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Theorem rrxprds 20893
Description: Expand the definition of the generalized real Euclidean spaces. (Contributed by Thierry Arnoux, 16-Jun-2019.)
Hypotheses
Ref Expression
rrxval.r  |-  H  =  (ℝ^ `  I )
rrxbase.b  |-  B  =  ( Base `  H
)
Assertion
Ref Expression
rrxprds  |-  ( I  e.  V  ->  H  =  (toCHil `  ( (RRfld X_s ( I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } ) )s  B ) ) )

Proof of Theorem rrxprds
StepHypRef Expression
1 rrxval.r . . 3  |-  H  =  (ℝ^ `  I )
21rrxval 20891 . 2  |-  ( I  e.  V  ->  H  =  (toCHil `  (RRfld freeLMod  I ) ) )
3 refld 18049 . . . . 5  |- RRfld  e. Field
4 eqid 2443 . . . . . 6  |-  (RRfld freeLMod  I )  =  (RRfld freeLMod  I )
5 eqid 2443 . . . . . 6  |-  ( Base `  (RRfld freeLMod  I ) )  =  ( Base `  (RRfld freeLMod  I ) )
64, 5frlmpws 18175 . . . . 5  |-  ( (RRfld 
e. Field  /\  I  e.  V
)  ->  (RRfld freeLMod  I )  =  ( ( (ringLMod ` RRfld )  ^s  I )s  ( Base `  (RRfld freeLMod  I ) ) ) )
73, 6mpan 670 . . . 4  |-  ( I  e.  V  ->  (RRfld freeLMod  I )  =  ( ( (ringLMod ` RRfld )  ^s  I )s  (
Base `  (RRfld freeLMod  I ) ) ) )
8 fvex 5701 . . . . . . 7  |-  ( (subringAlg  ` RRfld
) `  RR )  e.  _V
9 rebase 18036 . . . . . . . . . . . 12  |-  RR  =  ( Base ` RRfld )
109fveq2i 5694 . . . . . . . . . . 11  |-  ( (subringAlg  ` RRfld
) `  RR )  =  ( (subringAlg  ` RRfld ) `  ( Base ` RRfld ) )
11 rlmval 17272 . . . . . . . . . . 11  |-  (ringLMod ` RRfld )  =  ( (subringAlg  ` RRfld ) `  ( Base ` RRfld ) )
1210, 11eqtr4i 2466 . . . . . . . . . 10  |-  ( (subringAlg  ` RRfld
) `  RR )  =  (ringLMod ` RRfld )
1312eqcomi 2447 . . . . . . . . 9  |-  (ringLMod ` RRfld )  =  ( (subringAlg  ` RRfld ) `  RR )
1413oveq1i 6101 . . . . . . . 8  |-  ( (ringLMod ` RRfld )  ^s  I )  =  ( ( (subringAlg  ` RRfld ) `  RR )  ^s  I )
159ressid 14233 . . . . . . . . . 10  |-  (RRfld  e. Field  -> 
(RRfld ↾s  RR )  = RRfld )
163, 15ax-mp 5 . . . . . . . . 9  |-  (RRfld ↾s  RR )  = RRfld
17 eqidd 2444 . . . . . . . . . . 11  |-  ( T. 
->  ( (subringAlg  ` RRfld ) `  RR )  =  (
(subringAlg  ` RRfld ) `  RR ) )
189eqimssi 3410 . . . . . . . . . . . 12  |-  RR  C_  ( Base ` RRfld )
1918a1i 11 . . . . . . . . . . 11  |-  ( T. 
->  RR  C_  ( Base ` RRfld
) )
2017, 19srasca 17262 . . . . . . . . . 10  |-  ( T. 
->  (RRfld ↾s 
RR )  =  (Scalar `  ( (subringAlg  ` RRfld ) `  RR ) ) )
2120trud 1378 . . . . . . . . 9  |-  (RRfld ↾s  RR )  =  (Scalar `  ( (subringAlg  ` RRfld
) `  RR )
)
2216, 21eqtr3i 2465 . . . . . . . 8  |- RRfld  =  (Scalar `  ( (subringAlg  ` RRfld ) `  RR ) )
2314, 22pwsval 14424 . . . . . . 7  |-  ( ( ( (subringAlg  ` RRfld ) `  RR )  e.  _V  /\  I  e.  V )  ->  ( (ringLMod ` RRfld )  ^s  I
)  =  (RRfld X_s (
I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } ) ) )
248, 23mpan 670 . . . . . 6  |-  ( I  e.  V  ->  (
(ringLMod ` RRfld )  ^s  I )  =  (RRfld X_s ( I  X.  {
( (subringAlg  ` RRfld ) `  RR ) } ) ) )
2524eqcomd 2448 . . . . 5  |-  ( I  e.  V  ->  (RRfld X_s ( I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } ) )  =  ( (ringLMod ` RRfld )  ^s  I
) )
26 rrxbase.b . . . . . . 7  |-  B  =  ( Base `  H
)
272fveq2d 5695 . . . . . . 7  |-  ( I  e.  V  ->  ( Base `  H )  =  ( Base `  (toCHil `  (RRfld freeLMod  I ) ) ) )
2826, 27syl5eq 2487 . . . . . 6  |-  ( I  e.  V  ->  B  =  ( Base `  (toCHil `  (RRfld freeLMod  I ) ) ) )
29 eqid 2443 . . . . . . 7  |-  (toCHil `  (RRfld freeLMod  I ) )  =  (toCHil `  (RRfld freeLMod  I ) )
3029, 5tchbas 20734 . . . . . 6  |-  ( Base `  (RRfld freeLMod  I ) )  =  ( Base `  (toCHil `  (RRfld freeLMod  I ) ) )
3128, 30syl6eqr 2493 . . . . 5  |-  ( I  e.  V  ->  B  =  ( Base `  (RRfld freeLMod  I ) ) )
3225, 31oveq12d 6109 . . . 4  |-  ( I  e.  V  ->  (
(RRfld X_s ( I  X.  {
( (subringAlg  ` RRfld ) `  RR ) } ) )s  B )  =  ( ( (ringLMod ` RRfld )  ^s  I )s  (
Base `  (RRfld freeLMod  I ) ) ) )
337, 32eqtr4d 2478 . . 3  |-  ( I  e.  V  ->  (RRfld freeLMod  I )  =  ( (RRfld X_s ( I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } ) )s  B ) )
3433fveq2d 5695 . 2  |-  ( I  e.  V  ->  (toCHil `  (RRfld freeLMod  I ) )  =  (toCHil `  ( (RRfld X_s ( I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } ) )s  B ) ) )
352, 34eqtrd 2475 1  |-  ( I  e.  V  ->  H  =  (toCHil `  ( (RRfld X_s ( I  X.  { ( (subringAlg  ` RRfld ) `  RR ) } ) )s  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369   T. wtru 1370    e. wcel 1756   _Vcvv 2972    C_ wss 3328   {csn 3877    X. cxp 4838   ` cfv 5418  (class class class)co 6091   RRcr 9281   Basecbs 14174   ↾s cress 14175  Scalarcsca 14241   X_scprds 14384    ^s cpws 14385  Fieldcfield 16833  subringAlg csra 17249  ringLModcrglmod 17250  RRfldcrefld 18034   freeLMod cfrlm 18171  toCHilctch 20686  ℝ^crrx 20887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360  ax-addf 9361  ax-mulf 9362
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-tpos 6745  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-map 7216  df-ixp 7264  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-sup 7691  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-rp 10992  df-fz 11438  df-seq 11807  df-exp 11866  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-starv 14253  df-sca 14254  df-vsca 14255  df-ip 14256  df-tset 14257  df-ple 14258  df-ds 14260  df-unif 14261  df-hom 14262  df-cco 14263  df-0g 14380  df-prds 14386  df-pws 14388  df-mnd 15415  df-grp 15545  df-minusg 15546  df-subg 15678  df-cmn 16279  df-mgp 16592  df-ur 16604  df-rng 16647  df-cring 16648  df-oppr 16715  df-dvdsr 16733  df-unit 16734  df-invr 16764  df-dvr 16775  df-drng 16834  df-field 16835  df-subrg 16863  df-sra 17253  df-rgmod 17254  df-cnfld 17819  df-refld 18035  df-dsmm 18157  df-frlm 18172  df-tng 20177  df-tch 20688  df-rrx 20889
This theorem is referenced by:  rrxip  20894
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