Theorem frlmbas 20099

Description: Base set of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by AV, 23-Jun-2019.)

Hypotheses

Ref	Expression
frlmval.f	|- F = ( R freeLMod I )
frlmbas.n	|- N = ( Base ` R )
frlmbas.z	|- .0. = ( 0g ` R )
frlmbas.b	|- B = { k e. ( N ^m I ) | k finSupp .0. }

Assertion

Ref	Expression
frlmbas	|- ( ( R e. V /\ I e. W ) -> B = ( Base ` F ) )

Distinct variable groups:   k, N   R, k   k, I   k, W   k, V   .0. , k

Allowed substitution hints:   B( k)   F( k)

Proof of Theorem frlmbas

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6201	. . . . 5 |- (ringLMod ` R ) e. _V
2		fnconstg 6093	. . . . 5 |- ( (ringLMod ` R ) e. _V -> ( I X. { (ringLMod ` R ) } ) Fn I )
3	1, 2	ax-mp 5	. . . 4 |- ( I X. { (ringLMod ` R ) } ) Fn I
4		eqid 2622	. . . . 5 |- ( R X_s ( I X. { (ringLMod ` R ) } ) ) = ( R X_s ( I X. { (ringLMod ` R ) } ) )
5		eqid 2622	. . . . 5 |- { k e. ( Base ` ( R X_s ( I X. { (ringLMod ` R ) } ) ) ) | dom ( k \ ( 0g o. ( I X. { (ringLMod ` R ) } ) ) ) e. Fin } = { k e. ( Base ` ( R X_s ( I X. { (ringLMod ` R ) } ) ) ) | dom ( k \ ( 0g o. ( I X. { (ringLMod ` R ) } ) ) ) e. Fin }
6	4, 5	dsmmbas2 20081	. . . 4 |- ( ( ( I X. { (ringLMod ` R ) } ) Fn I /\ I e. W ) -> { k e. ( Base ` ( R X_s ( I X. { (ringLMod ` R ) } ) ) ) | dom ( k \ ( 0g o. ( I X. { (ringLMod ` R ) } ) ) ) e. Fin } = ( Base ` ( R (+)m ( I X. { (ringLMod ` R ) } ) ) ) )
7	3, 6	mpan 706	. . 3 |- ( I e. W -> { k e. ( Base ` ( R X_s ( I X. { (ringLMod ` R ) } ) ) ) | dom ( k \ ( 0g o. ( I X. { (ringLMod ` R ) } ) ) ) e. Fin } = ( Base ` ( R (+)m ( I X. { (ringLMod ` R ) } ) ) ) )
8	7	adantl 482	. 2 |- ( ( R e. V /\ I e. W ) -> { k e. ( Base ` ( R X_s ( I X. { (ringLMod ` R ) } ) ) ) | dom ( k \ ( 0g o. ( I X. { (ringLMod ` R ) } ) ) ) e. Fin } = ( Base ` ( R (+)m ( I X. { (ringLMod ` R ) } ) ) ) )
9		frlmbas.b	. . 3 |- B = { k e. ( N ^m I ) | k finSupp .0. }
10		fvco2 6273	. . . . . . . . . . . . 13 |- ( ( ( I X. { (ringLMod ` R ) } ) Fn I /\ x e. I ) -> ( ( 0g o. ( I X. { (ringLMod ` R ) } ) ) ` x ) = ( 0g ` ( ( I X. { (ringLMod ` R ) } ) ` x ) ) )
11	3, 10	mpan 706	. . . . . . . . . . . 12 |- ( x e. I -> ( ( 0g o. ( I X. { (ringLMod ` R ) } ) ) ` x ) = ( 0g ` ( ( I X. { (ringLMod ` R ) } ) ` x ) ) )
12	11	adantl 482	. . . . . . . . . . 11 |- ( ( ( ( R e. V /\ I e. W ) /\ k e. ( N ^m I ) ) /\ x e. I ) -> ( ( 0g o. ( I X. { (ringLMod ` R ) } ) ) ` x ) = ( 0g ` ( ( I X. { (ringLMod ` R ) } ) ` x ) ) )
13	1	fvconst2 6469	. . . . . . . . . . . . . 14 |- ( x e. I -> ( ( I X. { (ringLMod ` R ) } ) ` x ) = (ringLMod ` R ) )
14	13	adantl 482	. . . . . . . . . . . . 13 |- ( ( ( ( R e. V /\ I e. W ) /\ k e. ( N ^m I ) ) /\ x e. I ) -> ( ( I X. { (ringLMod ` R ) } ) ` x ) = (ringLMod ` R ) )
15	14	fveq2d 6195	. . . . . . . . . . . 12 |- ( ( ( ( R e. V /\ I e. W ) /\ k e. ( N ^m I ) ) /\ x e. I ) -> ( 0g ` ( ( I X. { (ringLMod ` R ) } ) ` x ) ) = ( 0g ` (ringLMod ` R ) ) )
16		frlmbas.z	. . . . . . . . . . . . 13 |- .0. = ( 0g ` R )
17		rlm0 19197	. . . . . . . . . . . . 13 |- ( 0g ` R ) = ( 0g ` (ringLMod ` R ) )
18	16, 17	eqtri 2644	. . . . . . . . . . . 12 |- .0. = ( 0g ` (ringLMod ` R ) )
19	15, 18	syl6eqr 2674	. . . . . . . . . . 11 |- ( ( ( ( R e. V /\ I e. W ) /\ k e. ( N ^m I ) ) /\ x e. I ) -> ( 0g ` ( ( I X. { (ringLMod ` R ) } ) ` x ) ) = .0. )
20	12, 19	eqtrd 2656	. . . . . . . . . 10 |- ( ( ( ( R e. V /\ I e. W ) /\ k e. ( N ^m I ) ) /\ x e. I ) -> ( ( 0g o. ( I X. { (ringLMod ` R ) } ) ) ` x ) = .0. )
21	20	neeq2d 2854	. . . . . . . . 9 |- ( ( ( ( R e. V /\ I e. W ) /\ k e. ( N ^m I ) ) /\ x e. I ) -> ( ( k ` x ) =/= ( ( 0g o. ( I X. { (ringLMod ` R ) } ) ) ` x ) <-> ( k ` x ) =/= .0. ) )
22	21	rabbidva 3188	. . . . . . . 8 |- ( ( ( R e. V /\ I e. W ) /\ k e. ( N ^m I ) ) -> { x e. I | ( k ` x ) =/= ( ( 0g o. ( I X. { (ringLMod ` R ) } ) ) ` x ) } = { x e. I | ( k ` x ) =/= .0. } )
23		elmapfn 7880	. . . . . . . . . 10 |- ( k e. ( N ^m I ) -> k Fn I )
24	23	adantl 482	. . . . . . . . 9 |- ( ( ( R e. V /\ I e. W ) /\ k e. ( N ^m I ) ) -> k Fn I )
25		fn0g 17262	. . . . . . . . . 10 |- 0g Fn _V
26		ssv 3625	. . . . . . . . . 10 |- ran ( I X. { (ringLMod ` R ) } ) C_ _V
27		fnco 5999	. . . . . . . . . 10 |- ( ( 0g Fn _V /\ ( I X. { (ringLMod ` R ) } ) Fn I /\ ran ( I X. { (ringLMod ` R ) } ) C_ _V ) -> ( 0g o. ( I X. { (ringLMod ` R ) } ) ) Fn I )
28	25, 3, 26, 27	mp3an 1424	. . . . . . . . 9 |- ( 0g o. ( I X. { (ringLMod ` R ) } ) ) Fn I
29		fndmdif 6321	. . . . . . . . 9 |- ( ( k Fn I /\ ( 0g o. ( I X. { (ringLMod ` R ) } ) ) Fn I ) -> dom ( k \ ( 0g o. ( I X. { (ringLMod ` R ) } ) ) ) = { x e. I | ( k ` x ) =/= ( ( 0g o. ( I X. { (ringLMod ` R ) } ) ) ` x ) } )
30	24, 28, 29	sylancl 694	. . . . . . . 8 |- ( ( ( R e. V /\ I e. W ) /\ k e. ( N ^m I ) ) -> dom ( k \ ( 0g o. ( I X. { (ringLMod ` R ) } ) ) ) = { x e. I | ( k ` x ) =/= ( ( 0g o. ( I X. { (ringLMod ` R ) } ) ) ` x ) } )
31		simplr 792	. . . . . . . . 9 |- ( ( ( R e. V /\ I e. W ) /\ k e. ( N ^m I ) ) -> I e. W )
32		fvex 6201	. . . . . . . . . . 11 |- ( 0g ` R ) e. _V
33	16, 32	eqeltri 2697	. . . . . . . . . 10 |- .0. e. _V
34	33	a1i 11	. . . . . . . . 9 |- ( ( ( R e. V /\ I e. W ) /\ k e. ( N ^m I ) ) -> .0. e. _V )
35		suppvalfn 7302	. . . . . . . . 9 |- ( ( k Fn I /\ I e. W /\ .0. e. _V ) -> ( k supp .0. ) = { x e. I | ( k ` x ) =/= .0. } )
36	24, 31, 34, 35	syl3anc 1326	. . . . . . . 8 |- ( ( ( R e. V /\ I e. W ) /\ k e. ( N ^m I ) ) -> ( k supp .0. ) = { x e. I | ( k ` x ) =/= .0. } )
37	22, 30, 36	3eqtr4d 2666	. . . . . . 7 |- ( ( ( R e. V /\ I e. W ) /\ k e. ( N ^m I ) ) -> dom ( k \ ( 0g o. ( I X. { (ringLMod ` R ) } ) ) ) = ( k supp .0. ) )
38	37	eleq1d 2686	. . . . . 6 |- ( ( ( R e. V /\ I e. W ) /\ k e. ( N ^m I ) ) -> ( dom ( k \ ( 0g o. ( I X. { (ringLMod ` R ) } ) ) ) e. Fin <-> ( k supp .0. ) e. Fin ) )
39		elmapfun 7881	. . . . . . . . 9 |- ( k e. ( N ^m I ) -> Fun k )
40		id 22	. . . . . . . . 9 |- ( k e. ( N ^m I ) -> k e. ( N ^m I ) )
41	33	a1i 11	. . . . . . . . 9 |- ( k e. ( N ^m I ) -> .0. e. _V )
42	39, 40, 41	3jca 1242	. . . . . . . 8 |- ( k e. ( N ^m I ) -> ( Fun k /\ k e. ( N ^m I ) /\ .0. e. _V ) )
43	42	adantl 482	. . . . . . 7 |- ( ( ( R e. V /\ I e. W ) /\ k e. ( N ^m I ) ) -> ( Fun k /\ k e. ( N ^m I ) /\ .0. e. _V ) )
44		funisfsupp 8280	. . . . . . 7 |- ( ( Fun k /\ k e. ( N ^m I ) /\ .0. e. _V ) -> ( k finSupp .0. <-> ( k supp .0. ) e. Fin ) )
45	43, 44	syl 17	. . . . . 6 |- ( ( ( R e. V /\ I e. W ) /\ k e. ( N ^m I ) ) -> ( k finSupp .0. <-> ( k supp .0. ) e. Fin ) )
46	38, 45	bitr4d 271	. . . . 5 |- ( ( ( R e. V /\ I e. W ) /\ k e. ( N ^m I ) ) -> ( dom ( k \ ( 0g o. ( I X. { (ringLMod ` R ) } ) ) ) e. Fin <-> k finSupp .0. ) )
47	46	rabbidva 3188	. . . 4 |- ( ( R e. V /\ I e. W ) -> { k e. ( N ^m I ) | dom ( k \ ( 0g o. ( I X. { (ringLMod ` R ) } ) ) ) e. Fin } = { k e. ( N ^m I ) | k finSupp .0. } )
48		eqid 2622	. . . . . . . . 9 |- ( (ringLMod ` R ) ^s I ) = ( (ringLMod ` R ) ^s I )
49		frlmbas.n	. . . . . . . . . 10 |- N = ( Base ` R )
50		rlmbas 19195	. . . . . . . . . 10 |- ( Base ` R ) = ( Base ` (ringLMod ` R ) )
51	49, 50	eqtri 2644	. . . . . . . . 9 |- N = ( Base ` (ringLMod ` R ) )
52	48, 51	pwsbas 16147	. . . . . . . 8 |- ( ( (ringLMod ` R ) e. _V /\ I e. W ) -> ( N ^m I ) = ( Base ` ( (ringLMod ` R ) ^s I ) ) )
53	1, 52	mpan 706	. . . . . . 7 |- ( I e. W -> ( N ^m I ) = ( Base ` ( (ringLMod ` R ) ^s I ) ) )
54	53	adantl 482	. . . . . 6 |- ( ( R e. V /\ I e. W ) -> ( N ^m I ) = ( Base ` ( (ringLMod ` R ) ^s I ) ) )
55		eqid 2622	. . . . . . . . . . 11 |- (Scalar ` (ringLMod ` R ) ) = (Scalar ` (ringLMod ` R ) )
56	48, 55	pwsval 16146	. . . . . . . . . 10 |- ( ( (ringLMod ` R ) e. _V /\ I e. W ) -> ( (ringLMod ` R ) ^s I ) = ( (Scalar ` (ringLMod ` R ) ) X_s ( I X. { (ringLMod ` R ) } ) ) )
57	1, 56	mpan 706	. . . . . . . . 9 |- ( I e. W -> ( (ringLMod ` R ) ^s I ) = ( (Scalar ` (ringLMod ` R ) ) X_s ( I X. { (ringLMod ` R ) } ) ) )
58	57	adantl 482	. . . . . . . 8 |- ( ( R e. V /\ I e. W ) -> ( (ringLMod ` R ) ^s I ) = ( (Scalar ` (ringLMod ` R ) ) X_s ( I X. { (ringLMod ` R ) } ) ) )
59		rlmsca 19200	. . . . . . . . . 10 |- ( R e. V -> R = (Scalar ` (ringLMod ` R ) ) )
60	59	adantr 481	. . . . . . . . 9 |- ( ( R e. V /\ I e. W ) -> R = (Scalar ` (ringLMod ` R ) ) )
61	60	oveq1d 6665	. . . . . . . 8 |- ( ( R e. V /\ I e. W ) -> ( R X_s ( I X. { (ringLMod ` R ) } ) ) = ( (Scalar ` (ringLMod ` R ) ) X_s ( I X. { (ringLMod ` R ) } ) ) )
62	58, 61	eqtr4d 2659	. . . . . . 7 |- ( ( R e. V /\ I e. W ) -> ( (ringLMod ` R ) ^s I ) = ( R X_s ( I X. { (ringLMod ` R ) } ) ) )
63	62	fveq2d 6195	. . . . . 6 |- ( ( R e. V /\ I e. W ) -> ( Base ` ( (ringLMod ` R ) ^s I ) ) = ( Base ` ( R X_s ( I X. { (ringLMod ` R ) } ) ) ) )
64	54, 63	eqtrd 2656	. . . . 5 |- ( ( R e. V /\ I e. W ) -> ( N ^m I ) = ( Base ` ( R X_s ( I X. { (ringLMod ` R ) } ) ) ) )
65		rabeq 3192	. . . . 5 |- ( ( N ^m I ) = ( Base ` ( R X_s ( I X. { (ringLMod ` R ) } ) ) ) -> { k e. ( N ^m I ) | dom ( k \ ( 0g o. ( I X. { (ringLMod ` R ) } ) ) ) e. Fin } = { k e. ( Base ` ( R X_s ( I X. { (ringLMod ` R ) } ) ) ) | dom ( k \ ( 0g o. ( I X. { (ringLMod ` R ) } ) ) ) e. Fin } )
66	64, 65	syl 17	. . . 4 |- ( ( R e. V /\ I e. W ) -> { k e. ( N ^m I ) | dom ( k \ ( 0g o. ( I X. { (ringLMod ` R ) } ) ) ) e. Fin } = { k e. ( Base ` ( R X_s ( I X. { (ringLMod ` R ) } ) ) ) | dom ( k \ ( 0g o. ( I X. { (ringLMod ` R ) } ) ) ) e. Fin } )
67	47, 66	eqtr3d 2658	. . 3 |- ( ( R e. V /\ I e. W ) -> { k e. ( N ^m I ) | k finSupp .0. } = { k e. ( Base ` ( R X_s ( I X. { (ringLMod ` R ) } ) ) ) | dom ( k \ ( 0g o. ( I X. { (ringLMod ` R ) } ) ) ) e. Fin } )
68	9, 67	syl5eq 2668	. 2 |- ( ( R e. V /\ I e. W ) -> B = { k e. ( Base ` ( R X_s ( I X. { (ringLMod ` R ) } ) ) ) | dom ( k \ ( 0g o. ( I X. { (ringLMod ` R ) } ) ) ) e. Fin } )
69		frlmval.f	. . . 4 |- F = ( R freeLMod I )
70	69	frlmval 20092	. . 3 |- ( ( R e. V /\ I e. W ) -> F = ( R (+)m ( I X. { (ringLMod ` R ) } ) ) )
71	70	fveq2d 6195	. 2 |- ( ( R e. V /\ I e. W ) -> ( Base ` F ) = ( Base ` ( R (+)m ( I X. { (ringLMod ` R ) } ) ) ) )
72	8, 68, 71	3eqtr4d 2666	1 |- ( ( R e. V /\ I e. W ) -> B = ( Base ` F ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   _Vcvv 3200   \ cdif 3571   C_ wss 3574   {csn 4177   class class class wbr 4653   X. cxp 5112   dom cdm 5114   ran crn 5115   o. ccom 5118   Fun wfun 5882   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   supp csupp 7295   ^m cmap 7857   Fincfn 7955   finSupp cfsupp 8275   Basecbs 15857  Scalarcsca 15944   0gc0g 16100   X__scprds 16106   ^_s cpws 16107  ringLModcrglmod 19169   (+)_m cdsmm 20075   freeLMod cfrlm 20090

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-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

This theorem is referenced by:  frlmelbas  20100  frlmfibas  20105  ellspd  20141  islindf4  20177  rrxbase  23176  rrxds  23181  frlmpwfi  37668