Theorem dsmmelbas 20083

Description: Membership in the finitely supported hull of a structure product in terms of the index set. (Contributed by Stefan O'Rear, 11-Jan-2015.)

Hypotheses

Ref	Expression
dsmmelbas.p	|- P = ( S X_s R )
dsmmelbas.c	|- C = ( S (+)m R )
dsmmelbas.b	|- B = ( Base ` P )
dsmmelbas.h	|- H = ( Base ` C )
dsmmelbas.i	|- ( ph -> I e. V )
dsmmelbas.r	|- ( ph -> R Fn I )

Assertion

Ref	Expression
dsmmelbas	|- ( ph -> ( X e. H <-> ( X e. B /\ { a e. I | ( X ` a ) =/= ( 0g ` ( R ` a ) ) } e. Fin ) ) )

Distinct variable groups:   S, a   R, a   X, a   I, a

Allowed substitution hints:   ph( a)   B( a)   C( a)   P( a)   H( a)   V( a)

Proof of Theorem dsmmelbas

Dummy variable b is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dsmmelbas.r	. . . . . 6 |- ( ph -> R Fn I )
2		dsmmelbas.i	. . . . . 6 |- ( ph -> I e. V )
3		fnex 6481	. . . . . 6 |- ( ( R Fn I /\ I e. V ) -> R e. _V )
4	1, 2, 3	syl2anc 693	. . . . 5 |- ( ph -> R e. _V )
5		eqid 2622	. . . . . 6 |- { b e. ( Base ` ( S X_s R ) ) | { a e. dom R | ( b ` a ) =/= ( 0g ` ( R ` a ) ) } e. Fin } = { b e. ( Base ` ( S X_s R ) ) | { a e. dom R | ( b ` a ) =/= ( 0g ` ( R ` a ) ) } e. Fin }
6	5	dsmmbase 20079	. . . . 5 |- ( R e. _V -> { b e. ( Base ` ( S X_s R ) ) | { a e. dom R | ( b ` a ) =/= ( 0g ` ( R ` a ) ) } e. Fin } = ( Base ` ( S (+)m R ) ) )
7	4, 6	syl 17	. . . 4 |- ( ph -> { b e. ( Base ` ( S X_s R ) ) | { a e. dom R | ( b ` a ) =/= ( 0g ` ( R ` a ) ) } e. Fin } = ( Base ` ( S (+)m R ) ) )
8		dsmmelbas.h	. . . . 5 |- H = ( Base ` C )
9		dsmmelbas.c	. . . . . 6 |- C = ( S (+)m R )
10	9	fveq2i 6194	. . . . 5 |- ( Base ` C ) = ( Base ` ( S (+)m R ) )
11	8, 10	eqtri 2644	. . . 4 |- H = ( Base ` ( S (+)m R ) )
12	7, 11	syl6reqr 2675	. . 3 |- ( ph -> H = { b e. ( Base ` ( S X_s R ) ) | { a e. dom R | ( b ` a ) =/= ( 0g ` ( R ` a ) ) } e. Fin } )
13	12	eleq2d 2687	. 2 |- ( ph -> ( X e. H <-> X e. { b e. ( Base ` ( S X_s R ) ) | { a e. dom R | ( b ` a ) =/= ( 0g ` ( R ` a ) ) } e. Fin } ) )
14		fveq1 6190	. . . . . . 7 |- ( b = X -> ( b ` a ) = ( X ` a ) )
15	14	neeq1d 2853	. . . . . 6 |- ( b = X -> ( ( b ` a ) =/= ( 0g ` ( R ` a ) ) <-> ( X ` a ) =/= ( 0g ` ( R ` a ) ) ) )
16	15	rabbidv 3189	. . . . 5 |- ( b = X -> { a e. dom R | ( b ` a ) =/= ( 0g ` ( R ` a ) ) } = { a e. dom R | ( X ` a ) =/= ( 0g ` ( R ` a ) ) } )
17	16	eleq1d 2686	. . . 4 |- ( b = X -> ( { a e. dom R | ( b ` a ) =/= ( 0g ` ( R ` a ) ) } e. Fin <-> { a e. dom R | ( X ` a ) =/= ( 0g ` ( R ` a ) ) } e. Fin ) )
18	17	elrab 3363	. . 3 |- ( X e. { b e. ( Base ` ( S X_s R ) ) | { a e. dom R | ( b ` a ) =/= ( 0g ` ( R ` a ) ) } e. Fin } <-> ( X e. ( Base ` ( S X_s R ) ) /\ { a e. dom R | ( X ` a ) =/= ( 0g ` ( R ` a ) ) } e. Fin ) )
19		dsmmelbas.b	. . . . . . 7 |- B = ( Base ` P )
20		dsmmelbas.p	. . . . . . . 8 |- P = ( S X_s R )
21	20	fveq2i 6194	. . . . . . 7 |- ( Base ` P ) = ( Base ` ( S X_s R ) )
22	19, 21	eqtr2i 2645	. . . . . 6 |- ( Base ` ( S X_s R ) ) = B
23	22	eleq2i 2693	. . . . 5 |- ( X e. ( Base ` ( S X_s R ) ) <-> X e. B )
24	23	a1i 11	. . . 4 |- ( ph -> ( X e. ( Base ` ( S X_s R ) ) <-> X e. B ) )
25		fndm 5990	. . . . . 6 |- ( R Fn I -> dom R = I )
26		rabeq 3192	. . . . . 6 |- ( dom R = I -> { a e. dom R | ( X ` a ) =/= ( 0g ` ( R ` a ) ) } = { a e. I | ( X ` a ) =/= ( 0g ` ( R ` a ) ) } )
27	1, 25, 26	3syl 18	. . . . 5 |- ( ph -> { a e. dom R | ( X ` a ) =/= ( 0g ` ( R ` a ) ) } = { a e. I | ( X ` a ) =/= ( 0g ` ( R ` a ) ) } )
28	27	eleq1d 2686	. . . 4 |- ( ph -> ( { a e. dom R | ( X ` a ) =/= ( 0g ` ( R ` a ) ) } e. Fin <-> { a e. I | ( X ` a ) =/= ( 0g ` ( R ` a ) ) } e. Fin ) )
29	24, 28	anbi12d 747	. . 3 |- ( ph -> ( ( X e. ( Base ` ( S X_s R ) ) /\ { a e. dom R | ( X ` a ) =/= ( 0g ` ( R ` a ) ) } e. Fin ) <-> ( X e. B /\ { a e. I | ( X ` a ) =/= ( 0g ` ( R ` a ) ) } e. Fin ) ) )
30	18, 29	syl5bb 272	. 2 |- ( ph -> ( X e. { b e. ( Base ` ( S X_s R ) ) | { a e. dom R | ( b ` a ) =/= ( 0g ` ( R ` a ) ) } e. Fin } <-> ( X e. B /\ { a e. I | ( X ` a ) =/= ( 0g ` ( R ` a ) ) } e. Fin ) ) )
31	13, 30	bitrd 268	1 |- ( ph -> ( X e. H <-> ( X e. B /\ { a e. I | ( X ` a ) =/= ( 0g ` ( R ` a ) ) } e. Fin ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   _Vcvv 3200   dom cdm 5114   Fn wfn 5883   ` 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-prds 16108  df-dsmm 20076

This theorem is referenced by:  dsmm0cl  20084  dsmmacl  20085  dsmmsubg  20087  dsmmlss  20088