Theorem ixpsnbasval 13962

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 6755	. . 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 13949	. . . . . . . . . . . 12 |- ringLMod Fn _V
4		elex 2771	. . . . . . . . . . . 12 |- ( R e. V -> R e. _V )
5		funfvex 5571	. . . . . . . . . . . . 13 |- ( ( Fun ringLMod /\ R e. dom ringLMod ) -> (ringLMod ` R ) e. _V )
6	5	funfni 5354	. . . . . . . . . . . 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 5733	. . . . . . . . . 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 5556	. . . . . . . 8 |- ( ( R e. V /\ X e. W ) -> ( ( { X } X. { (ringLMod ` R ) } ) ` X ) = ( { <. X , (ringLMod ` R ) >. } ` X ) )
12		fvsng 5754	. . . . . . . . 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 2226	. . . . . . 7 |- ( ( R e. V /\ X e. W ) -> ( ( { X } X. { (ringLMod ` R ) } ) ` X ) = (ringLMod ` R ) )
15	14	fveq2d 5558	. . . . . 6 |- ( ( R e. V /\ X e. W ) -> ( Base ` ( ( { X } X. { (ringLMod ` R ) } ) ` X ) ) = ( Base ` (ringLMod ` R ) ) )
16		csbfv2g 5593	. . . . . . . 8 |- ( X e. W -> [_ X / x ]_ ( Base ` ( ( { X } X. { (ringLMod ` R ) } ) ` x ) ) = ( Base ` [_ X / x ]_ ( ( { X } X. { (ringLMod ` R ) } ) ` x ) ) )
17		csbfvg 5594	. . . . . . . . 9 |- ( X e. W -> [_ X / x ]_ ( ( { X } X. { (ringLMod ` R ) } ) ` x ) = ( ( { X } X. { (ringLMod ` R ) } ) ` X ) )
18	17	fveq2d 5558	. . . . . . . 8 |- ( X e. W -> ( Base ` [_ X / x ]_ ( ( { X } X. { (ringLMod ` R ) } ) ` x ) ) = ( Base ` ( ( { X } X. { (ringLMod ` R ) } ) ` X ) ) )
19	16, 18	eqtrd 2226	. . . . . . 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 13951	. . . . . . 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 2236	. . . . 5 |- ( ( R e. V /\ X e. W ) -> [_ X / x ]_ ( Base ` ( ( { X } X. { (ringLMod ` R ) } ) ` x ) ) = ( Base ` R ) )
24	23	eleq2d 2263	. . . 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 2311	. 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 2226	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 2164   {cab 2179   _Vcvv 2760   [_csb 3080   {csn 3618   <.cop 3621   X. cxp 4657   Fn wfn 5249   ` cfv 5254   X_cixp 6752   Basecbs 12618  ringLModcrglmod 13930

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 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-pre-ltirr 7984  ax-pre-lttrn 7986  ax-pre-ltadd 7988

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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-ixp 6753  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-5 9044  df-6 9045  df-7 9046  df-8 9047  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-iress 12626  df-mulr 12709  df-sca 12711  df-vsca 12712  df-ip 12713  df-sra 13931  df-rgmod 13932

This theorem is referenced by: (None)