Theorem rrxprds 23177

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

Step	Hyp	Ref	Expression
1		rrxval.r	. . 3 |- H = (ℝ^ ` I )
2	1	rrxval 23175	. 2 |- ( I e. V -> H = (toCHil ` (RRfld freeLMod I ) ) )
3		refld 19965	. . . . 5 |- RRfld e. Field
4		eqid 2622	. . . . . 6 |- (RRfld freeLMod I ) = (RRfld freeLMod I )
5		eqid 2622	. . . . . 6 |- ( Base ` (RRfld freeLMod I ) ) = ( Base ` (RRfld freeLMod I ) )
6	4, 5	frlmpws 20094	. . . . 5 |- ( (RRfld e. Field /\ I e. V ) -> (RRfld freeLMod I ) = ( ( (ringLMod ` RRfld ) ^s I ) ↾s ( Base ` (RRfld freeLMod I ) ) ) )
7	3, 6	mpan 706	. . . 4 |- ( I e. V -> (RRfld freeLMod I ) = ( ( (ringLMod ` RRfld ) ^s I ) ↾s ( Base ` (RRfld freeLMod I ) ) ) )
8		fvex 6201	. . . . . . 7 |- ( (subringAlg ` RRfld ) ` RR ) e. _V
9		rlmval 19191	. . . . . . . . . 10 |- (ringLMod ` RRfld ) = ( (subringAlg ` RRfld ) ` ( Base ` RRfld ) )
10		rebase 19952	. . . . . . . . . . 11 |- RR = ( Base ` RRfld )
11	10	fveq2i 6194	. . . . . . . . . 10 |- ( (subringAlg ` RRfld ) ` RR ) = ( (subringAlg ` RRfld ) ` ( Base ` RRfld ) )
12	9, 11	eqtr4i 2647	. . . . . . . . 9 |- (ringLMod ` RRfld ) = ( (subringAlg ` RRfld ) ` RR )
13	12	oveq1i 6660	. . . . . . . 8 |- ( (ringLMod ` RRfld ) ^s I ) = ( ( (subringAlg ` RRfld ) ` RR ) ^s I )
14	10	ressid 15935	. . . . . . . . . 10 |- (RRfld e. Field -> (RRfld ↾s RR ) = RRfld )
15	3, 14	ax-mp 5	. . . . . . . . 9 |- (RRfld ↾s RR ) = RRfld
16		eqidd 2623	. . . . . . . . . . 11 |- ( T. -> ( (subringAlg ` RRfld ) ` RR ) = ( (subringAlg ` RRfld ) ` RR ) )
17	10	eqimssi 3659	. . . . . . . . . . . 12 |- RR C_ ( Base ` RRfld )
18	17	a1i 11	. . . . . . . . . . 11 |- ( T. -> RR C_ ( Base ` RRfld ) )
19	16, 18	srasca 19181	. . . . . . . . . 10 |- ( T. -> (RRfld ↾s RR ) = (Scalar ` ( (subringAlg ` RRfld ) ` RR ) ) )
20	19	trud 1493	. . . . . . . . 9 |- (RRfld ↾s RR ) = (Scalar ` ( (subringAlg ` RRfld ) ` RR ) )
21	15, 20	eqtr3i 2646	. . . . . . . 8 |- RRfld = (Scalar ` ( (subringAlg ` RRfld ) ` RR ) )
22	13, 21	pwsval 16146	. . . . . . 7 |- ( ( ( (subringAlg ` RRfld ) ` RR ) e. _V /\ I e. V ) -> ( (ringLMod ` RRfld ) ^s I ) = (RRfld X_s ( I X. { ( (subringAlg ` RRfld ) ` RR ) } ) ) )
23	8, 22	mpan 706	. . . . . 6 |- ( I e. V -> ( (ringLMod ` RRfld ) ^s I ) = (RRfld X_s ( I X. { ( (subringAlg ` RRfld ) ` RR ) } ) ) )
24	23	eqcomd 2628	. . . . 5 |- ( I e. V -> (RRfld X_s ( I X. { ( (subringAlg ` RRfld ) ` RR ) } ) ) = ( (ringLMod ` RRfld ) ^s I ) )
25	2	fveq2d 6195	. . . . . 6 |- ( I e. V -> ( Base ` H ) = ( Base ` (toCHil ` (RRfld freeLMod I ) ) ) )
26		rrxbase.b	. . . . . 6 |- B = ( Base ` H )
27		eqid 2622	. . . . . . 7 |- (toCHil ` (RRfld freeLMod I ) ) = (toCHil ` (RRfld freeLMod I ) )
28	27, 5	tchbas 23018	. . . . . 6 |- ( Base ` (RRfld freeLMod I ) ) = ( Base ` (toCHil ` (RRfld freeLMod I ) ) )
29	25, 26, 28	3eqtr4g 2681	. . . . 5 |- ( I e. V -> B = ( Base ` (RRfld freeLMod I ) ) )
30	24, 29	oveq12d 6668	. . . 4 |- ( I e. V -> ( (RRfld X_s ( I X. { ( (subringAlg ` RRfld ) ` RR ) } ) ) ↾s B ) = ( ( (ringLMod ` RRfld ) ^s I ) ↾s ( Base ` (RRfld freeLMod I ) ) ) )
31	7, 30	eqtr4d 2659	. . 3 |- ( I e. V -> (RRfld freeLMod I ) = ( (RRfld X_s ( I X. { ( (subringAlg ` RRfld ) ` RR ) } ) ) ↾s B ) )
32	31	fveq2d 6195	. 2 |- ( I e. V -> (toCHil ` (RRfld freeLMod I ) ) = (toCHil ` ( (RRfld X_s ( I X. { ( (subringAlg ` RRfld ) ` RR ) } ) ) ↾s B ) ) )
33	2, 32	eqtrd 2656	1 |- ( I e. V -> H = (toCHil ` ( (RRfld X_s ( I X. { ( (subringAlg ` RRfld ) ` RR ) } ) ) ↾s B ) ) )

Syntax hints:   -> wi 4   = wceq 1483   T. wtru 1484   e. wcel 1990   _Vcvv 3200   C_ wss 3574   {csn 4177   X. cxp 5112   ` cfv 5888  (class class class)co 6650   RRcr 9935   Basecbs 15857   ↾_s cress 15858  Scalarcsca 15944   X__scprds 16106   ^_s cpws 16107  Fieldcfield 18748  subringAlg csra 19168  ringLModcrglmod 19169  RRfldcrefld 19950   freeLMod cfrlm 20090  toCHilctch 22967  ℝ^crrx 23171

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-rp 11833  df-fz 12327  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-0g 16102  df-prds 16108  df-pws 16110  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-subg 17591  df-cmn 18195  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-drng 18749  df-field 18750  df-subrg 18778  df-sra 19172  df-rgmod 19173  df-cnfld 19747  df-refld 19951  df-dsmm 20076  df-frlm 20091  df-tng 22389  df-tch 22969  df-rrx 23173

This theorem is referenced by:  rrxip  23178