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Theorem rrxds 21553
Description: The distance over generalized Euclidean spaces. Compare with df-rrn 29912. (Contributed by Thierry Arnoux, 20-Jun-2019.) (Proof shortened by AV, 20-Jul-2019.)
Hypotheses
Ref Expression
rrxval.r  |-  H  =  (ℝ^ `  I )
rrxbase.b  |-  B  =  ( Base `  H
)
Assertion
Ref Expression
rrxds  |-  ( I  e.  V  ->  (
f  e.  B , 
g  e.  B  |->  ( sqr `  (RRfld  gsumg  ( x  e.  I  |->  ( ( ( f `  x
)  -  ( g `
 x ) ) ^ 2 ) ) ) ) )  =  ( dist `  H
) )
Distinct variable groups:    f, g, x, B    f, I, g, x    f, V, g, x
Allowed substitution hints:    H( x, f, g)

Proof of Theorem rrxds
Dummy variable  h is distinct from all other variables.
StepHypRef Expression
1 rrxval.r . . . 4  |-  H  =  (ℝ^ `  I )
21rrxval 21547 . . 3  |-  ( I  e.  V  ->  H  =  (toCHil `  (RRfld freeLMod  I ) ) )
32fveq2d 5861 . 2  |-  ( I  e.  V  ->  ( dist `  H )  =  ( dist `  (toCHil `  (RRfld freeLMod  I ) ) ) )
4 recrng 18417 . . . . 5  |- RRfld  e.  *Ring
5 srngrng 17277 . . . . 5  |-  (RRfld  e.  *Ring  -> RRfld 
e.  Ring )
64, 5ax-mp 5 . . . 4  |- RRfld  e.  Ring
7 eqid 2460 . . . . 5  |-  (RRfld freeLMod  I )  =  (RRfld freeLMod  I )
87frlmlmod 18540 . . . 4  |-  ( (RRfld 
e.  Ring  /\  I  e.  V )  ->  (RRfld freeLMod  I )  e.  LMod )
96, 8mpan 670 . . 3  |-  ( I  e.  V  ->  (RRfld freeLMod  I )  e.  LMod )
10 lmodgrp 17295 . . 3  |-  ( (RRfld freeLMod  I )  e.  LMod  ->  (RRfld freeLMod  I )  e.  Grp )
11 eqid 2460 . . . 4  |-  (toCHil `  (RRfld freeLMod  I ) )  =  (toCHil `  (RRfld freeLMod  I ) )
12 eqid 2460 . . . 4  |-  ( norm `  (toCHil `  (RRfld freeLMod  I ) ) )  =  (
norm `  (toCHil `  (RRfld freeLMod  I ) ) )
13 eqid 2460 . . . 4  |-  ( -g `  (RRfld freeLMod  I ) )  =  ( -g `  (RRfld freeLMod  I ) )
1411, 12, 13tchds 21402 . . 3  |-  ( (RRfld freeLMod  I )  e.  Grp  ->  ( ( norm `  (toCHil `  (RRfld freeLMod  I ) ) )  o.  ( -g `  (RRfld freeLMod  I ) ) )  =  ( dist `  (toCHil `  (RRfld freeLMod  I ) ) ) )
159, 10, 143syl 20 . 2  |-  ( I  e.  V  ->  (
( norm `  (toCHil `  (RRfld freeLMod  I ) ) )  o.  ( -g `  (RRfld freeLMod  I ) ) )  =  ( dist `  (toCHil `  (RRfld freeLMod  I ) ) ) )
16 eqid 2460 . . . . . . . 8  |-  ( Base `  (RRfld freeLMod  I ) )  =  ( Base `  (RRfld freeLMod  I ) )
1716, 13grpsubf 15911 . . . . . . 7  |-  ( (RRfld freeLMod  I )  e.  Grp  ->  (
-g `  (RRfld freeLMod  I ) ) : ( (
Base `  (RRfld freeLMod  I ) )  X.  ( Base `  (RRfld freeLMod  I ) ) ) --> ( Base `  (RRfld freeLMod  I ) ) )
189, 10, 173syl 20 . . . . . 6  |-  ( I  e.  V  ->  ( -g `  (RRfld freeLMod  I ) ) : ( ( Base `  (RRfld freeLMod  I ) )  X.  ( Base `  (RRfld freeLMod  I ) ) ) --> (
Base `  (RRfld freeLMod  I ) ) )
19 rrxbase.b . . . . . . . . . 10  |-  B  =  ( Base `  H
)
201, 19rrxbase 21548 . . . . . . . . 9  |-  ( I  e.  V  ->  B  =  { h  e.  ( RR  ^m  I )  |  h finSupp  0 }
)
21 rebase 18402 . . . . . . . . . . 11  |-  RR  =  ( Base ` RRfld )
22 re0g 18408 . . . . . . . . . . 11  |-  0  =  ( 0g ` RRfld )
23 eqid 2460 . . . . . . . . . . 11  |-  { h  e.  ( RR  ^m  I
)  |  h finSupp  0 }  =  { h  e.  ( RR  ^m  I
)  |  h finSupp  0 }
247, 21, 22, 23frlmbas 18546 . . . . . . . . . 10  |-  ( (RRfld 
e.  Ring  /\  I  e.  V )  ->  { h  e.  ( RR  ^m  I
)  |  h finSupp  0 }  =  ( Base `  (RRfld freeLMod  I ) ) )
256, 24mpan 670 . . . . . . . . 9  |-  ( I  e.  V  ->  { h  e.  ( RR  ^m  I
)  |  h finSupp  0 }  =  ( Base `  (RRfld freeLMod  I ) ) )
2620, 25eqtrd 2501 . . . . . . . 8  |-  ( I  e.  V  ->  B  =  ( Base `  (RRfld freeLMod  I ) ) )
2726, 26xpeq12d 5017 . . . . . . 7  |-  ( I  e.  V  ->  ( B  X.  B )  =  ( ( Base `  (RRfld freeLMod  I ) )  X.  ( Base `  (RRfld freeLMod  I ) ) ) )
2827, 26feq23d 5717 . . . . . 6  |-  ( I  e.  V  ->  (
( -g `  (RRfld freeLMod  I ) ) : ( B  X.  B ) --> B  <-> 
( -g `  (RRfld freeLMod  I ) ) : ( (
Base `  (RRfld freeLMod  I ) )  X.  ( Base `  (RRfld freeLMod  I ) ) ) --> ( Base `  (RRfld freeLMod  I ) ) ) )
2918, 28mpbird 232 . . . . 5  |-  ( I  e.  V  ->  ( -g `  (RRfld freeLMod  I ) ) : ( B  X.  B ) --> B )
3029fovrnda 6421 . . . 4  |-  ( ( I  e.  V  /\  ( f  e.  B  /\  g  e.  B
) )  ->  (
f ( -g `  (RRfld freeLMod  I ) ) g )  e.  B )
31 ffn 5722 . . . . . 6  |-  ( (
-g `  (RRfld freeLMod  I ) ) : ( B  X.  B ) --> B  ->  ( -g `  (RRfld freeLMod  I ) )  Fn  ( B  X.  B ) )
3229, 31syl 16 . . . . 5  |-  ( I  e.  V  ->  ( -g `  (RRfld freeLMod  I ) )  Fn  ( B  X.  B ) )
33 fnov 6385 . . . . 5  |-  ( (
-g `  (RRfld freeLMod  I ) )  Fn  ( B  X.  B )  <->  ( -g `  (RRfld freeLMod  I ) )  =  ( f  e.  B ,  g  e.  B  |->  ( f ( -g `  (RRfld freeLMod  I ) ) g ) ) )
3432, 33sylib 196 . . . 4  |-  ( I  e.  V  ->  ( -g `  (RRfld freeLMod  I ) )  =  ( f  e.  B ,  g  e.  B  |->  ( f (
-g `  (RRfld freeLMod  I ) ) g ) ) )
351, 19rrxnm 21551 . . . . 5  |-  ( I  e.  V  ->  (
h  e.  B  |->  ( sqr `  (RRfld  gsumg  ( x  e.  I  |->  ( ( h `  x ) ^ 2 ) ) ) ) )  =  ( norm `  H
) )
362fveq2d 5861 . . . . 5  |-  ( I  e.  V  ->  ( norm `  H )  =  ( norm `  (toCHil `  (RRfld freeLMod  I ) ) ) )
3735, 36eqtr2d 2502 . . . 4  |-  ( I  e.  V  ->  ( norm `  (toCHil `  (RRfld freeLMod  I ) ) )  =  ( h  e.  B  |->  ( sqr `  (RRfld  gsumg  (
x  e.  I  |->  ( ( h `  x
) ^ 2 ) ) ) ) ) )
38 fveq1 5856 . . . . . . . 8  |-  ( h  =  ( f (
-g `  (RRfld freeLMod  I ) ) g )  -> 
( h `  x
)  =  ( ( f ( -g `  (RRfld freeLMod  I ) ) g ) `
 x ) )
3938oveq1d 6290 . . . . . . 7  |-  ( h  =  ( f (
-g `  (RRfld freeLMod  I ) ) g )  -> 
( ( h `  x ) ^ 2 )  =  ( ( ( f ( -g `  (RRfld freeLMod  I ) ) g ) `  x ) ^ 2 ) )
4039mpteq2dv 4527 . . . . . 6  |-  ( h  =  ( f (
-g `  (RRfld freeLMod  I ) ) g )  -> 
( x  e.  I  |->  ( ( h `  x ) ^ 2 ) )  =  ( x  e.  I  |->  ( ( ( f (
-g `  (RRfld freeLMod  I ) ) g ) `  x ) ^ 2 ) ) )
4140oveq2d 6291 . . . . 5  |-  ( h  =  ( f (
-g `  (RRfld freeLMod  I ) ) g )  -> 
(RRfld  gsumg  ( x  e.  I  |->  ( ( h `  x ) ^ 2 ) ) )  =  (RRfld  gsumg  ( x  e.  I  |->  ( ( ( f ( -g `  (RRfld freeLMod  I ) ) g ) `
 x ) ^
2 ) ) ) )
4241fveq2d 5861 . . . 4  |-  ( h  =  ( f (
-g `  (RRfld freeLMod  I ) ) g )  -> 
( sqr `  (RRfld  gsumg  (
x  e.  I  |->  ( ( h `  x
) ^ 2 ) ) ) )  =  ( sqr `  (RRfld  gsumg  (
x  e.  I  |->  ( ( ( f (
-g `  (RRfld freeLMod  I ) ) g ) `  x ) ^ 2 ) ) ) ) )
4330, 34, 37, 42fmpt2co 6856 . . 3  |-  ( I  e.  V  ->  (
( norm `  (toCHil `  (RRfld freeLMod  I ) ) )  o.  ( -g `  (RRfld freeLMod  I ) ) )  =  ( f  e.  B ,  g  e.  B  |->  ( sqr `  (RRfld  gsumg  (
x  e.  I  |->  ( ( ( f (
-g `  (RRfld freeLMod  I ) ) g ) `  x ) ^ 2 ) ) ) ) ) )
44 simp1 991 . . . . . . . . . . . . . 14  |-  ( ( I  e.  V  /\  f  e.  B  /\  g  e.  B )  ->  I  e.  V )
45 simprl 755 . . . . . . . . . . . . . . . 16  |-  ( ( I  e.  V  /\  ( f  e.  B  /\  g  e.  B
) )  ->  f  e.  B )
4626adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( I  e.  V  /\  ( f  e.  B  /\  g  e.  B
) )  ->  B  =  ( Base `  (RRfld freeLMod  I ) ) )
4745, 46eleqtrd 2550 . . . . . . . . . . . . . . 15  |-  ( ( I  e.  V  /\  ( f  e.  B  /\  g  e.  B
) )  ->  f  e.  ( Base `  (RRfld freeLMod  I ) ) )
48473impb 1187 . . . . . . . . . . . . . 14  |-  ( ( I  e.  V  /\  f  e.  B  /\  g  e.  B )  ->  f  e.  ( Base `  (RRfld freeLMod  I ) ) )
497, 21, 16frlmbasmap 18553 . . . . . . . . . . . . . 14  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  f  e.  ( RR  ^m  I ) )
5044, 48, 49syl2anc 661 . . . . . . . . . . . . 13  |-  ( ( I  e.  V  /\  f  e.  B  /\  g  e.  B )  ->  f  e.  ( RR 
^m  I ) )
51 elmapi 7430 . . . . . . . . . . . . 13  |-  ( f  e.  ( RR  ^m  I )  ->  f : I --> RR )
5250, 51syl 16 . . . . . . . . . . . 12  |-  ( ( I  e.  V  /\  f  e.  B  /\  g  e.  B )  ->  f : I --> RR )
53 ffn 5722 . . . . . . . . . . . 12  |-  ( f : I --> RR  ->  f  Fn  I )
5452, 53syl 16 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  f  e.  B  /\  g  e.  B )  ->  f  Fn  I )
55 simprr 756 . . . . . . . . . . . . . . . 16  |-  ( ( I  e.  V  /\  ( f  e.  B  /\  g  e.  B
) )  ->  g  e.  B )
5655, 46eleqtrd 2550 . . . . . . . . . . . . . . 15  |-  ( ( I  e.  V  /\  ( f  e.  B  /\  g  e.  B
) )  ->  g  e.  ( Base `  (RRfld freeLMod  I ) ) )
57563impb 1187 . . . . . . . . . . . . . 14  |-  ( ( I  e.  V  /\  f  e.  B  /\  g  e.  B )  ->  g  e.  ( Base `  (RRfld freeLMod  I ) ) )
587, 21, 16frlmbasmap 18553 . . . . . . . . . . . . . 14  |-  ( ( I  e.  V  /\  g  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  g  e.  ( RR  ^m  I ) )
5944, 57, 58syl2anc 661 . . . . . . . . . . . . 13  |-  ( ( I  e.  V  /\  f  e.  B  /\  g  e.  B )  ->  g  e.  ( RR 
^m  I ) )
60 elmapi 7430 . . . . . . . . . . . . 13  |-  ( g  e.  ( RR  ^m  I )  ->  g : I --> RR )
6159, 60syl 16 . . . . . . . . . . . 12  |-  ( ( I  e.  V  /\  f  e.  B  /\  g  e.  B )  ->  g : I --> RR )
62 ffn 5722 . . . . . . . . . . . 12  |-  ( g : I --> RR  ->  g  Fn  I )
6361, 62syl 16 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  f  e.  B  /\  g  e.  B )  ->  g  Fn  I )
64 inidm 3700 . . . . . . . . . . 11  |-  ( I  i^i  I )  =  I
65 eqidd 2461 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  f  e.  B  /\  g  e.  B
)  /\  x  e.  I )  ->  (
f `  x )  =  ( f `  x ) )
66 eqidd 2461 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  f  e.  B  /\  g  e.  B
)  /\  x  e.  I )  ->  (
g `  x )  =  ( g `  x ) )
6754, 63, 44, 44, 64, 65, 66offval 6522 . . . . . . . . . 10  |-  ( ( I  e.  V  /\  f  e.  B  /\  g  e.  B )  ->  ( f  oF ( -g ` RRfld ) g )  =  ( x  e.  I  |->  ( ( f `  x ) ( -g ` RRfld ) ( g `  x ) ) ) )
686a1i 11 . . . . . . . . . . . 12  |-  ( ( I  e.  V  /\  ( f  e.  B  /\  g  e.  B
) )  -> RRfld  e.  Ring )
69 simpl 457 . . . . . . . . . . . 12  |-  ( ( I  e.  V  /\  ( f  e.  B  /\  g  e.  B
) )  ->  I  e.  V )
70 eqid 2460 . . . . . . . . . . . 12  |-  ( -g ` RRfld
)  =  ( -g ` RRfld
)
717, 16, 68, 69, 47, 56, 70, 13frlmsubgval 18558 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  ( f  e.  B  /\  g  e.  B
) )  ->  (
f ( -g `  (RRfld freeLMod  I ) ) g )  =  ( f  oF ( -g ` RRfld ) g ) )
72713impb 1187 . . . . . . . . . 10  |-  ( ( I  e.  V  /\  f  e.  B  /\  g  e.  B )  ->  ( f ( -g `  (RRfld freeLMod  I ) ) g )  =  ( f  oF ( -g ` RRfld
) g ) )
7352ffvelrnda 6012 . . . . . . . . . . . 12  |-  ( ( ( I  e.  V  /\  f  e.  B  /\  g  e.  B
)  /\  x  e.  I )  ->  (
f `  x )  e.  RR )
7461ffvelrnda 6012 . . . . . . . . . . . 12  |-  ( ( ( I  e.  V  /\  f  e.  B  /\  g  e.  B
)  /\  x  e.  I )  ->  (
g `  x )  e.  RR )
7570resubgval 18405 . . . . . . . . . . . 12  |-  ( ( ( f `  x
)  e.  RR  /\  ( g `  x
)  e.  RR )  ->  ( ( f `
 x )  -  ( g `  x
) )  =  ( ( f `  x
) ( -g ` RRfld ) ( g `  x ) ) )
7673, 74, 75syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  f  e.  B  /\  g  e.  B
)  /\  x  e.  I )  ->  (
( f `  x
)  -  ( g `
 x ) )  =  ( ( f `
 x ) (
-g ` RRfld ) (
g `  x )
) )
7776mpteq2dva 4526 . . . . . . . . . 10  |-  ( ( I  e.  V  /\  f  e.  B  /\  g  e.  B )  ->  ( x  e.  I  |->  ( ( f `  x )  -  (
g `  x )
) )  =  ( x  e.  I  |->  ( ( f `  x
) ( -g ` RRfld ) ( g `  x ) ) ) )
7867, 72, 773eqtr4d 2511 . . . . . . . . 9  |-  ( ( I  e.  V  /\  f  e.  B  /\  g  e.  B )  ->  ( f ( -g `  (RRfld freeLMod  I ) ) g )  =  ( x  e.  I  |->  ( ( f `  x )  -  ( g `  x ) ) ) )
7973, 74resubcld 9976 . . . . . . . . 9  |-  ( ( ( I  e.  V  /\  f  e.  B  /\  g  e.  B
)  /\  x  e.  I )  ->  (
( f `  x
)  -  ( g `
 x ) )  e.  RR )
8078, 79fvmpt2d 5950 . . . . . . . 8  |-  ( ( ( I  e.  V  /\  f  e.  B  /\  g  e.  B
)  /\  x  e.  I )  ->  (
( f ( -g `  (RRfld freeLMod  I ) ) g ) `  x )  =  ( ( f `
 x )  -  ( g `  x
) ) )
8180oveq1d 6290 . . . . . . 7  |-  ( ( ( I  e.  V  /\  f  e.  B  /\  g  e.  B
)  /\  x  e.  I )  ->  (
( ( f (
-g `  (RRfld freeLMod  I ) ) g ) `  x ) ^ 2 )  =  ( ( ( f `  x
)  -  ( g `
 x ) ) ^ 2 ) )
8281mpteq2dva 4526 . . . . . 6  |-  ( ( I  e.  V  /\  f  e.  B  /\  g  e.  B )  ->  ( x  e.  I  |->  ( ( ( f ( -g `  (RRfld freeLMod  I ) ) g ) `
 x ) ^
2 ) )  =  ( x  e.  I  |->  ( ( ( f `
 x )  -  ( g `  x
) ) ^ 2 ) ) )
8382oveq2d 6291 . . . . 5  |-  ( ( I  e.  V  /\  f  e.  B  /\  g  e.  B )  ->  (RRfld  gsumg  ( x  e.  I  |->  ( ( ( f ( -g `  (RRfld freeLMod  I ) ) g ) `
 x ) ^
2 ) ) )  =  (RRfld  gsumg  ( x  e.  I  |->  ( ( ( f `
 x )  -  ( g `  x
) ) ^ 2 ) ) ) )
8483fveq2d 5861 . . . 4  |-  ( ( I  e.  V  /\  f  e.  B  /\  g  e.  B )  ->  ( sqr `  (RRfld  gsumg  (
x  e.  I  |->  ( ( ( f (
-g `  (RRfld freeLMod  I ) ) g ) `  x ) ^ 2 ) ) ) )  =  ( sqr `  (RRfld  gsumg  (
x  e.  I  |->  ( ( ( f `  x )  -  (
g `  x )
) ^ 2 ) ) ) ) )
8584mpt2eq3dva 6336 . . 3  |-  ( I  e.  V  ->  (
f  e.  B , 
g  e.  B  |->  ( sqr `  (RRfld  gsumg  ( x  e.  I  |->  ( ( ( f ( -g `  (RRfld freeLMod  I ) ) g ) `  x ) ^ 2 ) ) ) ) )  =  ( f  e.  B ,  g  e.  B  |->  ( sqr `  (RRfld  gsumg  (
x  e.  I  |->  ( ( ( f `  x )  -  (
g `  x )
) ^ 2 ) ) ) ) ) )
8643, 85eqtrd 2501 . 2  |-  ( I  e.  V  ->  (
( norm `  (toCHil `  (RRfld freeLMod  I ) ) )  o.  ( -g `  (RRfld freeLMod  I ) ) )  =  ( f  e.  B ,  g  e.  B  |->  ( sqr `  (RRfld  gsumg  (
x  e.  I  |->  ( ( ( f `  x )  -  (
g `  x )
) ^ 2 ) ) ) ) ) )
873, 15, 863eqtr2rd 2508 1  |-  ( I  e.  V  ->  (
f  e.  B , 
g  e.  B  |->  ( sqr `  (RRfld  gsumg  ( x  e.  I  |->  ( ( ( f `  x
)  -  ( g `
 x ) ) ^ 2 ) ) ) ) )  =  ( dist `  H
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   {crab 2811   class class class wbr 4440    |-> cmpt 4498    X. cxp 4990    o. ccom 4996    Fn wfn 5574   -->wf 5575   ` cfv 5579  (class class class)co 6275    |-> cmpt2 6277    oFcof 6513    ^m cmap 7410   finSupp cfsupp 7818   RRcr 9480   0cc0 9481    - cmin 9794   2c2 10574   ^cexp 12122   sqrcsqr 13016   Basecbs 14479   distcds 14553    gsumg cgsu 14685   Grpcgrp 15716   -gcsg 15719   Ringcrg 16979   *Ringcsr 17269   LModclmod 17288  RRfldcrefld 18400   freeLMod cfrlm 18537   normcnm 20825  toCHilctch 21342  ℝ^crrx 21543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-tpos 6945  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-ixp 7460  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fsupp 7819  df-sup 7890  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-rp 11210  df-fz 11662  df-seq 12064  df-exp 12123  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-starv 14559  df-sca 14560  df-vsca 14561  df-ip 14562  df-tset 14563  df-ple 14564  df-ds 14566  df-unif 14567  df-hom 14568  df-cco 14569  df-0g 14686  df-prds 14692  df-pws 14694  df-mnd 15721  df-mhm 15770  df-grp 15851  df-minusg 15852  df-sbg 15853  df-subg 15986  df-ghm 16053  df-cmn 16589  df-mgp 16925  df-ur 16937  df-rng 16981  df-cring 16982  df-oppr 17049  df-dvdsr 17067  df-unit 17068  df-invr 17098  df-dvr 17109  df-rnghom 17141  df-drng 17174  df-field 17175  df-subrg 17203  df-staf 17270  df-srng 17271  df-lmod 17290  df-lss 17355  df-sra 17594  df-rgmod 17595  df-cnfld 18185  df-refld 18401  df-dsmm 18523  df-frlm 18538  df-nm 20831  df-tng 20833  df-tch 21344  df-rrx 21545
This theorem is referenced by:  rrxmval  21560  rrxmfval  21561
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