Theorem gsumpr 42139

Description: Group sum of a pair. (Contributed by AV, 6-Dec-2018.) (Proof shortened by AV, 28-Jul-2019.)

Hypotheses

Ref	Expression
gsumpr.b	|- B = ( Base ` G )
gsumpr.p	|- .+ = ( +g ` G )
gsumpr.s	|- ( k = M -> A = C )
gsumpr.t	|- ( k = N -> A = D )

Assertion

Ref	Expression
gsumpr	|- ( ( G e. CMnd /\ ( M e. V /\ N e. W /\ M =/= N ) /\ ( C e. B /\ D e. B ) ) -> ( G gsumg ( k e. { M , N } |-> A ) ) = ( C .+ D ) )

Distinct variable groups:   B, k   C, k   D, k   k, G   k, M   k, N   k, V   k, W

Allowed substitution hints:   A( k)   .+ ( k)

Proof of Theorem gsumpr

Step	Hyp	Ref	Expression
1		gsumpr.b	. . 3 |- B = ( Base ` G )
2		gsumpr.p	. . 3 |- .+ = ( +g ` G )
3		simp1 1061	. . 3 |- ( ( G e. CMnd /\ ( M e. V /\ N e. W /\ M =/= N ) /\ ( C e. B /\ D e. B ) ) -> G e. CMnd )
4		prfi 8235	. . . 4 |- { M , N } e. Fin
5	4	a1i 11	. . 3 |- ( ( G e. CMnd /\ ( M e. V /\ N e. W /\ M =/= N ) /\ ( C e. B /\ D e. B ) ) -> { M , N } e. Fin )
6		vex 3203	. . . . . 6 |- k e. _V
7	6	elpr 4198	. . . . 5 |- ( k e. { M , N } <-> ( k = M \/ k = N ) )
8		gsumpr.s	. . . . . . 7 |- ( k = M -> A = C )
9		eleq1a 2696	. . . . . . . . 9 |- ( C e. B -> ( A = C -> A e. B ) )
10	9	adantr 481	. . . . . . . 8 |- ( ( C e. B /\ D e. B ) -> ( A = C -> A e. B ) )
11	10	3ad2ant3 1084	. . . . . . 7 |- ( ( G e. CMnd /\ ( M e. V /\ N e. W /\ M =/= N ) /\ ( C e. B /\ D e. B ) ) -> ( A = C -> A e. B ) )
12	8, 11	syl5com 31	. . . . . 6 |- ( k = M -> ( ( G e. CMnd /\ ( M e. V /\ N e. W /\ M =/= N ) /\ ( C e. B /\ D e. B ) ) -> A e. B ) )
13		gsumpr.t	. . . . . . 7 |- ( k = N -> A = D )
14		eleq1a 2696	. . . . . . . . 9 |- ( D e. B -> ( A = D -> A e. B ) )
15	14	adantl 482	. . . . . . . 8 |- ( ( C e. B /\ D e. B ) -> ( A = D -> A e. B ) )
16	15	3ad2ant3 1084	. . . . . . 7 |- ( ( G e. CMnd /\ ( M e. V /\ N e. W /\ M =/= N ) /\ ( C e. B /\ D e. B ) ) -> ( A = D -> A e. B ) )
17	13, 16	syl5com 31	. . . . . 6 |- ( k = N -> ( ( G e. CMnd /\ ( M e. V /\ N e. W /\ M =/= N ) /\ ( C e. B /\ D e. B ) ) -> A e. B ) )
18	12, 17	jaoi 394	. . . . 5 |- ( ( k = M \/ k = N ) -> ( ( G e. CMnd /\ ( M e. V /\ N e. W /\ M =/= N ) /\ ( C e. B /\ D e. B ) ) -> A e. B ) )
19	7, 18	sylbi 207	. . . 4 |- ( k e. { M , N } -> ( ( G e. CMnd /\ ( M e. V /\ N e. W /\ M =/= N ) /\ ( C e. B /\ D e. B ) ) -> A e. B ) )
20	19	impcom 446	. . 3 |- ( ( ( G e. CMnd /\ ( M e. V /\ N e. W /\ M =/= N ) /\ ( C e. B /\ D e. B ) ) /\ k e. { M , N } ) -> A e. B )
21		disjsn2 4247	. . . . 5 |- ( M =/= N -> ( { M } i^i { N } ) = (/) )
22	21	3ad2ant3 1084	. . . 4 |- ( ( M e. V /\ N e. W /\ M =/= N ) -> ( { M } i^i { N } ) = (/) )
23	22	3ad2ant2 1083	. . 3 |- ( ( G e. CMnd /\ ( M e. V /\ N e. W /\ M =/= N ) /\ ( C e. B /\ D e. B ) ) -> ( { M } i^i { N } ) = (/) )
24		df-pr 4180	. . . 4 |- { M , N } = ( { M } u. { N } )
25	24	a1i 11	. . 3 |- ( ( G e. CMnd /\ ( M e. V /\ N e. W /\ M =/= N ) /\ ( C e. B /\ D e. B ) ) -> { M , N } = ( { M } u. { N } ) )
26		eqid 2622	. . 3 |- ( k e. { M , N } |-> A ) = ( k e. { M , N } |-> A )
27	1, 2, 3, 5, 20, 23, 25, 26	gsummptfidmsplitres 18331	. 2 |- ( ( G e. CMnd /\ ( M e. V /\ N e. W /\ M =/= N ) /\ ( C e. B /\ D e. B ) ) -> ( G gsumg ( k e. { M , N } |-> A ) ) = ( ( G gsumg ( ( k e. { M , N } |-> A ) |` { M } ) ) .+ ( G gsumg ( ( k e. { M , N } |-> A ) |` { N } ) ) ) )
28		snsspr1 4345	. . . . . 6 |- { M } C_ { M , N }
29		resmpt 5449	. . . . . 6 |- ( { M } C_ { M , N } -> ( ( k e. { M , N } |-> A ) |` { M } ) = ( k e. { M } |-> A ) )
30	28, 29	mp1i 13	. . . . 5 |- ( ( G e. CMnd /\ ( M e. V /\ N e. W /\ M =/= N ) /\ ( C e. B /\ D e. B ) ) -> ( ( k e. { M , N } |-> A ) |` { M } ) = ( k e. { M } |-> A ) )
31	30	oveq2d 6666	. . . 4 |- ( ( G e. CMnd /\ ( M e. V /\ N e. W /\ M =/= N ) /\ ( C e. B /\ D e. B ) ) -> ( G gsumg ( ( k e. { M , N } |-> A ) |` { M } ) ) = ( G gsumg ( k e. { M } |-> A ) ) )
32		cmnmnd 18208	. . . . 5 |- ( G e. CMnd -> G e. Mnd )
33		simp1 1061	. . . . 5 |- ( ( M e. V /\ N e. W /\ M =/= N ) -> M e. V )
34		simpl 473	. . . . 5 |- ( ( C e. B /\ D e. B ) -> C e. B )
35	1, 8	gsumsn 18354	. . . . 5 |- ( ( G e. Mnd /\ M e. V /\ C e. B ) -> ( G gsumg ( k e. { M } |-> A ) ) = C )
36	32, 33, 34, 35	syl3an 1368	. . . 4 |- ( ( G e. CMnd /\ ( M e. V /\ N e. W /\ M =/= N ) /\ ( C e. B /\ D e. B ) ) -> ( G gsumg ( k e. { M } |-> A ) ) = C )
37	31, 36	eqtrd 2656	. . 3 |- ( ( G e. CMnd /\ ( M e. V /\ N e. W /\ M =/= N ) /\ ( C e. B /\ D e. B ) ) -> ( G gsumg ( ( k e. { M , N } |-> A ) |` { M } ) ) = C )
38		snsspr2 4346	. . . . . 6 |- { N } C_ { M , N }
39		resmpt 5449	. . . . . 6 |- ( { N } C_ { M , N } -> ( ( k e. { M , N } |-> A ) |` { N } ) = ( k e. { N } |-> A ) )
40	38, 39	mp1i 13	. . . . 5 |- ( ( G e. CMnd /\ ( M e. V /\ N e. W /\ M =/= N ) /\ ( C e. B /\ D e. B ) ) -> ( ( k e. { M , N } |-> A ) |` { N } ) = ( k e. { N } |-> A ) )
41	40	oveq2d 6666	. . . 4 |- ( ( G e. CMnd /\ ( M e. V /\ N e. W /\ M =/= N ) /\ ( C e. B /\ D e. B ) ) -> ( G gsumg ( ( k e. { M , N } |-> A ) |` { N } ) ) = ( G gsumg ( k e. { N } |-> A ) ) )
42		simp2 1062	. . . . 5 |- ( ( M e. V /\ N e. W /\ M =/= N ) -> N e. W )
43		simpr 477	. . . . 5 |- ( ( C e. B /\ D e. B ) -> D e. B )
44	1, 13	gsumsn 18354	. . . . 5 |- ( ( G e. Mnd /\ N e. W /\ D e. B ) -> ( G gsumg ( k e. { N } |-> A ) ) = D )
45	32, 42, 43, 44	syl3an 1368	. . . 4 |- ( ( G e. CMnd /\ ( M e. V /\ N e. W /\ M =/= N ) /\ ( C e. B /\ D e. B ) ) -> ( G gsumg ( k e. { N } |-> A ) ) = D )
46	41, 45	eqtrd 2656	. . 3 |- ( ( G e. CMnd /\ ( M e. V /\ N e. W /\ M =/= N ) /\ ( C e. B /\ D e. B ) ) -> ( G gsumg ( ( k e. { M , N } |-> A ) |` { N } ) ) = D )
47	37, 46	oveq12d 6668	. 2 |- ( ( G e. CMnd /\ ( M e. V /\ N e. W /\ M =/= N ) /\ ( C e. B /\ D e. B ) ) -> ( ( G gsumg ( ( k e. { M , N } |-> A ) |` { M } ) ) .+ ( G gsumg ( ( k e. { M , N } |-> A ) |` { N } ) ) ) = ( C .+ D ) )
48	27, 47	eqtrd 2656	1 |- ( ( G e. CMnd /\ ( M e. V /\ N e. W /\ M =/= N ) /\ ( C e. B /\ D e. B ) ) -> ( G gsumg ( k e. { M , N } |-> A ) ) = ( C .+ D ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   {cpr 4179   |-> cmpt 4729   |` cres 5116   ` cfv 5888  (class class class)co 6650   Fincfn 7955   Basecbs 15857   +g cplusg 15941   gsum_g cgsu 16101   Mndcmnd 17294  CMndccmn 18193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-gsum 16103  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195

This theorem is referenced by:  lincvalpr  42207  zlmodzxzldeplem3  42291