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Theorem isinftm 27557
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 eleq1 2539 . . . . . 6  |-  ( x  =  X  ->  (
x  e.  B  <->  X  e.  B ) )
2 eleq1 2539 . . . . . 6  |-  ( y  =  Y  ->  (
y  e.  B  <->  Y  e.  B ) )
31, 2bi2anan9 871 . . . . 5  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( ( x  e.  B  /\  y  e.  B )  <->  ( X  e.  B  /\  Y  e.  B ) ) )
4 simpl 457 . . . . . . 7  |-  ( ( x  =  X  /\  y  =  Y )  ->  x  =  X )
54breq2d 4465 . . . . . 6  |-  ( ( x  =  X  /\  y  =  Y )  ->  (  .0.  .<  x  <->  .0. 
.<  X ) )
64oveq2d 6311 . . . . . . . 8  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( n  .x.  x
)  =  ( n 
.x.  X ) )
7 simpr 461 . . . . . . . 8  |-  ( ( x  =  X  /\  y  =  Y )  ->  y  =  Y )
86, 7breq12d 4466 . . . . . . 7  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( ( n  .x.  x )  .<  y  <->  ( n  .x.  X ) 
.<  Y ) )
98ralbidv 2906 . . . . . 6  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( A. n  e.  NN  ( n  .x.  x )  .<  y  <->  A. n  e.  NN  (
n  .x.  X )  .<  Y ) )
105, 9anbi12d 710 . . . . 5  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( (  .0.  .<  x  /\  A. n  e.  NN  ( n  .x.  x )  .<  y
)  <->  (  .0.  .<  X  /\  A. n  e.  NN  ( n  .x.  X )  .<  Y ) ) )
113, 10anbi12d 710 . . . 4  |-  ( ( 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 ) ) ) )
12 eqid 2467 . . . 4  |-  { <. 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 ) ) }
1311, 12brabga 4767 . . 3  |-  ( ( 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 ) ) ) )
14133adant1 1014 . 2  |-  ( ( 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 ) ) ) )
15 elex 3127 . . . . 5  |-  ( W  e.  V  ->  W  e.  _V )
16153ad2ant1 1017 . . . 4  |-  ( ( W  e.  V  /\  X  e.  B  /\  Y  e.  B )  ->  W  e.  _V )
17 fveq2 5872 . . . . . . . . . 10  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
18 inftm.b . . . . . . . . . 10  |-  B  =  ( Base `  W
)
1917, 18syl6eqr 2526 . . . . . . . . 9  |-  ( w  =  W  ->  ( Base `  w )  =  B )
2019eleq2d 2537 . . . . . . . 8  |-  ( w  =  W  ->  (
x  e.  ( Base `  w )  <->  x  e.  B ) )
2119eleq2d 2537 . . . . . . . 8  |-  ( w  =  W  ->  (
y  e.  ( Base `  w )  <->  y  e.  B ) )
2220, 21anbi12d 710 . . . . . . 7  |-  ( w  =  W  ->  (
( x  e.  (
Base `  w )  /\  y  e.  ( Base `  w ) )  <-> 
( x  e.  B  /\  y  e.  B
) ) )
23 fveq2 5872 . . . . . . . . . 10  |-  ( w  =  W  ->  ( 0g `  w )  =  ( 0g `  W
) )
24 inftm.0 . . . . . . . . . 10  |-  .0.  =  ( 0g `  W )
2523, 24syl6eqr 2526 . . . . . . . . 9  |-  ( w  =  W  ->  ( 0g `  w )  =  .0.  )
26 fveq2 5872 . . . . . . . . . 10  |-  ( w  =  W  ->  ( lt `  w )  =  ( lt `  W
) )
27 inftm.l . . . . . . . . . 10  |-  .<  =  ( lt `  W )
2826, 27syl6eqr 2526 . . . . . . . . 9  |-  ( w  =  W  ->  ( lt `  w )  = 
.<  )
29 eqidd 2468 . . . . . . . . 9  |-  ( w  =  W  ->  x  =  x )
3025, 28, 29breq123d 4467 . . . . . . . 8  |-  ( w  =  W  ->  (
( 0g `  w
) ( lt `  w ) x  <->  .0.  .<  x
) )
31 fveq2 5872 . . . . . . . . . . . 12  |-  ( w  =  W  ->  (.g `  w )  =  (.g `  W ) )
32 inftm.x . . . . . . . . . . . 12  |-  .x.  =  (.g
`  W )
3331, 32syl6eqr 2526 . . . . . . . . . . 11  |-  ( w  =  W  ->  (.g `  w )  =  .x.  )
3433oveqd 6312 . . . . . . . . . 10  |-  ( w  =  W  ->  (
n (.g `  w ) x )  =  ( n 
.x.  x ) )
35 eqidd 2468 . . . . . . . . . 10  |-  ( w  =  W  ->  y  =  y )
3634, 28, 35breq123d 4467 . . . . . . . . 9  |-  ( w  =  W  ->  (
( n (.g `  w
) x ) ( lt `  w ) y  <->  ( n  .x.  x )  .<  y
) )
3736ralbidv 2906 . . . . . . . 8  |-  ( w  =  W  ->  ( A. n  e.  NN  ( n (.g `  w
) x ) ( lt `  w ) y  <->  A. n  e.  NN  ( n  .x.  x ) 
.<  y ) )
3830, 37anbi12d 710 . . . . . . 7  |-  ( 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
) ) )
3922, 38anbi12d 710 . . . . . 6  |-  ( 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 ) ) ) )
4039opabbidv 4516 . . . . 5  |-  ( 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
) ) } )
41 df-inftm 27554 . . . . 5  |- <<<  =  (
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 ) ) } )
42 fvex 5882 . . . . . . . 8  |-  ( Base `  W )  e.  _V
4318, 42eqeltri 2551 . . . . . . 7  |-  B  e. 
_V
4443, 43xpex 6599 . . . . . 6  |-  ( B  X.  B )  e. 
_V
45 opabssxp 5080 . . . . . 6  |-  { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B
)  /\  (  .0.  .<  x  /\  A. n  e.  NN  ( n  .x.  x )  .<  y
) ) }  C_  ( B  X.  B
)
4644, 45ssexi 4598 . . . . 5  |-  { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B
)  /\  (  .0.  .<  x  /\  A. n  e.  NN  ( n  .x.  x )  .<  y
) ) }  e.  _V
4740, 41, 46fvmpt 5957 . . . 4  |-  ( W  e.  _V  ->  (<<< `  W )  =  { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B )  /\  (  .0.  .<  x  /\  A. n  e.  NN  (
n  .x.  x )  .<  y ) ) } )
4816, 47syl 16 . . 3  |-  ( ( 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 ) ) } )
4948breqd 4464 . 2  |-  ( ( 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 ) )
50 3simpc 995 . . 3  |-  ( ( W  e.  V  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  e.  B  /\  Y  e.  B
) )
5150biantrurd 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 ) ) ) )
5214, 49, 513bitr4d 285 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 973    = wceq 1379    e. wcel 1767   A.wral 2817   _Vcvv 3118   class class class wbr 4453   {copab 4510    X. cxp 5003   ` cfv 5594  (class class class)co 6295   NNcn 10548   Basecbs 14506   0gc0g 14711   ltcplt 15444  .gcmg 15927  <<<cinftm 27552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6298  df-inftm 27554
This theorem is referenced by:  pnfinf  27559  isarchi2  27561
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