Theorem ixpsnbasval 14098

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 6769	. . 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 14085	. . . . . . . . . . . 12 |- ringLMod Fn _V
4		elex 2774	. . . . . . . . . . . 12 |- ( R e. V -> R e. _V )
5		funfvex 5578	. . . . . . . . . . . . 13 |- ( ( Fun ringLMod /\ R e. dom ringLMod ) -> (ringLMod ` R ) e. _V )
6	5	funfni 5361	. . . . . . . . . . . 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 5740	. . . . . . . . . 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 5563	. . . . . . . 8 |- ( ( R e. V /\ X e. W ) -> ( ( { X } X. { (ringLMod ` R ) } ) ` X ) = ( { <. X , (ringLMod ` R ) >. } ` X ) )
12		fvsng 5761	. . . . . . . . 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 2229	. . . . . . 7 |- ( ( R e. V /\ X e. W ) -> ( ( { X } X. { (ringLMod ` R ) } ) ` X ) = (ringLMod ` R ) )
15	14	fveq2d 5565	. . . . . 6 |- ( ( R e. V /\ X e. W ) -> ( Base ` ( ( { X } X. { (ringLMod ` R ) } ) ` X ) ) = ( Base ` (ringLMod ` R ) ) )
16		csbfv2g 5600	. . . . . . . 8 |- ( X e. W -> [_ X / x ]_ ( Base ` ( ( { X } X. { (ringLMod ` R ) } ) ` x ) ) = ( Base ` [_ X / x ]_ ( ( { X } X. { (ringLMod ` R ) } ) ` x ) ) )
17		csbfvg 5601	. . . . . . . . 9 |- ( X e. W -> [_ X / x ]_ ( ( { X } X. { (ringLMod ` R ) } ) ` x ) = ( ( { X } X. { (ringLMod ` R ) } ) ` X ) )
18	17	fveq2d 5565	. . . . . . . 8 |- ( X e. W -> ( Base ` [_ X / x ]_ ( ( { X } X. { (ringLMod ` R ) } ) ` x ) ) = ( Base ` ( ( { X } X. { (ringLMod ` R ) } ) ` X ) ) )
19	16, 18	eqtrd 2229	. . . . . . 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 14087	. . . . . . 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 2239	. . . . 5 |- ( ( R e. V /\ X e. W ) -> [_ X / x ]_ ( Base ` ( ( { X } X. { (ringLMod ` R ) } ) ` x ) ) = ( Base ` R ) )
24	23	eleq2d 2266	. . . 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 2314	. 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 2229	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 2167   {cab 2182   _Vcvv 2763   [_csb 3084   {csn 3623   <.cop 3626   X. cxp 4662   Fn wfn 5254   ` cfv 5259   X_cixp 6766   Basecbs 12703  ringLModcrglmod 14066

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-pre-ltirr 8008  ax-pre-lttrn 8010  ax-pre-ltadd 8012

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-ixp 6767  df-pnf 8080  df-mnf 8081  df-ltxr 8083  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-5 9069  df-6 9070  df-7 9071  df-8 9072  df-ndx 12706  df-slot 12707  df-base 12709  df-sets 12710  df-iress 12711  df-mulr 12794  df-sca 12796  df-vsca 12797  df-ip 12798  df-sra 14067  df-rgmod 14068

This theorem is referenced by: (None)