Theorem islmodfg 37639

Description: Property of a finitely generated left module. (Contributed by Stefan O'Rear, 1-Jan-2015.)

Hypotheses

Ref	Expression
islmodfg.b	|- B = ( Base ` W )
islmodfg.n	|- N = ( LSpan ` W )

Assertion

Ref	Expression
islmodfg	|- ( W e. LMod -> ( W e. LFinGen <-> E. b e. ~P B ( b e. Fin /\ ( N ` b ) = B ) ) )

Distinct variable groups:   W, b   B, b   N, b

Proof of Theorem islmodfg

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-lfig 37638	. . . 4 |- LFinGen = { a e. LMod | ( Base ` a ) e. ( ( LSpan ` a ) " ( ~P ( Base ` a ) i^i Fin ) ) }
2	1	eleq2i 2693	. . 3 |- ( W e. LFinGen <-> W e. { a e. LMod | ( Base ` a ) e. ( ( LSpan ` a ) " ( ~P ( Base ` a ) i^i Fin ) ) } )
3		fveq2 6191	. . . . 5 |- ( a = W -> ( Base ` a ) = ( Base ` W ) )
4		fveq2 6191	. . . . . . 7 |- ( a = W -> ( LSpan ` a ) = ( LSpan ` W ) )
5		islmodfg.n	. . . . . . 7 |- N = ( LSpan ` W )
6	4, 5	syl6eqr 2674	. . . . . 6 |- ( a = W -> ( LSpan ` a ) = N )
7	3	pweqd 4163	. . . . . . 7 |- ( a = W -> ~P ( Base ` a ) = ~P ( Base ` W ) )
8	7	ineq1d 3813	. . . . . 6 |- ( a = W -> ( ~P ( Base ` a ) i^i Fin ) = ( ~P ( Base ` W ) i^i Fin ) )
9	6, 8	imaeq12d 5467	. . . . 5 |- ( a = W -> ( ( LSpan ` a ) " ( ~P ( Base ` a ) i^i Fin ) ) = ( N " ( ~P ( Base ` W ) i^i Fin ) ) )
10	3, 9	eleq12d 2695	. . . 4 |- ( a = W -> ( ( Base ` a ) e. ( ( LSpan ` a ) " ( ~P ( Base ` a ) i^i Fin ) ) <-> ( Base ` W ) e. ( N " ( ~P ( Base ` W ) i^i Fin ) ) ) )
11	10	elrab3 3364	. . 3 |- ( W e. LMod -> ( W e. { a e. LMod | ( Base ` a ) e. ( ( LSpan ` a ) " ( ~P ( Base ` a ) i^i Fin ) ) } <-> ( Base ` W ) e. ( N " ( ~P ( Base ` W ) i^i Fin ) ) ) )
12	2, 11	syl5bb 272	. 2 |- ( W e. LMod -> ( W e. LFinGen <-> ( Base ` W ) e. ( N " ( ~P ( Base ` W ) i^i Fin ) ) ) )
13		eqid 2622	. . . . . 6 |- ( Base ` W ) = ( Base ` W )
14		eqid 2622	. . . . . 6 |- ( LSubSp ` W ) = ( LSubSp ` W )
15	13, 14, 5	lspf 18974	. . . . 5 |- ( W e. LMod -> N : ~P ( Base ` W ) --> ( LSubSp ` W ) )
16		ffn 6045	. . . . 5 |- ( N : ~P ( Base ` W ) --> ( LSubSp ` W ) -> N Fn ~P ( Base ` W ) )
17	15, 16	syl 17	. . . 4 |- ( W e. LMod -> N Fn ~P ( Base ` W ) )
18		inss1 3833	. . . 4 |- ( ~P ( Base ` W ) i^i Fin ) C_ ~P ( Base ` W )
19		fvelimab 6253	. . . 4 |- ( ( N Fn ~P ( Base ` W ) /\ ( ~P ( Base ` W ) i^i Fin ) C_ ~P ( Base ` W ) ) -> ( ( Base ` W ) e. ( N " ( ~P ( Base ` W ) i^i Fin ) ) <-> E. b e. ( ~P ( Base ` W ) i^i Fin ) ( N ` b ) = ( Base ` W ) ) )
20	17, 18, 19	sylancl 694	. . 3 |- ( W e. LMod -> ( ( Base ` W ) e. ( N " ( ~P ( Base ` W ) i^i Fin ) ) <-> E. b e. ( ~P ( Base ` W ) i^i Fin ) ( N ` b ) = ( Base ` W ) ) )
21		elin 3796	. . . . . . 7 |- ( b e. ( ~P ( Base ` W ) i^i Fin ) <-> ( b e. ~P ( Base ` W ) /\ b e. Fin ) )
22		islmodfg.b	. . . . . . . . . . 11 |- B = ( Base ` W )
23	22	eqcomi 2631	. . . . . . . . . 10 |- ( Base ` W ) = B
24	23	pweqi 4162	. . . . . . . . 9 |- ~P ( Base ` W ) = ~P B
25	24	eleq2i 2693	. . . . . . . 8 |- ( b e. ~P ( Base ` W ) <-> b e. ~P B )
26	25	anbi1i 731	. . . . . . 7 |- ( ( b e. ~P ( Base ` W ) /\ b e. Fin ) <-> ( b e. ~P B /\ b e. Fin ) )
27	21, 26	bitri 264	. . . . . 6 |- ( b e. ( ~P ( Base ` W ) i^i Fin ) <-> ( b e. ~P B /\ b e. Fin ) )
28	23	eqeq2i 2634	. . . . . 6 |- ( ( N ` b ) = ( Base ` W ) <-> ( N ` b ) = B )
29	27, 28	anbi12i 733	. . . . 5 |- ( ( b e. ( ~P ( Base ` W ) i^i Fin ) /\ ( N ` b ) = ( Base ` W ) ) <-> ( ( b e. ~P B /\ b e. Fin ) /\ ( N ` b ) = B ) )
30		anass 681	. . . . 5 |- ( ( ( b e. ~P B /\ b e. Fin ) /\ ( N ` b ) = B ) <-> ( b e. ~P B /\ ( b e. Fin /\ ( N ` b ) = B ) ) )
31	29, 30	bitri 264	. . . 4 |- ( ( b e. ( ~P ( Base ` W ) i^i Fin ) /\ ( N ` b ) = ( Base ` W ) ) <-> ( b e. ~P B /\ ( b e. Fin /\ ( N ` b ) = B ) ) )
32	31	rexbii2 3039	. . 3 |- ( E. b e. ( ~P ( Base ` W ) i^i Fin ) ( N ` b ) = ( Base ` W ) <-> E. b e. ~P B ( b e. Fin /\ ( N ` b ) = B ) )
33	20, 32	syl6bb 276	. 2 |- ( W e. LMod -> ( ( Base ` W ) e. ( N " ( ~P ( Base ` W ) i^i Fin ) ) <-> E. b e. ~P B ( b e. Fin /\ ( N ` b ) = B ) ) )
34	12, 33	bitrd 268	1 |- ( W e. LMod -> ( W e. LFinGen <-> E. b e. ~P B ( b e. Fin /\ ( N ` b ) = B ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   {crab 2916   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888   Fincfn 7955   Basecbs 15857   LModclmod 18863   LSubSpclss 18932   LSpanclspn 18971  LFinGenclfig 37637

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-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-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-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  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-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mgp 18490  df-ur 18502  df-ring 18549  df-lmod 18865  df-lss 18933  df-lsp 18972  df-lfig 37638

This theorem is referenced by:  islssfg  37640  lnrfg  37689