Theorem pwsle 16152

Description: Ordering in a structure power. (Contributed by Mario Carneiro, 16-Aug-2015.)

Hypotheses

Ref	Expression
pwsle.y	|- Y = ( R ^s I )
pwsle.v	|- B = ( Base ` Y )
pwsle.o	|- O = ( le ` R )
pwsle.l	|- .<_ = ( le ` Y )

Assertion

Ref	Expression
pwsle	|- ( ( R e. V /\ I e. W ) -> .<_ = ( oR O i^i ( B X. B ) ) )

Proof of Theorem pwsle

Dummy variables f g x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3203	. . . . . . 7 |- f e. _V
2		vex 3203	. . . . . . 7 |- g e. _V
3	1, 2	prss 4351	. . . . . 6 |- ( ( f e. B /\ g e. B ) <-> { f , g } C_ B )
4		pwsle.v	. . . . . . . 8 |- B = ( Base ` Y )
5		pwsle.y	. . . . . . . . . 10 |- Y = ( R ^s I )
6		eqid 2622	. . . . . . . . . 10 |- (Scalar ` R ) = (Scalar ` R )
7	5, 6	pwsval 16146	. . . . . . . . 9 |- ( ( R e. V /\ I e. W ) -> Y = ( (Scalar ` R ) X_s ( I X. { R } ) ) )
8	7	fveq2d 6195	. . . . . . . 8 |- ( ( R e. V /\ I e. W ) -> ( Base ` Y ) = ( Base ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) )
9	4, 8	syl5eq 2668	. . . . . . 7 |- ( ( R e. V /\ I e. W ) -> B = ( Base ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) )
10	9	sseq2d 3633	. . . . . 6 |- ( ( R e. V /\ I e. W ) -> ( { f , g } C_ B <-> { f , g } C_ ( Base ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) ) )
11	3, 10	syl5bb 272	. . . . 5 |- ( ( R e. V /\ I e. W ) -> ( ( f e. B /\ g e. B ) <-> { f , g } C_ ( Base ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) ) )
12	11	anbi1d 741	. . . 4 |- ( ( R e. V /\ I e. W ) -> ( ( ( f e. B /\ g e. B ) /\ A. x e. I ( f ` x ) ( le ` ( ( I X. { R } ) ` x ) ) ( g ` x ) ) <-> ( { f , g } C_ ( Base ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) /\ A. x e. I ( f ` x ) ( le ` ( ( I X. { R } ) ` x ) ) ( g ` x ) ) ) )
13		simpll 790	. . . . . . . . . . . 12 |- ( ( ( R e. V /\ I e. W ) /\ ( f e. B /\ g e. B ) ) -> R e. V )
14		fvconst2g 6467	. . . . . . . . . . . 12 |- ( ( R e. V /\ x e. I ) -> ( ( I X. { R } ) ` x ) = R )
15	13, 14	sylan 488	. . . . . . . . . . 11 |- ( ( ( ( R e. V /\ I e. W ) /\ ( f e. B /\ g e. B ) ) /\ x e. I ) -> ( ( I X. { R } ) ` x ) = R )
16	15	fveq2d 6195	. . . . . . . . . 10 |- ( ( ( ( R e. V /\ I e. W ) /\ ( f e. B /\ g e. B ) ) /\ x e. I ) -> ( le ` ( ( I X. { R } ) ` x ) ) = ( le ` R ) )
17		pwsle.o	. . . . . . . . . 10 |- O = ( le ` R )
18	16, 17	syl6eqr 2674	. . . . . . . . 9 |- ( ( ( ( R e. V /\ I e. W ) /\ ( f e. B /\ g e. B ) ) /\ x e. I ) -> ( le ` ( ( I X. { R } ) ` x ) ) = O )
19	18	breqd 4664	. . . . . . . 8 |- ( ( ( ( R e. V /\ I e. W ) /\ ( f e. B /\ g e. B ) ) /\ x e. I ) -> ( ( f ` x ) ( le ` ( ( I X. { R } ) ` x ) ) ( g ` x ) <-> ( f ` x ) O ( g ` x ) ) )
20	19	ralbidva 2985	. . . . . . 7 |- ( ( ( R e. V /\ I e. W ) /\ ( f e. B /\ g e. B ) ) -> ( A. x e. I ( f ` x ) ( le ` ( ( I X. { R } ) ` x ) ) ( g ` x ) <-> A. x e. I ( f ` x ) O ( g ` x ) ) )
21		eqid 2622	. . . . . . . . . 10 |- ( Base ` R ) = ( Base ` R )
22		simplr 792	. . . . . . . . . 10 |- ( ( ( R e. V /\ I e. W ) /\ ( f e. B /\ g e. B ) ) -> I e. W )
23		simprl 794	. . . . . . . . . 10 |- ( ( ( R e. V /\ I e. W ) /\ ( f e. B /\ g e. B ) ) -> f e. B )
24	5, 21, 4, 13, 22, 23	pwselbas 16149	. . . . . . . . 9 |- ( ( ( R e. V /\ I e. W ) /\ ( f e. B /\ g e. B ) ) -> f : I --> ( Base ` R ) )
25		ffn 6045	. . . . . . . . 9 |- ( f : I --> ( Base ` R ) -> f Fn I )
26	24, 25	syl 17	. . . . . . . 8 |- ( ( ( R e. V /\ I e. W ) /\ ( f e. B /\ g e. B ) ) -> f Fn I )
27		simprr 796	. . . . . . . . . 10 |- ( ( ( R e. V /\ I e. W ) /\ ( f e. B /\ g e. B ) ) -> g e. B )
28	5, 21, 4, 13, 22, 27	pwselbas 16149	. . . . . . . . 9 |- ( ( ( R e. V /\ I e. W ) /\ ( f e. B /\ g e. B ) ) -> g : I --> ( Base ` R ) )
29		ffn 6045	. . . . . . . . 9 |- ( g : I --> ( Base ` R ) -> g Fn I )
30	28, 29	syl 17	. . . . . . . 8 |- ( ( ( R e. V /\ I e. W ) /\ ( f e. B /\ g e. B ) ) -> g Fn I )
31		inidm 3822	. . . . . . . 8 |- ( I i^i I ) = I
32		eqidd 2623	. . . . . . . 8 |- ( ( ( ( R e. V /\ I e. W ) /\ ( f e. B /\ g e. B ) ) /\ x e. I ) -> ( f ` x ) = ( f ` x ) )
33		eqidd 2623	. . . . . . . 8 |- ( ( ( ( R e. V /\ I e. W ) /\ ( f e. B /\ g e. B ) ) /\ x e. I ) -> ( g ` x ) = ( g ` x ) )
34	26, 30, 22, 22, 31, 32, 33	ofrfval 6905	. . . . . . 7 |- ( ( ( R e. V /\ I e. W ) /\ ( f e. B /\ g e. B ) ) -> ( f oR O g <-> A. x e. I ( f ` x ) O ( g ` x ) ) )
35	20, 34	bitr4d 271	. . . . . 6 |- ( ( ( R e. V /\ I e. W ) /\ ( f e. B /\ g e. B ) ) -> ( A. x e. I ( f ` x ) ( le ` ( ( I X. { R } ) ` x ) ) ( g ` x ) <-> f oR O g ) )
36	35	pm5.32da 673	. . . . 5 |- ( ( R e. V /\ I e. W ) -> ( ( ( f e. B /\ g e. B ) /\ A. x e. I ( f ` x ) ( le ` ( ( I X. { R } ) ` x ) ) ( g ` x ) ) <-> ( ( f e. B /\ g e. B ) /\ f oR O g ) ) )
37		brinxp2 5180	. . . . . 6 |- ( f ( oR O i^i ( B X. B ) ) g <-> ( f e. B /\ g e. B /\ f oR O g ) )
38		df-3an 1039	. . . . . 6 |- ( ( f e. B /\ g e. B /\ f oR O g ) <-> ( ( f e. B /\ g e. B ) /\ f oR O g ) )
39	37, 38	bitri 264	. . . . 5 |- ( f ( oR O i^i ( B X. B ) ) g <-> ( ( f e. B /\ g e. B ) /\ f oR O g ) )
40	36, 39	syl6bbr 278	. . . 4 |- ( ( R e. V /\ I e. W ) -> ( ( ( f e. B /\ g e. B ) /\ A. x e. I ( f ` x ) ( le ` ( ( I X. { R } ) ` x ) ) ( g ` x ) ) <-> f ( oR O i^i ( B X. B ) ) g ) )
41	12, 40	bitr3d 270	. . 3 |- ( ( R e. V /\ I e. W ) -> ( ( { f , g } C_ ( Base ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) /\ A. x e. I ( f ` x ) ( le ` ( ( I X. { R } ) ` x ) ) ( g ` x ) ) <-> f ( oR O i^i ( B X. B ) ) g ) )
42	41	opabbidv 4716	. 2 |- ( ( R e. V /\ I e. W ) -> { <. f , g >. | ( { f , g } C_ ( Base ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) /\ A. x e. I ( f ` x ) ( le ` ( ( I X. { R } ) ` x ) ) ( g ` x ) ) } = { <. f , g >. | f ( oR O i^i ( B X. B ) ) g } )
43		pwsle.l	. . . 4 |- .<_ = ( le ` Y )
44	7	fveq2d 6195	. . . 4 |- ( ( R e. V /\ I e. W ) -> ( le ` Y ) = ( le ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) )
45	43, 44	syl5eq 2668	. . 3 |- ( ( R e. V /\ I e. W ) -> .<_ = ( le ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) )
46		eqid 2622	. . . 4 |- ( (Scalar ` R ) X_s ( I X. { R } ) ) = ( (Scalar ` R ) X_s ( I X. { R } ) )
47		fvexd 6203	. . . 4 |- ( ( R e. V /\ I e. W ) -> (Scalar ` R ) e. _V )
48		simpr 477	. . . . 5 |- ( ( R e. V /\ I e. W ) -> I e. W )
49		snex 4908	. . . . 5 |- { R } e. _V
50		xpexg 6960	. . . . 5 |- ( ( I e. W /\ { R } e. _V ) -> ( I X. { R } ) e. _V )
51	48, 49, 50	sylancl 694	. . . 4 |- ( ( R e. V /\ I e. W ) -> ( I X. { R } ) e. _V )
52		eqid 2622	. . . 4 |- ( Base ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) = ( Base ` ( (Scalar ` R ) X_s ( I X. { R } ) ) )
53		snnzg 4308	. . . . . 6 |- ( R e. V -> { R } =/= (/) )
54	53	adantr 481	. . . . 5 |- ( ( R e. V /\ I e. W ) -> { R } =/= (/) )
55		dmxp 5344	. . . . 5 |- ( { R } =/= (/) -> dom ( I X. { R } ) = I )
56	54, 55	syl 17	. . . 4 |- ( ( R e. V /\ I e. W ) -> dom ( I X. { R } ) = I )
57		eqid 2622	. . . 4 |- ( le ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) = ( le ` ( (Scalar ` R ) X_s ( I X. { R } ) ) )
58	46, 47, 51, 52, 56, 57	prdsle 16122	. . 3 |- ( ( R e. V /\ I e. W ) -> ( le ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) = { <. f , g >. | ( { f , g } C_ ( Base ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) /\ A. x e. I ( f ` x ) ( le ` ( ( I X. { R } ) ` x ) ) ( g ` x ) ) } )
59	45, 58	eqtrd 2656	. 2 |- ( ( R e. V /\ I e. W ) -> .<_ = { <. f , g >. | ( { f , g } C_ ( Base ` ( (Scalar ` R ) X_s ( I X. { R } ) ) ) /\ A. x e. I ( f ` x ) ( le ` ( ( I X. { R } ) ` x ) ) ( g ` x ) ) } )
60		inss2 3834	. . . . 5 |- ( oR O i^i ( B X. B ) ) C_ ( B X. B )
61		relxp 5227	. . . . 5 |- Rel ( B X. B )
62		relss 5206	. . . . 5 |- ( ( oR O i^i ( B X. B ) ) C_ ( B X. B ) -> ( Rel ( B X. B ) -> Rel ( oR O i^i ( B X. B ) ) ) )
63	60, 61, 62	mp2 9	. . . 4 |- Rel ( oR O i^i ( B X. B ) )
64	63	a1i 11	. . 3 |- ( ( R e. V /\ I e. W ) -> Rel ( oR O i^i ( B X. B ) ) )
65		dfrel4v 5584	. . 3 |- ( Rel ( oR O i^i ( B X. B ) ) <-> ( oR O i^i ( B X. B ) ) = { <. f , g >. | f ( oR O i^i ( B X. B ) ) g } )
66	64, 65	sylib 208	. 2 |- ( ( R e. V /\ I e. W ) -> ( oR O i^i ( B X. B ) ) = { <. f , g >. | f ( oR O i^i ( B X. B ) ) g } )
67	42, 59, 66	3eqtr4d 2666	1 |- ( ( R e. V /\ I e. W ) -> .<_ = ( oR O i^i ( B X. B ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   {cpr 4179   class class class wbr 4653   {copab 4712   X. cxp 5112   dom cdm 5114   Rel wrel 5119   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oRcofr 6896   Basecbs 15857  Scalarcsca 15944   lecple 15948   X__scprds 16106   ^_s cpws 16107

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

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-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-ofr 6898  df-om 7066  df-1st 7168  df-2nd 7169  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-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-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-prds 16108  df-pws 16110

This theorem is referenced by:  pwsleval  16153