_ y / n ]_ A , .0. ) = [_ X / n ]_ A )
27		nfcv 2616	. . . . . 6 |- F/_ y if ( n = X , A , .0. )
28		nfv 1712	. . . . . . 7 |- F/ n y = X
29		nfcsb1v 3436	. . . . . . 7 |- F/_ n [_ y / n ]_ A
30		nfcv 2616	. . . . . . 7 |- F/_ n .0.
31	28, 29, 30	nfif 3958	. . . . . 6 |- F/_ n if ( y = X , [_ y / n ]_ A , .0. )
32		eqeq1 2458	. . . . . . 7 |- ( n = y -> ( n = X <-> y = X ) )
33		csbeq1a 3429	. . . . . . 7 |- ( n = y -> A = [_ y / n ]_ A )
34	32, 33	ifbieq1d 3952	. . . . . 6 |- ( n = y -> if ( n = X , A , .0. ) = if ( y = X , [_ y / n ]_ A , .0. ) )
35	27, 31, 34	cbvmpt 4529	. . . . 5 |- ( n e. I |-> if ( n = X , A , .0. ) ) = ( y e. I |-> if ( y = X , [_ y / n ]_ A , .0. ) )
36	12, 35	eqtri 2483	. . . 4 |- F = ( y e. I |-> if ( y = X , [_ y / n ]_ A , .0. ) )
37	26, 36	fvmptg 5929	. . 3 |- ( ( X e. I /\ [_ X / n ]_ A e. ( Base ` G ) ) -> ( F ` X ) = [_ X / n ]_ A )
38	5, 23, 37	syl2anc 659	. 2 |- ( ph -> ( F ` X ) = [_ X / n ]_ A )
39	21, 38	eqtrd 2495	1 |- ( ph -> ( G gsumg F ) = [_ X / n ]_ A )

Syntax hints:   -> wi 4   /\ wa 367   = wceq 1398   e. wcel 1823   =/= wne 2649   A.wral 2804   [_csb 3420   \ cdif 3458   ifcif 3929   {csn 4016   |-> cmpt 4497   ` cfv 5570  (class class class)co 6270   supp csupp 6891   Basecbs 14716   0gc0g 14929   gsum_g cgsu 14930   Mndcmnd 16118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-seq 12090  df-hash 12388  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-0g 14931  df-gsum 14932  df-mre 15075  df-mrc 15076  df-acs 15078  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-submnd 16166  df-mulg 16259  df-cntz 16554  df-cmn 16999

This theorem is referenced by:  gsummptif1n0  17189  gsummoncoe1  18541  scmatscm  19182  idpm2idmp  19469  mp2pm2mplem4  19477  monmat2matmon  19492