Theorem smadiadetlem0 20467

Description: Lemma 0 for smadiadet 20476: The products of the Leibniz' formula vanish for all permutations fixing the index of the row containing the 0's and the 1 to the column with the 1. (Contributed by AV, 3-Jan-2019.)

Hypotheses

Ref	Expression
marep01ma.a	|- A = ( N Mat R )
marep01ma.b	|- B = ( Base ` A )
marep01ma.r	|- R e. CRing
marep01ma.0	|- .0. = ( 0g ` R )
marep01ma.1	|- .1. = ( 1r ` R )
smadiadetlem.p	|- P = ( Base ` ( SymGrp ` N ) )
smadiadetlem.g	|- G = (mulGrp ` R )

Assertion

Ref	Expression
smadiadetlem0	|- ( ( M e. B /\ K e. N /\ L e. N ) -> ( Q e. ( P \ { q e. P | ( q ` K ) = L } ) -> ( G gsumg ( n e. N |-> ( n ( i e. N , j e. N |-> if ( i = K , if ( j = L , .1. , .0. ) , ( i M j ) ) ) ( Q ` n ) ) ) ) = .0. ) )

Distinct variable groups:   i, j, n, B   i, q, K, j, n   i, L, j, n, q   i, M, j, n   i, N, j, n   P, i, j, n, q   Q, i, j, n, q   R, i, j, n   .1. , i, j, n   .0. , i, j, n   n, G

Allowed substitution hints:   A( i, j, n, q)   B( q)   R( q)   .1. ( q)   G( i, j, q)   M( q)   N( q)   .0. ( q)

Proof of Theorem smadiadetlem0

Dummy variable m is distinct from all other variables.

Step	Hyp	Ref	Expression
1		smadiadetlem.g	. . 3 |- G = (mulGrp ` R )
2		marep01ma.0	. . 3 |- .0. = ( 0g ` R )
3		marep01ma.r	. . . 4 |- R e. CRing
4	3	a1i 11	. . 3 |- ( ( ( M e. B /\ K e. N /\ L e. N ) /\ Q e. ( P \ { q e. P | ( q ` K ) = L } ) ) -> R e. CRing )
5		marep01ma.a	. . . . . . 7 |- A = ( N Mat R )
6		marep01ma.b	. . . . . . 7 |- B = ( Base ` A )
7	5, 6	matrcl 20218	. . . . . 6 |- ( M e. B -> ( N e. Fin /\ R e. _V ) )
8	7	simpld 475	. . . . 5 |- ( M e. B -> N e. Fin )
9	8	3ad2ant1 1082	. . . 4 |- ( ( M e. B /\ K e. N /\ L e. N ) -> N e. Fin )
10	9	adantr 481	. . 3 |- ( ( ( M e. B /\ K e. N /\ L e. N ) /\ Q e. ( P \ { q e. P | ( q ` K ) = L } ) ) -> N e. Fin )
11		crngring 18558	. . . . . . 7 |- ( R e. CRing -> R e. Ring )
12	3, 11	mp1i 13	. . . . . 6 |- ( ( ( M e. B /\ K e. N /\ L e. N ) /\ Q e. ( P \ { q e. P | ( q ` K ) = L } ) ) -> R e. Ring )
13		eldifi 3732	. . . . . . 7 |- ( Q e. ( P \ { q e. P | ( q ` K ) = L } ) -> Q e. P )
14	13	adantl 482	. . . . . 6 |- ( ( ( M e. B /\ K e. N /\ L e. N ) /\ Q e. ( P \ { q e. P | ( q ` K ) = L } ) ) -> Q e. P )
15		marep01ma.1	. . . . . . . . 9 |- .1. = ( 1r ` R )
16	5, 6, 3, 2, 15	marep01ma 20466	. . . . . . . 8 |- ( M e. B -> ( i e. N , j e. N |-> if ( i = K , if ( j = L , .1. , .0. ) , ( i M j ) ) ) e. B )
17	16	3ad2ant1 1082	. . . . . . 7 |- ( ( M e. B /\ K e. N /\ L e. N ) -> ( i e. N , j e. N |-> if ( i = K , if ( j = L , .1. , .0. ) , ( i M j ) ) ) e. B )
18	17	adantr 481	. . . . . 6 |- ( ( ( M e. B /\ K e. N /\ L e. N ) /\ Q e. ( P \ { q e. P | ( q ` K ) = L } ) ) -> ( i e. N , j e. N |-> if ( i = K , if ( j = L , .1. , .0. ) , ( i M j ) ) ) e. B )
19		smadiadetlem.p	. . . . . . 7 |- P = ( Base ` ( SymGrp ` N ) )
20	5, 6, 19	matepm2cl 20269	. . . . . 6 |- ( ( R e. Ring /\ Q e. P /\ ( i e. N , j e. N |-> if ( i = K , if ( j = L , .1. , .0. ) , ( i M j ) ) ) e. B ) -> A. m e. N ( m ( i e. N , j e. N |-> if ( i = K , if ( j = L , .1. , .0. ) , ( i M j ) ) ) ( Q ` m ) ) e. ( Base ` R ) )
21	12, 14, 18, 20	syl3anc 1326	. . . . 5 |- ( ( ( M e. B /\ K e. N /\ L e. N ) /\ Q e. ( P \ { q e. P | ( q ` K ) = L } ) ) -> A. m e. N ( m ( i e. N , j e. N |-> if ( i = K , if ( j = L , .1. , .0. ) , ( i M j ) ) ) ( Q ` m ) ) e. ( Base ` R ) )
22		id 22	. . . . . . . 8 |- ( m = n -> m = n )
23		fveq2 6191	. . . . . . . 8 |- ( m = n -> ( Q ` m ) = ( Q ` n ) )
24	22, 23	oveq12d 6668	. . . . . . 7 |- ( m = n -> ( m ( i e. N , j e. N |-> if ( i = K , if ( j = L , .1. , .0. ) , ( i M j ) ) ) ( Q ` m ) ) = ( n ( i e. N , j e. N |-> if ( i = K , if ( j = L , .1. , .0. ) , ( i M j ) ) ) ( Q ` n ) ) )
25	24	eleq1d 2686	. . . . . 6 |- ( m = n -> ( ( m ( i e. N , j e. N |-> if ( i = K , if ( j = L , .1. , .0. ) , ( i M j ) ) ) ( Q ` m ) ) e. ( Base ` R ) <-> ( n ( i e. N , j e. N |-> if ( i = K , if ( j = L , .1. , .0. ) , ( i M j ) ) ) ( Q ` n ) ) e. ( Base ` R ) ) )
26	25	rspccv 3306	. . . . 5 |- ( A. m e. N ( m ( i e. N , j e. N |-> if ( i = K , if ( j = L , .1. , .0. ) , ( i M j ) ) ) ( Q ` m ) ) e. ( Base ` R ) -> ( n e. N -> ( n ( i e. N , j e. N |-> if ( i = K , if ( j = L , .1. , .0. ) , ( i M j ) ) ) ( Q ` n ) ) e. ( Base ` R ) ) )
27	21, 26	syl 17	. . . 4 |- ( ( ( M e. B /\ K e. N /\ L e. N ) /\ Q e. ( P \ { q e. P | ( q ` K ) = L } ) ) -> ( n e. N -> ( n ( i e. N , j e. N |-> if ( i = K , if ( j = L , .1. , .0. ) , ( i M j ) ) ) ( Q ` n ) ) e. ( Base ` R ) ) )
28	27	imp 445	. . 3 |- ( ( ( ( M e. B /\ K e. N /\ L e. N ) /\ Q e. ( P \ { q e. P | ( q ` K ) = L } ) ) /\ n e. N ) -> ( n ( i e. N , j e. N |-> if ( i = K , if ( j = L , .1. , .0. ) , ( i M j ) ) ) ( Q ` n ) ) e. ( Base ` R ) )
29		id 22	. . . . 5 |- ( n = m -> n = m )
30		fveq2 6191	. . . . 5 |- ( n = m -> ( Q ` n ) = ( Q ` m ) )
31	29, 30	oveq12d 6668	. . . 4 |- ( n = m -> ( n ( i e. N , j e. N |-> if ( i = K , if ( j = L , .1. , .0. ) , ( i M j ) ) ) ( Q ` n ) ) = ( m ( i e. N , j e. N |-> if ( i = K , if ( j = L , .1. , .0. ) , ( i M j ) ) ) ( Q ` m ) ) )
32	31	adantl 482	. . 3 |- ( ( ( ( M e. B /\ K e. N /\ L e. N ) /\ Q e. ( P \ { q e. P | ( q ` K ) = L } ) ) /\ n = m ) -> ( n ( i e. N , j e. N |-> if ( i = K , if ( j = L , .1. , .0. ) , ( i M j ) ) ) ( Q ` n ) ) = ( m ( i e. N , j e. N |-> if ( i = K , if ( j = L , .1. , .0. ) , ( i M j ) ) ) ( Q ` m ) ) )
33	19, 2, 15	symgmatr01 20460	. . . . 5 |- ( ( K e. N /\ L e. N ) -> ( Q e. ( P \ { q e. P | ( q ` K ) = L } ) -> E. m e. N ( m ( i e. N , j e. N |-> if ( i = K , if ( j = L , .1. , .0. ) , ( i M j ) ) ) ( Q ` m ) ) = .0. ) )
34	33	3adant1 1079	. . . 4 |- ( ( M e. B /\ K e. N /\ L e. N ) -> ( Q e. ( P \ { q e. P | ( q ` K ) = L } ) -> E. m e. N ( m ( i e. N , j e. N |-> if ( i = K , if ( j = L , .1. , .0. ) , ( i M j ) ) ) ( Q ` m ) ) = .0. ) )
35	34	imp 445	. . 3 |- ( ( ( M e. B /\ K e. N /\ L e. N ) /\ Q e. ( P \ { q e. P | ( q ` K ) = L } ) ) -> E. m e. N ( m ( i e. N , j e. N |-> if ( i = K , if ( j = L , .1. , .0. ) , ( i M j ) ) ) ( Q ` m ) ) = .0. )
36	1, 2, 4, 10, 28, 32, 35	gsummgp0 18608	. 2 |- ( ( ( M e. B /\ K e. N /\ L e. N ) /\ Q e. ( P \ { q e. P | ( q ` K ) = L } ) ) -> ( G gsumg ( n e. N |-> ( n ( i e. N , j e. N |-> if ( i = K , if ( j = L , .1. , .0. ) , ( i M j ) ) ) ( Q ` n ) ) ) ) = .0. )
37	36	ex 450	1 |- ( ( M e. B /\ K e. N /\ L e. N ) -> ( Q e. ( P \ { q e. P | ( q ` K ) = L } ) -> ( G gsumg ( n e. N |-> ( n ( i e. N , j e. N |-> if ( i = K , if ( j = L , .1. , .0. ) , ( i M j ) ) ) ( Q ` n ) ) ) ) = .0. ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   \ cdif 3571   ifcif 4086   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Fincfn 7955   Basecbs 15857   0gc0g 16100   gsum_g cgsu 16101   SymGrpcsymg 17797  mulGrpcmgp 18489   1rcur 18501   Ringcrg 18547   CRingccrg 18548   Mat cmat 20213

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-ot 4186  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-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  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-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-0g 16102  df-gsum 16103  df-prds 16108  df-pws 16110  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-mulg 17541  df-cntz 17750  df-symg 17798  df-cmn 18195  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-sra 19172  df-rgmod 19173  df-dsmm 20076  df-frlm 20091  df-mat 20214

This theorem is referenced by:  smadiadetlem1a  20469