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Theorem rrxds 22115
Description: The distance over generalized Euclidean spaces. Compare with df-rrn 31584. (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 22109 . . 3  |-  ( I  e.  V  ->  H  =  (toCHil `  (RRfld freeLMod  I ) ) )
32fveq2d 5852 . 2  |-  ( I  e.  V  ->  ( dist `  H )  =  ( dist `  (toCHil `  (RRfld freeLMod  I ) ) ) )
4 recrng 18953 . . . . 5  |- RRfld  e.  *Ring
5 srngring 17819 . . . . 5  |-  (RRfld  e.  *Ring  -> RRfld 
e.  Ring )
64, 5ax-mp 5 . . . 4  |- RRfld  e.  Ring
7 eqid 2402 . . . . 5  |-  (RRfld freeLMod  I )  =  (RRfld freeLMod  I )
87frlmlmod 19076 . . . 4  |-  ( (RRfld 
e.  Ring  /\  I  e.  V )  ->  (RRfld freeLMod  I )  e.  LMod )
96, 8mpan 668 . . 3  |-  ( I  e.  V  ->  (RRfld freeLMod  I )  e.  LMod )
10 lmodgrp 17837 . . 3  |-  ( (RRfld freeLMod  I )  e.  LMod  ->  (RRfld freeLMod  I )  e.  Grp )
11 eqid 2402 . . . 4  |-  (toCHil `  (RRfld freeLMod  I ) )  =  (toCHil `  (RRfld freeLMod  I ) )
12 eqid 2402 . . . 4  |-  ( norm `  (toCHil `  (RRfld freeLMod  I ) ) )  =  (
norm `  (toCHil `  (RRfld freeLMod  I ) ) )
13 eqid 2402 . . . 4  |-  ( -g `  (RRfld freeLMod  I ) )  =  ( -g `  (RRfld freeLMod  I ) )
1411, 12, 13tchds 21964 . . 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 2402 . . . . . . . 8  |-  ( Base `  (RRfld freeLMod  I ) )  =  ( Base `  (RRfld freeLMod  I ) )
1716, 13grpsubf 16439 . . . . . . 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 22110 . . . . . . . . 9  |-  ( I  e.  V  ->  B  =  { h  e.  ( RR  ^m  I )  |  h finSupp  0 }
)
21 rebase 18938 . . . . . . . . . . 11  |-  RR  =  ( Base ` RRfld )
22 re0g 18944 . . . . . . . . . . 11  |-  0  =  ( 0g ` RRfld )
23 eqid 2402 . . . . . . . . . . 11  |-  { h  e.  ( RR  ^m  I
)  |  h finSupp  0 }  =  { h  e.  ( RR  ^m  I
)  |  h finSupp  0 }
247, 21, 22, 23frlmbas 19082 . . . . . . . . . 10  |-  ( (RRfld 
e.  Ring  /\  I  e.  V )  ->  { h  e.  ( RR  ^m  I
)  |  h finSupp  0 }  =  ( Base `  (RRfld freeLMod  I ) ) )
256, 24mpan 668 . . . . . . . . 9  |-  ( I  e.  V  ->  { h  e.  ( RR  ^m  I
)  |  h finSupp  0 }  =  ( Base `  (RRfld freeLMod  I ) ) )
2620, 25eqtrd 2443 . . . . . . . 8  |-  ( I  e.  V  ->  B  =  ( Base `  (RRfld freeLMod  I ) ) )
2726sqxpeqd 4848 . . . . . . 7  |-  ( I  e.  V  ->  ( B  X.  B )  =  ( ( Base `  (RRfld freeLMod  I ) )  X.  ( Base `  (RRfld freeLMod  I ) ) ) )
2827, 26feq23d 5708 . . . . . 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 6426 . . . 4  |-  ( ( I  e.  V  /\  ( f  e.  B  /\  g  e.  B
) )  ->  (
f ( -g `  (RRfld freeLMod  I ) ) g )  e.  B )
31 ffn 5713 . . . . . 6  |-  ( (
-g `  (RRfld freeLMod  I ) ) : ( B  X.  B ) --> B  ->  ( -g `  (RRfld freeLMod  I ) )  Fn  ( B  X.  B ) )
3229, 31syl 17 . . . . 5  |-  ( I  e.  V  ->  ( -g `  (RRfld freeLMod  I ) )  Fn  ( B  X.  B ) )
33 fnov 6390 . . . . 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 22113 . . . . 5  |-  ( I  e.  V  ->  (
h  e.  B  |->  ( sqr `  (RRfld  gsumg  ( x  e.  I  |->  ( ( h `  x ) ^ 2 ) ) ) ) )  =  ( norm `  H
) )
362fveq2d 5852 . . . . 5  |-  ( I  e.  V  ->  ( norm `  H )  =  ( norm `  (toCHil `  (RRfld freeLMod  I ) ) ) )
3735, 36eqtr2d 2444 . . . 4  |-  ( I  e.  V  ->  ( norm `  (toCHil `  (RRfld freeLMod  I ) ) )  =  ( h  e.  B  |->  ( sqr `  (RRfld  gsumg  (
x  e.  I  |->  ( ( h `  x
) ^ 2 ) ) ) ) ) )
38 fveq1 5847 . . . . . . . 8  |-  ( h  =  ( f (
-g `  (RRfld freeLMod  I ) ) g )  -> 
( h `  x
)  =  ( ( f ( -g `  (RRfld freeLMod  I ) ) g ) `
 x ) )
3938oveq1d 6292 . . . . . . 7  |-  ( h  =  ( f (
-g `  (RRfld freeLMod  I ) ) g )  -> 
( ( h `  x ) ^ 2 )  =  ( ( ( f ( -g `  (RRfld freeLMod  I ) ) g ) `  x ) ^ 2 ) )
4039mpteq2dv 4481 . . . . . 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 6293 . . . . 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 5852 . . . 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 6866 . . 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 997 . . . . . . . . . . . . . 14  |-  ( ( I  e.  V  /\  f  e.  B  /\  g  e.  B )  ->  I  e.  V )
45 simprl 756 . . . . . . . . . . . . . . . 16  |-  ( ( I  e.  V  /\  ( f  e.  B  /\  g  e.  B
) )  ->  f  e.  B )
4626adantr 463 . . . . . . . . . . . . . . . 16  |-  ( ( I  e.  V  /\  ( f  e.  B  /\  g  e.  B
) )  ->  B  =  ( Base `  (RRfld freeLMod  I ) ) )
4745, 46eleqtrd 2492 . . . . . . . . . . . . . . 15  |-  ( ( I  e.  V  /\  ( f  e.  B  /\  g  e.  B
) )  ->  f  e.  ( Base `  (RRfld freeLMod  I ) ) )
48473impb 1193 . . . . . . . . . . . . . 14  |-  ( ( I  e.  V  /\  f  e.  B  /\  g  e.  B )  ->  f  e.  ( Base `  (RRfld freeLMod  I ) ) )
497, 21, 16frlmbasmap 19087 . . . . . . . . . . . . . 14  |-  ( ( I  e.  V  /\  f  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  f  e.  ( RR  ^m  I ) )
5044, 48, 49syl2anc 659 . . . . . . . . . . . . 13  |-  ( ( I  e.  V  /\  f  e.  B  /\  g  e.  B )  ->  f  e.  ( RR 
^m  I ) )
51 elmapi 7477 . . . . . . . . . . . . 13  |-  ( f  e.  ( RR  ^m  I )  ->  f : I --> RR )
5250, 51syl 17 . . . . . . . . . . . 12  |-  ( ( I  e.  V  /\  f  e.  B  /\  g  e.  B )  ->  f : I --> RR )
53 ffn 5713 . . . . . . . . . . . 12  |-  ( f : I --> RR  ->  f  Fn  I )
5452, 53syl 17 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  f  e.  B  /\  g  e.  B )  ->  f  Fn  I )
55 simprr 758 . . . . . . . . . . . . . . . 16  |-  ( ( I  e.  V  /\  ( f  e.  B  /\  g  e.  B
) )  ->  g  e.  B )
5655, 46eleqtrd 2492 . . . . . . . . . . . . . . 15  |-  ( ( I  e.  V  /\  ( f  e.  B  /\  g  e.  B
) )  ->  g  e.  ( Base `  (RRfld freeLMod  I ) ) )
57563impb 1193 . . . . . . . . . . . . . 14  |-  ( ( I  e.  V  /\  f  e.  B  /\  g  e.  B )  ->  g  e.  ( Base `  (RRfld freeLMod  I ) ) )
587, 21, 16frlmbasmap 19087 . . . . . . . . . . . . . 14  |-  ( ( I  e.  V  /\  g  e.  ( Base `  (RRfld freeLMod  I ) ) )  ->  g  e.  ( RR  ^m  I ) )
5944, 57, 58syl2anc 659 . . . . . . . . . . . . 13  |-  ( ( I  e.  V  /\  f  e.  B  /\  g  e.  B )  ->  g  e.  ( RR 
^m  I ) )
60 elmapi 7477 . . . . . . . . . . . . 13  |-  ( g  e.  ( RR  ^m  I )  ->  g : I --> RR )
6159, 60syl 17 . . . . . . . . . . . 12  |-  ( ( I  e.  V  /\  f  e.  B  /\  g  e.  B )  ->  g : I --> RR )
62 ffn 5713 . . . . . . . . . . . 12  |-  ( g : I --> RR  ->  g  Fn  I )
6361, 62syl 17 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  f  e.  B  /\  g  e.  B )  ->  g  Fn  I )
64 inidm 3647 . . . . . . . . . . 11  |-  ( I  i^i  I )  =  I
65 eqidd 2403 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  f  e.  B  /\  g  e.  B
)  /\  x  e.  I )  ->  (
f `  x )  =  ( f `  x ) )
66 eqidd 2403 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  f  e.  B  /\  g  e.  B
)  /\  x  e.  I )  ->  (
g `  x )  =  ( g `  x ) )
6754, 63, 44, 44, 64, 65, 66offval 6527 . . . . . . . . . 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 455 . . . . . . . . . . . 12  |-  ( ( I  e.  V  /\  ( f  e.  B  /\  g  e.  B
) )  ->  I  e.  V )
70 eqid 2402 . . . . . . . . . . . 12  |-  ( -g ` RRfld
)  =  ( -g ` RRfld
)
717, 16, 68, 69, 47, 56, 70, 13frlmsubgval 19092 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  ( f  e.  B  /\  g  e.  B
) )  ->  (
f ( -g `  (RRfld freeLMod  I ) ) g )  =  ( f  oF ( -g ` RRfld ) g ) )
72713impb 1193 . . . . . . . . . 10  |-  ( ( I  e.  V  /\  f  e.  B  /\  g  e.  B )  ->  ( f ( -g `  (RRfld freeLMod  I ) ) g )  =  ( f  oF ( -g ` RRfld
) g ) )
7352ffvelrnda 6008 . . . . . . . . . . . 12  |-  ( ( ( I  e.  V  /\  f  e.  B  /\  g  e.  B
)  /\  x  e.  I )  ->  (
f `  x )  e.  RR )
7461ffvelrnda 6008 . . . . . . . . . . . 12  |-  ( ( ( I  e.  V  /\  f  e.  B  /\  g  e.  B
)  /\  x  e.  I )  ->  (
g `  x )  e.  RR )
7570resubgval 18941 . . . . . . . . . . . 12  |-  ( ( ( f `  x
)  e.  RR  /\  ( g `  x
)  e.  RR )  ->  ( ( f `
 x )  -  ( g `  x
) )  =  ( ( f `  x
) ( -g ` RRfld ) ( g `  x ) ) )
7673, 74, 75syl2anc 659 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  f  e.  B  /\  g  e.  B
)  /\  x  e.  I )  ->  (
( f `  x
)  -  ( g `
 x ) )  =  ( ( f `
 x ) (
-g ` RRfld ) (
g `  x )
) )
7776mpteq2dva 4480 . . . . . . . . . 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 2453 . . . . . . . . 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 10027 . . . . . . . . 9  |-  ( ( ( I  e.  V  /\  f  e.  B  /\  g  e.  B
)  /\  x  e.  I )  ->  (
( f `  x
)  -  ( g `
 x ) )  e.  RR )
8078, 79fvmpt2d 5942 . . . . . . . 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 6292 . . . . . . 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 4480 . . . . . 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 6293 . . . . 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 5852 . . . 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 6341 . . 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 2443 . 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 2450 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842   {crab 2757   class class class wbr 4394    |-> cmpt 4452    X. cxp 4820    o. ccom 4826    Fn wfn 5563   -->wf 5564   ` cfv 5568  (class class class)co 6277    |-> cmpt2 6279    oFcof 6518    ^m cmap 7456   finSupp cfsupp 7862   RRcr 9520   0cc0 9521    - cmin 9840   2c2 10625   ^cexp 12208   sqrcsqrt 13213   Basecbs 14839   distcds 14916    gsumg cgsu 15053   Grpcgrp 16375   -gcsg 16377   Ringcrg 17516   *Ringcsr 17811   LModclmod 17830  RRfldcrefld 18936   freeLMod cfrlm 19073   normcnm 21387  toCHilctch 21904  ℝ^crrx 22105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-pre-sup 9599  ax-addf 9600  ax-mulf 9601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6520  df-om 6683  df-1st 6783  df-2nd 6784  df-supp 6902  df-tpos 6957  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-oadd 7170  df-er 7347  df-map 7458  df-ixp 7507  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-fsupp 7863  df-sup 7934  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-nn 10576  df-2 10634  df-3 10635  df-4 10636  df-5 10637  df-6 10638  df-7 10639  df-8 10640  df-9 10641  df-10 10642  df-n0 10836  df-z 10905  df-dec 11019  df-uz 11127  df-rp 11265  df-fz 11725  df-seq 12150  df-exp 12209  df-cj 13079  df-re 13080  df-im 13081  df-sqrt 13215  df-abs 13216  df-struct 14841  df-ndx 14842  df-slot 14843  df-base 14844  df-sets 14845  df-ress 14846  df-plusg 14920  df-mulr 14921  df-starv 14922  df-sca 14923  df-vsca 14924  df-ip 14925  df-tset 14926  df-ple 14927  df-ds 14929  df-unif 14930  df-hom 14931  df-cco 14932  df-0g 15054  df-prds 15060  df-pws 15062  df-mgm 16194  df-sgrp 16233  df-mnd 16243  df-mhm 16288  df-grp 16379  df-minusg 16380  df-sbg 16381  df-subg 16520  df-ghm 16587  df-cmn 17122  df-mgp 17460  df-ur 17472  df-ring 17518  df-cring 17519  df-oppr 17590  df-dvdsr 17608  df-unit 17609  df-invr 17639  df-dvr 17650  df-rnghom 17682  df-drng 17716  df-field 17717  df-subrg 17745  df-staf 17812  df-srng 17813  df-lmod 17832  df-lss 17897  df-sra 18136  df-rgmod 18137  df-cnfld 18739  df-refld 18937  df-dsmm 19059  df-frlm 19074  df-nm 21393  df-tng 21395  df-tch 21906  df-rrx 22107
This theorem is referenced by:  rrxmval  22122  rrxmfval  22123
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