Theorem ixpsnbasval 14803

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 6983	. . 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 14790	. . . . . . . . . . . 12 |- ringLMod Fn _V
4		elex 2833	. . . . . . . . . . . 12 |- ( R e. V -> R e. _V )
5		funfvex 5712	. . . . . . . . . . . . 13 |- ( ( Fun ringLMod /\ R e. dom ringLMod ) -> (ringLMod ` R ) e. _V )
6	5	funfni 5483	. . . . . . . . . . . 12 |- ( (ringLMod Fn _V /\ R e. _V ) -> (ringLMod ` R ) e. _V )
7	3, 4, 6	sylancr 418	. . . . . . . . . . 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 5884	. . . . . . . . . 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 5697	. . . . . . . 8 |- ( ( R e. V /\ X e. W ) -> ( ( { X } X. { (ringLMod ` R ) } ) ` X ) = ( { <. X , (ringLMod ` R ) >. } ` X ) )
12		fvsng 5911	. . . . . . . . 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 2271	. . . . . . 7 |- ( ( R e. V /\ X e. W ) -> ( ( { X } X. { (ringLMod ` R ) } ) ` X ) = (ringLMod ` R ) )
15	14	fveq2d 5699	. . . . . 6 |- ( ( R e. V /\ X e. W ) -> ( Base ` ( ( { X } X. { (ringLMod ` R ) } ) ` X ) ) = ( Base ` (ringLMod ` R ) ) )
16		csbfv2g 5737	. . . . . . . 8 |- ( X e. W -> [_ X / x ]_ ( Base ` ( ( { X } X. { (ringLMod ` R ) } ) ` x ) ) = ( Base ` [_ X / x ]_ ( ( { X } X. { (ringLMod ` R ) } ) ` x ) ) )
17		csbfvg 5738	. . . . . . . . 9 |- ( X e. W -> [_ X / x ]_ ( ( { X } X. { (ringLMod ` R ) } ) ` x ) = ( ( { X } X. { (ringLMod ` R ) } ) ` X ) )
18	17	fveq2d 5699	. . . . . . . 8 |- ( X e. W -> ( Base ` [_ X / x ]_ ( ( { X } X. { (ringLMod ` R ) } ) ` x ) ) = ( Base ` ( ( { X } X. { (ringLMod ` R ) } ) ` X ) ) )
19	16, 18	eqtrd 2271	. . . . . . 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 14792	. . . . . . 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 2281	. . . . 5 |- ( ( R e. V /\ X e. W ) -> [_ X / x ]_ ( Base ` ( ( { X } X. { (ringLMod ` R ) } ) ` x ) ) = ( Base ` R ) )
24	23	eleq2d 2308	. . . 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 468	. . 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 2358	. 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 2271	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 ) ) } )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   {cab 2224   _Vcvv 2821   [_csb 3147   {csn 3709   <.cop 3712   X. cxp 4772   Fn wfn 5372   ` cfv 5377   X_cixp 6980   Basecbs 13352  ringLModcrglmod 14771

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-lttrn 8293  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-ixp 6981  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-7 9368  df-8 9369  df-ndx 13355  df-slot 13356  df-base 13358  df-sets 13359  df-iress 13360  df-mulr 13445  df-sca 13447  df-vsca 13448  df-ip 13449  df-sra 14772  df-rgmod 14773

This theorem is used by: (None)