Theorem ixpsnbasval 13779

Description: The value of an infinite Cartesian product of the base of a left module over a ring with a singleton. (Contributed by AV, 3-Dec-2018.)

Assertion

Ref	Expression
ixpsnbasval	|- ( ( R e. V /\ X e. W ) -> X_ x e. { X } ( Base ` ( ( { X } X. { (ringLMod ` R ) } ) ` x ) ) = { f | ( f Fn { X } /\ ( f ` X ) e. ( Base ` R ) ) } )

Distinct variable groups:   R, f, x   f, V   f, W   f, X, x

Allowed substitution hints:   V( x)   W( x)

Proof of Theorem ixpsnbasval

Step	Hyp	Ref	Expression
1		ixpsnval 6726	. . 3 |- ( X e. W -> X_ x e. { X } ( Base ` ( ( { X } X. { (ringLMod ` R ) } ) ` x ) ) = { f | ( f Fn { X } /\ ( f ` X ) e. [_ X / x ]_ ( Base ` ( ( { X } X. { (ringLMod ` R ) } ) ` x ) ) ) } )
2	1	adantl 277	. 2 |- ( ( R e. V /\ X e. W ) -> X_ x e. { X } ( Base ` ( ( { X } X. { (ringLMod ` R ) } ) ` x ) ) = { f | ( f Fn { X } /\ ( f ` X ) e. [_ X / x ]_ ( Base ` ( ( { X } X. { (ringLMod ` R ) } ) ` x ) ) ) } )
3		rlmfn 13766	. . . . . . . . . . . 12 |- ringLMod Fn _V
4		elex 2763	. . . . . . . . . . . 12 |- ( R e. V -> R e. _V )
5		funfvex 5551	. . . . . . . . . . . . 13 |- ( ( Fun ringLMod /\ R e. dom ringLMod ) -> (ringLMod ` R ) e. _V )
6	5	funfni 5335	. . . . . . . . . . . 12 |- ( (ringLMod Fn _V /\ R e. _V ) -> (ringLMod ` R ) e. _V )
7	3, 4, 6	sylancr 414	. . . . . . . . . . 11 |- ( R e. V -> (ringLMod ` R ) e. _V )
8	7	anim1ci 341	. . . . . . . . . 10 |- ( ( R e. V /\ X e. W ) -> ( X e. W /\ (ringLMod ` R ) e. _V ) )
9		xpsng 5711	. . . . . . . . . 10 |- ( ( X e. W /\ (ringLMod ` R ) e. _V ) -> ( { X } X. { (ringLMod ` R ) } ) = { <. X , (ringLMod ` R ) >. } )
10	8, 9	syl 14	. . . . . . . . 9 |- ( ( R e. V /\ X e. W ) -> ( { X } X. { (ringLMod ` R ) } ) = { <. X , (ringLMod ` R ) >. } )
11	10	fveq1d 5536	. . . . . . . 8 |- ( ( R e. V /\ X e. W ) -> ( ( { X } X. { (ringLMod ` R ) } ) ` X ) = ( { <. X , (ringLMod ` R ) >. } ` X ) )
12		fvsng 5732	. . . . . . . . 9 |- ( ( X e. W /\ (ringLMod ` R ) e. _V ) -> ( { <. X , (ringLMod ` R ) >. } ` X ) = (ringLMod ` R ) )
13	8, 12	syl 14	. . . . . . . 8 |- ( ( R e. V /\ X e. W ) -> ( { <. X , (ringLMod ` R ) >. } ` X ) = (ringLMod ` R ) )
14	11, 13	eqtrd 2222	. . . . . . 7 |- ( ( R e. V /\ X e. W ) -> ( ( { X } X. { (ringLMod ` R ) } ) ` X ) = (ringLMod ` R ) )
15	14	fveq2d 5538	. . . . . 6 |- ( ( R e. V /\ X e. W ) -> ( Base ` ( ( { X } X. { (ringLMod ` R ) } ) ` X ) ) = ( Base ` (ringLMod ` R ) ) )
16		csbfv2g 5572	. . . . . . . 8 |- ( X e. W -> [_ X / x ]_ ( Base ` ( ( { X } X. { (ringLMod ` R ) } ) ` x ) ) = ( Base ` [_ X / x ]_ ( ( { X } X. { (ringLMod ` R ) } ) ` x ) ) )
17		csbfvg 5573	. . . . . . . . 9 |- ( X e. W -> [_ X / x ]_ ( ( { X } X. { (ringLMod ` R ) } ) ` x ) = ( ( { X } X. { (ringLMod ` R ) } ) ` X ) )
18	17	fveq2d 5538	. . . . . . . 8 |- ( X e. W -> ( Base ` [_ X / x ]_ ( ( { X } X. { (ringLMod ` R ) } ) ` x ) ) = ( Base ` ( ( { X } X. { (ringLMod ` R ) } ) ` X ) ) )
19	16, 18	eqtrd 2222	. . . . . . 7 |- ( X e. W -> [_ X / x ]_ ( Base ` ( ( { X } X. { (ringLMod ` R ) } ) ` x ) ) = ( Base ` ( ( { X } X. { (ringLMod ` R ) } ) ` X ) ) )
20	19	adantl 277	. . . . . 6 |- ( ( R e. V /\ X e. W ) -> [_ X / x ]_ ( Base ` ( ( { X } X. { (ringLMod ` R ) } ) ` x ) ) = ( Base ` ( ( { X } X. { (ringLMod ` R ) } ) ` X ) ) )
21		rlmbasg 13768	. . . . . . 7 |- ( R e. V -> ( Base ` R ) = ( Base ` (ringLMod ` R ) ) )
22	21	adantr 276	. . . . . 6 |- ( ( R e. V /\ X e. W ) -> ( Base ` R ) = ( Base ` (ringLMod ` R ) ) )
23	15, 20, 22	3eqtr4d 2232	. . . . 5 |- ( ( R e. V /\ X e. W ) -> [_ X / x ]_ ( Base ` ( ( { X } X. { (ringLMod ` R ) } ) ` x ) ) = ( Base ` R ) )
24	23	eleq2d 2259	. . . 4 |- ( ( R e. V /\ X e. W ) -> ( ( f ` X ) e. [_ X / x ]_ ( Base ` ( ( { X } X. { (ringLMod ` R ) } ) ` x ) ) <-> ( f ` X ) e. ( Base ` R ) ) )
25	24	anbi2d 464	. . 3 |- ( ( R e. V /\ X e. W ) -> ( ( f Fn { X } /\ ( f ` X ) e. [_ X / x ]_ ( Base ` ( ( { X } X. { (ringLMod ` R ) } ) ` x ) ) ) <-> ( f Fn { X } /\ ( f ` X ) e. ( Base ` R ) ) ) )
26	25	abbidv 2307	. 2 |- ( ( R e. V /\ X e. W ) -> { f | ( f Fn { X } /\ ( f ` X ) e. [_ X / x ]_ ( Base ` ( ( { X } X. { (ringLMod ` R ) } ) ` x ) ) ) } = { f | ( f Fn { X } /\ ( f ` X ) e. ( Base ` R ) ) } )
27	2, 26	eqtrd 2222	1 |- ( ( R e. V /\ X e. W ) -> X_ x e. { X } ( Base ` ( ( { X } X. { (ringLMod ` R ) } ) ` x ) ) = { f | ( f Fn { X } /\ ( f ` X ) e. ( Base ` R ) ) } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2160   {cab 2175   _Vcvv 2752   [_csb 3072   {csn 3607   <.cop 3610   X. cxp 4642   Fn wfn 5230   ` cfv 5235   X_cixp 6723   Basecbs 12511  ringLModcrglmod 13747

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7931  ax-resscn 7932  ax-1cn 7933  ax-1re 7934  ax-icn 7935  ax-addcl 7936  ax-addrcl 7937  ax-mulcl 7938  ax-addcom 7940  ax-addass 7942  ax-i2m1 7945  ax-0lt1 7946  ax-0id 7948  ax-rnegex 7949  ax-pre-ltirr 7952  ax-pre-lttrn 7954  ax-pre-ltadd 7956

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5898  df-oprab 5899  df-mpo 5900  df-ixp 6724  df-pnf 8023  df-mnf 8024  df-ltxr 8026  df-inn 8949  df-2 9007  df-3 9008  df-4 9009  df-5 9010  df-6 9011  df-7 9012  df-8 9013  df-ndx 12514  df-slot 12515  df-base 12517  df-sets 12518  df-iress 12519  df-mulr 12600  df-sca 12602  df-vsca 12603  df-ip 12604  df-sra 13748  df-rgmod 13749

This theorem is referenced by: (None)