Theorem dsmmbas2 20081

Description: Base set of the direct sum module using the fndmin 6324 abbreviation. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Hypotheses

Ref	Expression
dsmmbas2.p	|- P = ( S X_s R )
dsmmbas2.b	|- B = { f e. ( Base ` P ) | dom ( f \ ( 0g o. R ) ) e. Fin }

Assertion

Ref	Expression
dsmmbas2	|- ( ( R Fn I /\ I e. V ) -> B = ( Base ` ( S (+)m R ) ) )

Distinct variable groups:   S, f   R, f   P, f   f, I   f, V

Allowed substitution hint:   B( f)

Proof of Theorem dsmmbas2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dsmmbas2.b	. 2 |- B = { f e. ( Base ` P ) | dom ( f \ ( 0g o. R ) ) e. Fin }
2		dsmmbas2.p	. . . . . 6 |- P = ( S X_s R )
3	2	fveq2i 6194	. . . . 5 |- ( Base ` P ) = ( Base ` ( S X_s R ) )
4		rabeq 3192	. . . . 5 |- ( ( Base ` P ) = ( Base ` ( S X_s R ) ) -> { f e. ( Base ` P ) | dom ( f \ ( 0g o. R ) ) e. Fin } = { f e. ( Base ` ( S X_s R ) ) | dom ( f \ ( 0g o. R ) ) e. Fin } )
5	3, 4	ax-mp 5	. . . 4 |- { f e. ( Base ` P ) | dom ( f \ ( 0g o. R ) ) e. Fin } = { f e. ( Base ` ( S X_s R ) ) | dom ( f \ ( 0g o. R ) ) e. Fin }
6		simpll 790	. . . . . . . . . 10 |- ( ( ( R Fn I /\ I e. V ) /\ f e. ( Base ` ( S X_s R ) ) ) -> R Fn I )
7		fvco2 6273	. . . . . . . . . 10 |- ( ( R Fn I /\ x e. I ) -> ( ( 0g o. R ) ` x ) = ( 0g ` ( R ` x ) ) )
8	6, 7	sylan 488	. . . . . . . . 9 |- ( ( ( ( R Fn I /\ I e. V ) /\ f e. ( Base ` ( S X_s R ) ) ) /\ x e. I ) -> ( ( 0g o. R ) ` x ) = ( 0g ` ( R ` x ) ) )
9	8	neeq2d 2854	. . . . . . . 8 |- ( ( ( ( R Fn I /\ I e. V ) /\ f e. ( Base ` ( S X_s R ) ) ) /\ x e. I ) -> ( ( f ` x ) =/= ( ( 0g o. R ) ` x ) <-> ( f ` x ) =/= ( 0g ` ( R ` x ) ) ) )
10	9	rabbidva 3188	. . . . . . 7 |- ( ( ( R Fn I /\ I e. V ) /\ f e. ( Base ` ( S X_s R ) ) ) -> { x e. I | ( f ` x ) =/= ( ( 0g o. R ) ` x ) } = { x e. I | ( f ` x ) =/= ( 0g ` ( R ` x ) ) } )
11		eqid 2622	. . . . . . . . 9 |- ( S X_s R ) = ( S X_s R )
12		eqid 2622	. . . . . . . . 9 |- ( Base ` ( S X_s R ) ) = ( Base ` ( S X_s R ) )
13		noel 3919	. . . . . . . . . . . 12 |- -. f e. (/)
14		reldmprds 16109	. . . . . . . . . . . . . . . 16 |- Rel dom X_s
15	14	ovprc1 6684	. . . . . . . . . . . . . . 15 |- ( -. S e. _V -> ( S X_s R ) = (/) )
16	15	fveq2d 6195	. . . . . . . . . . . . . 14 |- ( -. S e. _V -> ( Base ` ( S X_s R ) ) = ( Base ` (/) ) )
17		base0 15912	. . . . . . . . . . . . . 14 |- (/) = ( Base ` (/) )
18	16, 17	syl6eqr 2674	. . . . . . . . . . . . 13 |- ( -. S e. _V -> ( Base ` ( S X_s R ) ) = (/) )
19	18	eleq2d 2687	. . . . . . . . . . . 12 |- ( -. S e. _V -> ( f e. ( Base ` ( S X_s R ) ) <-> f e. (/) ) )
20	13, 19	mtbiri 317	. . . . . . . . . . 11 |- ( -. S e. _V -> -. f e. ( Base ` ( S X_s R ) ) )
21	20	con4i 113	. . . . . . . . . 10 |- ( f e. ( Base ` ( S X_s R ) ) -> S e. _V )
22	21	adantl 482	. . . . . . . . 9 |- ( ( ( R Fn I /\ I e. V ) /\ f e. ( Base ` ( S X_s R ) ) ) -> S e. _V )
23		simplr 792	. . . . . . . . 9 |- ( ( ( R Fn I /\ I e. V ) /\ f e. ( Base ` ( S X_s R ) ) ) -> I e. V )
24		simpr 477	. . . . . . . . 9 |- ( ( ( R Fn I /\ I e. V ) /\ f e. ( Base ` ( S X_s R ) ) ) -> f e. ( Base ` ( S X_s R ) ) )
25	11, 12, 22, 23, 6, 24	prdsbasfn 16131	. . . . . . . 8 |- ( ( ( R Fn I /\ I e. V ) /\ f e. ( Base ` ( S X_s R ) ) ) -> f Fn I )
26		fn0g 17262	. . . . . . . . . . . 12 |- 0g Fn _V
27		dffn2 6047	. . . . . . . . . . . 12 |- ( 0g Fn _V <-> 0g : _V --> _V )
28	26, 27	mpbi 220	. . . . . . . . . . 11 |- 0g : _V --> _V
29		dffn2 6047	. . . . . . . . . . . 12 |- ( R Fn I <-> R : I --> _V )
30	29	biimpi 206	. . . . . . . . . . 11 |- ( R Fn I -> R : I --> _V )
31		fco 6058	. . . . . . . . . . 11 |- ( ( 0g : _V --> _V /\ R : I --> _V ) -> ( 0g o. R ) : I --> _V )
32	28, 30, 31	sylancr 695	. . . . . . . . . 10 |- ( R Fn I -> ( 0g o. R ) : I --> _V )
33		ffn 6045	. . . . . . . . . 10 |- ( ( 0g o. R ) : I --> _V -> ( 0g o. R ) Fn I )
34	32, 33	syl 17	. . . . . . . . 9 |- ( R Fn I -> ( 0g o. R ) Fn I )
35	34	ad2antrr 762	. . . . . . . 8 |- ( ( ( R Fn I /\ I e. V ) /\ f e. ( Base ` ( S X_s R ) ) ) -> ( 0g o. R ) Fn I )
36		fndmdif 6321	. . . . . . . 8 |- ( ( f Fn I /\ ( 0g o. R ) Fn I ) -> dom ( f \ ( 0g o. R ) ) = { x e. I | ( f ` x ) =/= ( ( 0g o. R ) ` x ) } )
37	25, 35, 36	syl2anc 693	. . . . . . 7 |- ( ( ( R Fn I /\ I e. V ) /\ f e. ( Base ` ( S X_s R ) ) ) -> dom ( f \ ( 0g o. R ) ) = { x e. I | ( f ` x ) =/= ( ( 0g o. R ) ` x ) } )
38		fndm 5990	. . . . . . . . 9 |- ( R Fn I -> dom R = I )
39		rabeq 3192	. . . . . . . . 9 |- ( dom R = I -> { x e. dom R | ( f ` x ) =/= ( 0g ` ( R ` x ) ) } = { x e. I | ( f ` x ) =/= ( 0g ` ( R ` x ) ) } )
40	38, 39	syl 17	. . . . . . . 8 |- ( R Fn I -> { x e. dom R | ( f ` x ) =/= ( 0g ` ( R ` x ) ) } = { x e. I | ( f ` x ) =/= ( 0g ` ( R ` x ) ) } )
41	40	ad2antrr 762	. . . . . . 7 |- ( ( ( R Fn I /\ I e. V ) /\ f e. ( Base ` ( S X_s R ) ) ) -> { x e. dom R | ( f ` x ) =/= ( 0g ` ( R ` x ) ) } = { x e. I | ( f ` x ) =/= ( 0g ` ( R ` x ) ) } )
42	10, 37, 41	3eqtr4d 2666	. . . . . 6 |- ( ( ( R Fn I /\ I e. V ) /\ f e. ( Base ` ( S X_s R ) ) ) -> dom ( f \ ( 0g o. R ) ) = { x e. dom R | ( f ` x ) =/= ( 0g ` ( R ` x ) ) } )
43	42	eleq1d 2686	. . . . 5 |- ( ( ( R Fn I /\ I e. V ) /\ f e. ( Base ` ( S X_s R ) ) ) -> ( dom ( f \ ( 0g o. R ) ) e. Fin <-> { x e. dom R | ( f ` x ) =/= ( 0g ` ( R ` x ) ) } e. Fin ) )
44	43	rabbidva 3188	. . . 4 |- ( ( R Fn I /\ I e. V ) -> { f e. ( Base ` ( S X_s R ) ) | dom ( f \ ( 0g o. R ) ) e. Fin } = { f e. ( Base ` ( S X_s R ) ) | { x e. dom R | ( f ` x ) =/= ( 0g ` ( R ` x ) ) } e. Fin } )
45	5, 44	syl5eq 2668	. . 3 |- ( ( R Fn I /\ I e. V ) -> { f e. ( Base ` P ) | dom ( f \ ( 0g o. R ) ) e. Fin } = { f e. ( Base ` ( S X_s R ) ) | { x e. dom R | ( f ` x ) =/= ( 0g ` ( R ` x ) ) } e. Fin } )
46		fnex 6481	. . . 4 |- ( ( R Fn I /\ I e. V ) -> R e. _V )
47		eqid 2622	. . . . 5 |- { f e. ( Base ` ( S X_s R ) ) | { x e. dom R | ( f ` x ) =/= ( 0g ` ( R ` x ) ) } e. Fin } = { f e. ( Base ` ( S X_s R ) ) | { x e. dom R | ( f ` x ) =/= ( 0g ` ( R ` x ) ) } e. Fin }
48	47	dsmmbase 20079	. . . 4 |- ( R e. _V -> { f e. ( Base ` ( S X_s R ) ) | { x e. dom R | ( f ` x ) =/= ( 0g ` ( R ` x ) ) } e. Fin } = ( Base ` ( S (+)m R ) ) )
49	46, 48	syl 17	. . 3 |- ( ( R Fn I /\ I e. V ) -> { f e. ( Base ` ( S X_s R ) ) | { x e. dom R | ( f ` x ) =/= ( 0g ` ( R ` x ) ) } e. Fin } = ( Base ` ( S (+)m R ) ) )
50	45, 49	eqtrd 2656	. 2 |- ( ( R Fn I /\ I e. V ) -> { f e. ( Base ` P ) | dom ( f \ ( 0g o. R ) ) e. Fin } = ( Base ` ( S (+)m R ) ) )
51	1, 50	syl5eq 2668	1 |- ( ( R Fn I /\ I e. V ) -> B = ( Base ` ( S (+)m R ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   _Vcvv 3200   \ cdif 3571   (/)c0 3915   dom cdm 5114   o. ccom 5118   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   Fincfn 7955   Basecbs 15857   0gc0g 16100   X__scprds 16106   (+)_m cdsmm 20075

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-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-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-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-dsmm 20076

This theorem is referenced by:  dsmmfi  20082  frlmbas  20099