Theorem mdetunilem2 20419

Description: Lemma for mdetuni 20428. (Contributed by SO, 15-Jul-2018.)

Hypotheses

Ref	Expression
mdetuni.a	|- A = ( N Mat R )
mdetuni.b	|- B = ( Base ` A )
mdetuni.k	|- K = ( Base ` R )
mdetuni.0g	|- .0. = ( 0g ` R )
mdetuni.1r	|- .1. = ( 1r ` R )
mdetuni.pg	|- .+ = ( +g ` R )
mdetuni.tg	|- .x. = ( .r ` R )
mdetuni.n	|- ( ph -> N e. Fin )
mdetuni.r	|- ( ph -> R e. Ring )
mdetuni.ff	|- ( ph -> D : B --> K )
mdetuni.al	|- ( ph -> A. x e. B A. y e. N A. z e. N ( ( y =/= z /\ A. w e. N ( y x w ) = ( z x w ) ) -> ( D ` x ) = .0. ) )
mdetuni.li	|- ( ph -> A. x e. B A. y e. B A. z e. B A. w e. N ( ( ( x |` ( { w } X. N ) ) = ( ( y |` ( { w } X. N ) ) oF .+ ( z |` ( { w } X. N ) ) ) /\ ( x |` ( ( N \ { w } ) X. N ) ) = ( y |` ( ( N \ { w } ) X. N ) ) /\ ( x |` ( ( N \ { w } ) X. N ) ) = ( z |` ( ( N \ { w } ) X. N ) ) ) -> ( D ` x ) = ( ( D ` y ) .+ ( D ` z ) ) ) )
mdetuni.sc	|- ( ph -> A. x e. B A. y e. K A. z e. B A. w e. N ( ( ( x |` ( { w } X. N ) ) = ( ( ( { w } X. N ) X. { y } ) oF .x. ( z |` ( { w } X. N ) ) ) /\ ( x |` ( ( N \ { w } ) X. N ) ) = ( z |` ( ( N \ { w } ) X. N ) ) ) -> ( D ` x ) = ( y .x. ( D ` z ) ) ) )
mdetunilem2.ph	|- ( ps -> ph )
mdetunilem2.eg	|- ( ps -> ( E e. N /\ G e. N /\ E =/= G ) )
mdetunilem2.f	|- ( ( ps /\ b e. N ) -> F e. K )
mdetunilem2.h	|- ( ( ps /\ a e. N /\ b e. N ) -> H e. K )

Assertion

Ref	Expression
mdetunilem2	|- ( ps -> ( D ` ( a e. N , b e. N |-> if ( a = E , F , if ( a = G , F , H ) ) ) ) = .0. )

Distinct variable groups:   ph, x, y, z, w, a, b   x, B, y, z, w, a, b   x, K, y, z, w, a, b   x, N, y, z, w, a, b   x, D, y, z, w, a, b   x, .x. , y, z, w   .+ , a, b, x, y, z, w   .0. , a, b, x, y, z, w   .1. , a, b, x, y, z, w   x, R, y, z, w   A, a, b, x, y, z, w   x, E, y, z, w   x, F, y, z, w   x, G, y, z, w   x, H, y, z, w   ps, a, b, x, y, z, w   E, a, b   G, a, b   F, a

Allowed substitution hints:   R( a, b)   .x. ( a, b)   F( b)   H( a, b)

Proof of Theorem mdetunilem2

Step	Hyp	Ref	Expression
1		mdetunilem2.ph	. 2 |- ( ps -> ph )
2		mdetuni.a	. . 3 |- A = ( N Mat R )
3		mdetuni.k	. . 3 |- K = ( Base ` R )
4		mdetuni.b	. . 3 |- B = ( Base ` A )
5		mdetuni.n	. . . 4 |- ( ph -> N e. Fin )
6	1, 5	syl 17	. . 3 |- ( ps -> N e. Fin )
7		mdetuni.r	. . . 4 |- ( ph -> R e. Ring )
8	1, 7	syl 17	. . 3 |- ( ps -> R e. Ring )
9		mdetunilem2.f	. . . . 5 |- ( ( ps /\ b e. N ) -> F e. K )
10	9	3adant2 1080	. . . 4 |- ( ( ps /\ a e. N /\ b e. N ) -> F e. K )
11		mdetunilem2.h	. . . . 5 |- ( ( ps /\ a e. N /\ b e. N ) -> H e. K )
12	10, 11	ifcld 4131	. . . 4 |- ( ( ps /\ a e. N /\ b e. N ) -> if ( a = G , F , H ) e. K )
13	10, 12	ifcld 4131	. . 3 |- ( ( ps /\ a e. N /\ b e. N ) -> if ( a = E , F , if ( a = G , F , H ) ) e. K )
14	2, 3, 4, 6, 8, 13	matbas2d 20229	. 2 |- ( ps -> ( a e. N , b e. N |-> if ( a = E , F , if ( a = G , F , H ) ) ) e. B )
15		eqidd 2623	. . . . 5 |- ( ( ps /\ w e. N ) -> ( a e. N , b e. N |-> if ( a = E , F , if ( a = G , F , H ) ) ) = ( a e. N , b e. N |-> if ( a = E , F , if ( a = G , F , H ) ) ) )
16		iftrue 4092	. . . . . . 7 |- ( a = E -> if ( a = E , F , if ( a = G , F , H ) ) = F )
17		csbeq1a 3542	. . . . . . 7 |- ( b = w -> F = [_ w / b ]_ F )
18	16, 17	sylan9eq 2676	. . . . . 6 |- ( ( a = E /\ b = w ) -> if ( a = E , F , if ( a = G , F , H ) ) = [_ w / b ]_ F )
19	18	adantl 482	. . . . 5 |- ( ( ( ps /\ w e. N ) /\ ( a = E /\ b = w ) ) -> if ( a = E , F , if ( a = G , F , H ) ) = [_ w / b ]_ F )
20		eqidd 2623	. . . . 5 |- ( ( ( ps /\ w e. N ) /\ a = E ) -> N = N )
21		mdetunilem2.eg	. . . . . . 7 |- ( ps -> ( E e. N /\ G e. N /\ E =/= G ) )
22	21	simp1d 1073	. . . . . 6 |- ( ps -> E e. N )
23	22	adantr 481	. . . . 5 |- ( ( ps /\ w e. N ) -> E e. N )
24		simpr 477	. . . . 5 |- ( ( ps /\ w e. N ) -> w e. N )
25		nfv 1843	. . . . . . 7 |- F/ b ( ps /\ w e. N )
26		nfcsb1v 3549	. . . . . . . 8 |- F/_ b [_ w / b ]_ F
27	26	nfel1 2779	. . . . . . 7 |- F/ b [_ w / b ]_ F e. K
28	25, 27	nfim 1825	. . . . . 6 |- F/ b ( ( ps /\ w e. N ) -> [_ w / b ]_ F e. K )
29		eleq1 2689	. . . . . . . 8 |- ( b = w -> ( b e. N <-> w e. N ) )
30	29	anbi2d 740	. . . . . . 7 |- ( b = w -> ( ( ps /\ b e. N ) <-> ( ps /\ w e. N ) ) )
31	17	eleq1d 2686	. . . . . . 7 |- ( b = w -> ( F e. K <-> [_ w / b ]_ F e. K ) )
32	30, 31	imbi12d 334	. . . . . 6 |- ( b = w -> ( ( ( ps /\ b e. N ) -> F e. K ) <-> ( ( ps /\ w e. N ) -> [_ w / b ]_ F e. K ) ) )
33	28, 32, 9	chvar 2262	. . . . 5 |- ( ( ps /\ w e. N ) -> [_ w / b ]_ F e. K )
34		nfv 1843	. . . . 5 |- F/ a ( ps /\ w e. N )
35		nfcv 2764	. . . . 5 |- F/_ b E
36		nfcv 2764	. . . . 5 |- F/_ a w
37		nfcv 2764	. . . . 5 |- F/_ a [_ w / b ]_ F
38	15, 19, 20, 23, 24, 33, 34, 25, 35, 36, 37, 26	ovmpt2dxf 6786	. . . 4 |- ( ( ps /\ w e. N ) -> ( E ( a e. N , b e. N |-> if ( a = E , F , if ( a = G , F , H ) ) ) w ) = [_ w / b ]_ F )
39	21	simp3d 1075	. . . . . . . . . . . . 13 |- ( ps -> E =/= G )
40	39	adantr 481	. . . . . . . . . . . 12 |- ( ( ps /\ w e. N ) -> E =/= G )
41		neeq2 2857	. . . . . . . . . . . 12 |- ( a = G -> ( E =/= a <-> E =/= G ) )
42	40, 41	syl5ibrcom 237	. . . . . . . . . . 11 |- ( ( ps /\ w e. N ) -> ( a = G -> E =/= a ) )
43	42	imp 445	. . . . . . . . . 10 |- ( ( ( ps /\ w e. N ) /\ a = G ) -> E =/= a )
44	43	necomd 2849	. . . . . . . . 9 |- ( ( ( ps /\ w e. N ) /\ a = G ) -> a =/= E )
45	44	neneqd 2799	. . . . . . . 8 |- ( ( ( ps /\ w e. N ) /\ a = G ) -> -. a = E )
46	45	adantrr 753	. . . . . . 7 |- ( ( ( ps /\ w e. N ) /\ ( a = G /\ b = w ) ) -> -. a = E )
47	46	iffalsed 4097	. . . . . 6 |- ( ( ( ps /\ w e. N ) /\ ( a = G /\ b = w ) ) -> if ( a = E , F , if ( a = G , F , H ) ) = if ( a = G , F , H ) )
48		iftrue 4092	. . . . . . . 8 |- ( a = G -> if ( a = G , F , H ) = F )
49	48, 17	sylan9eq 2676	. . . . . . 7 |- ( ( a = G /\ b = w ) -> if ( a = G , F , H ) = [_ w / b ]_ F )
50	49	adantl 482	. . . . . 6 |- ( ( ( ps /\ w e. N ) /\ ( a = G /\ b = w ) ) -> if ( a = G , F , H ) = [_ w / b ]_ F )
51	47, 50	eqtrd 2656	. . . . 5 |- ( ( ( ps /\ w e. N ) /\ ( a = G /\ b = w ) ) -> if ( a = E , F , if ( a = G , F , H ) ) = [_ w / b ]_ F )
52		eqidd 2623	. . . . 5 |- ( ( ( ps /\ w e. N ) /\ a = G ) -> N = N )
53	21	simp2d 1074	. . . . . 6 |- ( ps -> G e. N )
54	53	adantr 481	. . . . 5 |- ( ( ps /\ w e. N ) -> G e. N )
55		nfcv 2764	. . . . 5 |- F/_ b G
56	15, 51, 52, 54, 24, 33, 34, 25, 55, 36, 37, 26	ovmpt2dxf 6786	. . . 4 |- ( ( ps /\ w e. N ) -> ( G ( a e. N , b e. N |-> if ( a = E , F , if ( a = G , F , H ) ) ) w ) = [_ w / b ]_ F )
57	38, 56	eqtr4d 2659	. . 3 |- ( ( ps /\ w e. N ) -> ( E ( a e. N , b e. N |-> if ( a = E , F , if ( a = G , F , H ) ) ) w ) = ( G ( a e. N , b e. N |-> if ( a = E , F , if ( a = G , F , H ) ) ) w ) )
58	57	ralrimiva 2966	. 2 |- ( ps -> A. w e. N ( E ( a e. N , b e. N |-> if ( a = E , F , if ( a = G , F , H ) ) ) w ) = ( G ( a e. N , b e. N |-> if ( a = E , F , if ( a = G , F , H ) ) ) w ) )
59		mdetuni.0g	. . 3 |- .0. = ( 0g ` R )
60		mdetuni.1r	. . 3 |- .1. = ( 1r ` R )
61		mdetuni.pg	. . 3 |- .+ = ( +g ` R )
62		mdetuni.tg	. . 3 |- .x. = ( .r ` R )
63		mdetuni.ff	. . 3 |- ( ph -> D : B --> K )
64		mdetuni.al	. . 3 |- ( ph -> A. x e. B A. y e. N A. z e. N ( ( y =/= z /\ A. w e. N ( y x w ) = ( z x w ) ) -> ( D ` x ) = .0. ) )
65		mdetuni.li	. . 3 |- ( ph -> A. x e. B A. y e. B A. z e. B A. w e. N ( ( ( x |` ( { w } X. N ) ) = ( ( y |` ( { w } X. N ) ) oF .+ ( z |` ( { w } X. N ) ) ) /\ ( x |` ( ( N \ { w } ) X. N ) ) = ( y |` ( ( N \ { w } ) X. N ) ) /\ ( x |` ( ( N \ { w } ) X. N ) ) = ( z |` ( ( N \ { w } ) X. N ) ) ) -> ( D ` x ) = ( ( D ` y ) .+ ( D ` z ) ) ) )
66		mdetuni.sc	. . 3 |- ( ph -> A. x e. B A. y e. K A. z e. B A. w e. N ( ( ( x |` ( { w } X. N ) ) = ( ( ( { w } X. N ) X. { y } ) oF .x. ( z |` ( { w } X. N ) ) ) /\ ( x |` ( ( N \ { w } ) X. N ) ) = ( z |` ( ( N \ { w } ) X. N ) ) ) -> ( D ` x ) = ( y .x. ( D ` z ) ) ) )
67	2, 4, 3, 59, 60, 61, 62, 5, 7, 63, 64, 65, 66	mdetunilem1 20418	. 2 |- ( ( ( ph /\ ( a e. N , b e. N |-> if ( a = E , F , if ( a = G , F , H ) ) ) e. B /\ A. w e. N ( E ( a e. N , b e. N |-> if ( a = E , F , if ( a = G , F , H ) ) ) w ) = ( G ( a e. N , b e. N |-> if ( a = E , F , if ( a = G , F , H ) ) ) w ) ) /\ ( E e. N /\ G e. N /\ E =/= G ) ) -> ( D ` ( a e. N , b e. N |-> if ( a = E , F , if ( a = G , F , H ) ) ) ) = .0. )
68	1, 14, 58, 21, 67	syl31anc 1329	1 |- ( ps -> ( D ` ( a e. N , b e. N |-> if ( a = E , F , if ( a = G , F , H ) ) ) ) = .0. )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   [_csb 3533   \ cdif 3571   ifcif 4086   {csn 4177   X. cxp 5112   |` cres 5116   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   oFcof 6895   Fincfn 7955   Basecbs 15857   +g cplusg 15941   .rcmulr 15942   0gc0g 16100   1rcur 18501   Ringcrg 18547   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-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-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-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-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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  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-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-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-prds 16108  df-pws 16110  df-sra 19172  df-rgmod 19173  df-dsmm 20076  df-frlm 20091  df-mat 20214

This theorem is referenced by:  mdetunilem6  20423  mdetunilem8  20425