Theorem ixpsnbasval 14470

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 6865	. . 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 14457	. . . . . . . . . . . 12 |- ringLMod Fn _V
4		elex 2812	. . . . . . . . . . . 12 |- ( R e. V -> R e. _V )
5		funfvex 5652	. . . . . . . . . . . . 13 |- ( ( Fun ringLMod /\ R e. dom ringLMod ) -> (ringLMod ` R ) e. _V )
6	5	funfni 5429	. . . . . . . . . . . 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 5818	. . . . . . . . . 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 5637	. . . . . . . 8 |- ( ( R e. V /\ X e. W ) -> ( ( { X } X. { (ringLMod ` R ) } ) ` X ) = ( { <. X , (ringLMod ` R ) >. } ` X ) )
12		fvsng 5845	. . . . . . . . 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 2262	. . . . . . 7 |- ( ( R e. V /\ X e. W ) -> ( ( { X } X. { (ringLMod ` R ) } ) ` X ) = (ringLMod ` R ) )
15	14	fveq2d 5639	. . . . . 6 |- ( ( R e. V /\ X e. W ) -> ( Base ` ( ( { X } X. { (ringLMod ` R ) } ) ` X ) ) = ( Base ` (ringLMod ` R ) ) )
16		csbfv2g 5676	. . . . . . . 8 |- ( X e. W -> [_ X / x ]_ ( Base ` ( ( { X } X. { (ringLMod ` R ) } ) ` x ) ) = ( Base ` [_ X / x ]_ ( ( { X } X. { (ringLMod ` R ) } ) ` x ) ) )
17		csbfvg 5677	. . . . . . . . 9 |- ( X e. W -> [_ X / x ]_ ( ( { X } X. { (ringLMod ` R ) } ) ` x ) = ( ( { X } X. { (ringLMod ` R ) } ) ` X ) )
18	17	fveq2d 5639	. . . . . . . 8 |- ( X e. W -> ( Base ` [_ X / x ]_ ( ( { X } X. { (ringLMod ` R ) } ) ` x ) ) = ( Base ` ( ( { X } X. { (ringLMod ` R ) } ) ` X ) ) )
19	16, 18	eqtrd 2262	. . . . . . 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 14459	. . . . . . 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 2272	. . . . 5 |- ( ( R e. V /\ X e. W ) -> [_ X / x ]_ ( Base ` ( ( { X } X. { (ringLMod ` R ) } ) ` x ) ) = ( Base ` R ) )
24	23	eleq2d 2299	. . . 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 2347	. 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 2262	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 1395   e. wcel 2200   {cab 2215   _Vcvv 2800   [_csb 3125   {csn 3667   <.cop 3670   X. cxp 4721   Fn wfn 5319   ` cfv 5324   X_cixp 6862   Basecbs 13072  ringLModcrglmod 14438

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-i2m1 8127  ax-0lt1 8128  ax-0id 8130  ax-rnegex 8131  ax-pre-ltirr 8134  ax-pre-lttrn 8136  ax-pre-ltadd 8138

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-ixp 6863  df-pnf 8206  df-mnf 8207  df-ltxr 8209  df-inn 9134  df-2 9192  df-3 9193  df-4 9194  df-5 9195  df-6 9196  df-7 9197  df-8 9198  df-ndx 13075  df-slot 13076  df-base 13078  df-sets 13079  df-iress 13080  df-mulr 13164  df-sca 13166  df-vsca 13167  df-ip 13168  df-sra 14439  df-rgmod 14440

This theorem is referenced by: (None)