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Theorem isinftm 26210
Description: Express  x is infinitesimal with respect to  y for a structure  W. (Contributed by Thierry Arnoux, 30-Jan-2018.)
Hypotheses
Ref Expression
inftm.b  |-  B  =  ( Base `  W
)
inftm.0  |-  .0.  =  ( 0g `  W )
inftm.x  |-  .x.  =  (.g
`  W )
inftm.l  |-  .<  =  ( lt `  W )
Assertion
Ref Expression
isinftm  |-  ( ( W  e.  V  /\  X  e.  B  /\  Y  e.  B )  ->  ( X (<<< `  W
) Y  <->  (  .0.  .<  X  /\  A. n  e.  NN  ( n  .x.  X )  .<  Y ) ) )
Distinct variable groups:    n, W    n, X    n, Y
Allowed substitution hints:    B( n)    .< ( n)    .x. ( n)    V( n)    .0. ( n)

Proof of Theorem isinftm
Dummy variables  x  w  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2993 . . . . . 6  |-  ( W  e.  V  ->  W  e.  _V )
213ad2ant1 1009 . . . . 5  |-  ( ( W  e.  V  /\  X  e.  B  /\  Y  e.  B )  ->  W  e.  _V )
3 fveq2 5703 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
4 inftm.b . . . . . . . . . . 11  |-  B  =  ( Base `  W
)
53, 4syl6eqr 2493 . . . . . . . . . 10  |-  ( w  =  W  ->  ( Base `  w )  =  B )
65eleq2d 2510 . . . . . . . . 9  |-  ( w  =  W  ->  (
x  e.  ( Base `  w )  <->  x  e.  B ) )
75eleq2d 2510 . . . . . . . . 9  |-  ( w  =  W  ->  (
y  e.  ( Base `  w )  <->  y  e.  B ) )
86, 7anbi12d 710 . . . . . . . 8  |-  ( w  =  W  ->  (
( x  e.  (
Base `  w )  /\  y  e.  ( Base `  w ) )  <-> 
( x  e.  B  /\  y  e.  B
) ) )
9 fveq2 5703 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( 0g `  w )  =  ( 0g `  W
) )
10 inftm.0 . . . . . . . . . . 11  |-  .0.  =  ( 0g `  W )
119, 10syl6eqr 2493 . . . . . . . . . 10  |-  ( w  =  W  ->  ( 0g `  w )  =  .0.  )
12 fveq2 5703 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( lt `  w )  =  ( lt `  W
) )
13 inftm.l . . . . . . . . . . 11  |-  .<  =  ( lt `  W )
1412, 13syl6eqr 2493 . . . . . . . . . 10  |-  ( w  =  W  ->  ( lt `  w )  = 
.<  )
15 eqidd 2444 . . . . . . . . . 10  |-  ( w  =  W  ->  x  =  x )
1611, 14, 15breq123d 4318 . . . . . . . . 9  |-  ( w  =  W  ->  (
( 0g `  w
) ( lt `  w ) x  <->  .0.  .<  x
) )
17 fveq2 5703 . . . . . . . . . . . . 13  |-  ( w  =  W  ->  (.g `  w )  =  (.g `  W ) )
18 inftm.x . . . . . . . . . . . . 13  |-  .x.  =  (.g
`  W )
1917, 18syl6eqr 2493 . . . . . . . . . . . 12  |-  ( w  =  W  ->  (.g `  w )  =  .x.  )
2019oveqd 6120 . . . . . . . . . . 11  |-  ( w  =  W  ->  (
n (.g `  w ) x )  =  ( n 
.x.  x ) )
21 eqidd 2444 . . . . . . . . . . 11  |-  ( w  =  W  ->  y  =  y )
2220, 14, 21breq123d 4318 . . . . . . . . . 10  |-  ( w  =  W  ->  (
( n (.g `  w
) x ) ( lt `  w ) y  <->  ( n  .x.  x )  .<  y
) )
2322ralbidv 2747 . . . . . . . . 9  |-  ( w  =  W  ->  ( A. n  e.  NN  ( n (.g `  w
) x ) ( lt `  w ) y  <->  A. n  e.  NN  ( n  .x.  x ) 
.<  y ) )
2416, 23anbi12d 710 . . . . . . . 8  |-  ( w  =  W  ->  (
( ( 0g `  w ) ( lt
`  w ) x  /\  A. n  e.  NN  ( n (.g `  w ) x ) ( lt `  w
) y )  <->  (  .0.  .<  x  /\  A. n  e.  NN  ( n  .x.  x )  .<  y
) ) )
258, 24anbi12d 710 . . . . . . 7  |-  ( w  =  W  ->  (
( ( x  e.  ( Base `  w
)  /\  y  e.  ( Base `  w )
)  /\  ( ( 0g `  w ) ( lt `  w ) x  /\  A. n  e.  NN  ( n (.g `  w ) x ) ( lt `  w
) y ) )  <-> 
( ( x  e.  B  /\  y  e.  B )  /\  (  .0.  .<  x  /\  A. n  e.  NN  (
n  .x.  x )  .<  y ) ) ) )
2625opabbidv 4367 . . . . . 6  |-  ( w  =  W  ->  { <. x ,  y >.  |  ( ( x  e.  (
Base `  w )  /\  y  e.  ( Base `  w ) )  /\  ( ( 0g
`  w ) ( lt `  w ) x  /\  A. n  e.  NN  ( n (.g `  w ) x ) ( lt `  w
) y ) ) }  =  { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B
)  /\  (  .0.  .<  x  /\  A. n  e.  NN  ( n  .x.  x )  .<  y
) ) } )
27 df-inftm 26207 . . . . . 6  |- <<<  =  (
w  e.  _V  |->  {
<. x ,  y >.  |  ( ( x  e.  ( Base `  w
)  /\  y  e.  ( Base `  w )
)  /\  ( ( 0g `  w ) ( lt `  w ) x  /\  A. n  e.  NN  ( n (.g `  w ) x ) ( lt `  w
) y ) ) } )
28 fvex 5713 . . . . . . . . 9  |-  ( Base `  W )  e.  _V
294, 28eqeltri 2513 . . . . . . . 8  |-  B  e. 
_V
3029, 29xpex 6520 . . . . . . 7  |-  ( B  X.  B )  e. 
_V
31 opabssxp 4923 . . . . . . 7  |-  { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B
)  /\  (  .0.  .<  x  /\  A. n  e.  NN  ( n  .x.  x )  .<  y
) ) }  C_  ( B  X.  B
)
3230, 31ssexi 4449 . . . . . 6  |-  { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B
)  /\  (  .0.  .<  x  /\  A. n  e.  NN  ( n  .x.  x )  .<  y
) ) }  e.  _V
3326, 27, 32fvmpt 5786 . . . . 5  |-  ( W  e.  _V  ->  (<<< `  W )  =  { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B )  /\  (  .0.  .<  x  /\  A. n  e.  NN  (
n  .x.  x )  .<  y ) ) } )
342, 33syl 16 . . . 4  |-  ( ( W  e.  V  /\  X  e.  B  /\  Y  e.  B )  ->  (<<< `  W )  =  { <. x ,  y
>.  |  ( (
x  e.  B  /\  y  e.  B )  /\  (  .0.  .<  x  /\  A. n  e.  NN  ( n  .x.  x ) 
.<  y ) ) } )
3534breqd 4315 . . 3  |-  ( ( W  e.  V  /\  X  e.  B  /\  Y  e.  B )  ->  ( X (<<< `  W
) Y  <->  X { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B )  /\  (  .0.  .<  x  /\  A. n  e.  NN  (
n  .x.  x )  .<  y ) ) } Y ) )
36 eleq1 2503 . . . . . . 7  |-  ( x  =  X  ->  (
x  e.  B  <->  X  e.  B ) )
37 eleq1 2503 . . . . . . 7  |-  ( y  =  Y  ->  (
y  e.  B  <->  Y  e.  B ) )
3836, 37bi2anan9 868 . . . . . 6  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( ( x  e.  B  /\  y  e.  B )  <->  ( X  e.  B  /\  Y  e.  B ) ) )
39 simpl 457 . . . . . . . 8  |-  ( ( x  =  X  /\  y  =  Y )  ->  x  =  X )
40 breq2 4308 . . . . . . . 8  |-  ( x  =  X  ->  (  .0.  .<  x  <->  .0.  .<  X ) )
4139, 40syl 16 . . . . . . 7  |-  ( ( x  =  X  /\  y  =  Y )  ->  (  .0.  .<  x  <->  .0. 
.<  X ) )
4239oveq2d 6119 . . . . . . . . 9  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( n  .x.  x
)  =  ( n 
.x.  X ) )
43 simpr 461 . . . . . . . . 9  |-  ( ( x  =  X  /\  y  =  Y )  ->  y  =  Y )
4442, 43breq12d 4317 . . . . . . . 8  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( ( n  .x.  x )  .<  y  <->  ( n  .x.  X ) 
.<  Y ) )
4544ralbidv 2747 . . . . . . 7  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( A. n  e.  NN  ( n  .x.  x )  .<  y  <->  A. n  e.  NN  (
n  .x.  X )  .<  Y ) )
4641, 45anbi12d 710 . . . . . 6  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( (  .0.  .<  x  /\  A. n  e.  NN  ( n  .x.  x )  .<  y
)  <->  (  .0.  .<  X  /\  A. n  e.  NN  ( n  .x.  X )  .<  Y ) ) )
4738, 46anbi12d 710 . . . . 5  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( ( ( x  e.  B  /\  y  e.  B )  /\  (  .0.  .<  x  /\  A. n  e.  NN  (
n  .x.  x )  .<  y ) )  <->  ( ( X  e.  B  /\  Y  e.  B )  /\  (  .0.  .<  X  /\  A. n  e.  NN  (
n  .x.  X )  .<  Y ) ) ) )
48 eqid 2443 . . . . 5  |-  { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B
)  /\  (  .0.  .<  x  /\  A. n  e.  NN  ( n  .x.  x )  .<  y
) ) }  =  { <. x ,  y
>.  |  ( (
x  e.  B  /\  y  e.  B )  /\  (  .0.  .<  x  /\  A. n  e.  NN  ( n  .x.  x ) 
.<  y ) ) }
4947, 48brabga 4615 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B
)  /\  (  .0.  .<  x  /\  A. n  e.  NN  ( n  .x.  x )  .<  y
) ) } Y  <->  ( ( X  e.  B  /\  Y  e.  B
)  /\  (  .0.  .<  X  /\  A. n  e.  NN  ( n  .x.  X )  .<  Y ) ) ) )
50493adant1 1006 . . 3  |-  ( ( W  e.  V  /\  X  e.  B  /\  Y  e.  B )  ->  ( X { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B
)  /\  (  .0.  .<  x  /\  A. n  e.  NN  ( n  .x.  x )  .<  y
) ) } Y  <->  ( ( X  e.  B  /\  Y  e.  B
)  /\  (  .0.  .<  X  /\  A. n  e.  NN  ( n  .x.  X )  .<  Y ) ) ) )
5135, 50bitrd 253 . 2  |-  ( ( W  e.  V  /\  X  e.  B  /\  Y  e.  B )  ->  ( X (<<< `  W
) Y  <->  ( ( X  e.  B  /\  Y  e.  B )  /\  (  .0.  .<  X  /\  A. n  e.  NN  (
n  .x.  X )  .<  Y ) ) ) )
52 3simpc 987 . . 3  |-  ( ( W  e.  V  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  e.  B  /\  Y  e.  B
) )
5352biantrurd 508 . 2  |-  ( ( W  e.  V  /\  X  e.  B  /\  Y  e.  B )  ->  ( (  .0.  .<  X  /\  A. n  e.  NN  ( n  .x.  X )  .<  Y )  <-> 
( ( X  e.  B  /\  Y  e.  B )  /\  (  .0.  .<  X  /\  A. n  e.  NN  (
n  .x.  X )  .<  Y ) ) ) )
5451, 53bitr4d 256 1  |-  ( ( W  e.  V  /\  X  e.  B  /\  Y  e.  B )  ->  ( X (<<< `  W
) Y  <->  (  .0.  .<  X  /\  A. n  e.  NN  ( n  .x.  X )  .<  Y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2727   _Vcvv 2984   class class class wbr 4304   {copab 4361    X. cxp 4850   ` cfv 5430  (class class class)co 6103   NNcn 10334   Basecbs 14186   0gc0g 14390   ltcplt 15123  .gcmg 15426  <<<cinftm 26205
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-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-iota 5393  df-fun 5432  df-fv 5438  df-ov 6106  df-inftm 26207
This theorem is referenced by:  pnfinf  26212  isarchi2  26214
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