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Theorem nninfwlporlemd 7170
Description: Given two countably infinite sequences of zeroes and ones, they are equal if and only if a sequence formed by pointwise comparing them is all ones. (Contributed by Jim Kingdon, 6-Dec-2024.)
Hypotheses
Ref Expression
nninfwlporlem.x  |-  ( ph  ->  X : om --> 2o )
nninfwlporlem.y  |-  ( ph  ->  Y : om --> 2o )
nninfwlporlem.d  |-  D  =  ( i  e.  om  |->  if ( ( X `  i )  =  ( Y `  i ) ,  1o ,  (/) ) )
Assertion
Ref Expression
nninfwlporlemd  |-  ( ph  ->  ( X  =  Y  <-> 
D  =  ( i  e.  om  |->  1o ) ) )
Distinct variable groups:    D, i    i, X    i, Y    ph, i

Proof of Theorem nninfwlporlemd
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 1n0 6433 . . . . . . . . 9  |-  1o  =/=  (/)
21neii 2349 . . . . . . . 8  |-  -.  1o  =  (/)
32intnan 929 . . . . . . 7  |-  -.  ( -.  ( X `  i
)  =  ( Y `
 i )  /\  1o  =  (/) )
43biorfi 746 . . . . . 6  |-  ( ( X `  i )  =  ( Y `  i )  <->  ( ( X `  i )  =  ( Y `  i )  \/  ( -.  ( X `  i
)  =  ( Y `
 i )  /\  1o  =  (/) ) ) )
5 eqid 2177 . . . . . . . 8  |-  1o  =  1o
65biantru 302 . . . . . . 7  |-  ( ( X `  i )  =  ( Y `  i )  <->  ( ( X `  i )  =  ( Y `  i )  /\  1o  =  1o ) )
76orbi1i 763 . . . . . 6  |-  ( ( ( X `  i
)  =  ( Y `
 i )  \/  ( -.  ( X `
 i )  =  ( Y `  i
)  /\  1o  =  (/) ) )  <->  ( (
( X `  i
)  =  ( Y `
 i )  /\  1o  =  1o )  \/  ( -.  ( X `
 i )  =  ( Y `  i
)  /\  1o  =  (/) ) ) )
84, 7bitri 184 . . . . 5  |-  ( ( X `  i )  =  ( Y `  i )  <->  ( (
( X `  i
)  =  ( Y `
 i )  /\  1o  =  1o )  \/  ( -.  ( X `
 i )  =  ( Y `  i
)  /\  1o  =  (/) ) ) )
9 eqcom 2179 . . . . . 6  |-  ( 1o  =  ( D `  i )  <->  ( D `  i )  =  1o )
10 nninfwlporlem.d . . . . . . . . . 10  |-  D  =  ( i  e.  om  |->  if ( ( X `  i )  =  ( Y `  i ) ,  1o ,  (/) ) )
11 fveq2 5516 . . . . . . . . . . . . 13  |-  ( i  =  j  ->  ( X `  i )  =  ( X `  j ) )
12 fveq2 5516 . . . . . . . . . . . . 13  |-  ( i  =  j  ->  ( Y `  i )  =  ( Y `  j ) )
1311, 12eqeq12d 2192 . . . . . . . . . . . 12  |-  ( i  =  j  ->  (
( X `  i
)  =  ( Y `
 i )  <->  ( X `  j )  =  ( Y `  j ) ) )
1413ifbid 3556 . . . . . . . . . . 11  |-  ( i  =  j  ->  if ( ( X `  i )  =  ( Y `  i ) ,  1o ,  (/) )  =  if (
( X `  j
)  =  ( Y `
 j ) ,  1o ,  (/) ) )
1514cbvmptv 4100 . . . . . . . . . 10  |-  ( i  e.  om  |->  if ( ( X `  i
)  =  ( Y `
 i ) ,  1o ,  (/) ) )  =  ( j  e. 
om  |->  if ( ( X `  j )  =  ( Y `  j ) ,  1o ,  (/) ) )
1610, 15eqtri 2198 . . . . . . . . 9  |-  D  =  ( j  e.  om  |->  if ( ( X `  j )  =  ( Y `  j ) ,  1o ,  (/) ) )
17 fveq2 5516 . . . . . . . . . . 11  |-  ( j  =  i  ->  ( X `  j )  =  ( X `  i ) )
18 fveq2 5516 . . . . . . . . . . 11  |-  ( j  =  i  ->  ( Y `  j )  =  ( Y `  i ) )
1917, 18eqeq12d 2192 . . . . . . . . . 10  |-  ( j  =  i  ->  (
( X `  j
)  =  ( Y `
 j )  <->  ( X `  i )  =  ( Y `  i ) ) )
2019ifbid 3556 . . . . . . . . 9  |-  ( j  =  i  ->  if ( ( X `  j )  =  ( Y `  j ) ,  1o ,  (/) )  =  if (
( X `  i
)  =  ( Y `
 i ) ,  1o ,  (/) ) )
21 simpr 110 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  om )  ->  i  e.  om )
22 1lt2o 6443 . . . . . . . . . . 11  |-  1o  e.  2o
2322a1i 9 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  om )  ->  1o  e.  2o )
24 0lt2o 6442 . . . . . . . . . . 11  |-  (/)  e.  2o
2524a1i 9 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  om )  ->  (/)  e.  2o )
26 2ssom 6525 . . . . . . . . . . . 12  |-  2o  C_  om
27 nninfwlporlem.x . . . . . . . . . . . . 13  |-  ( ph  ->  X : om --> 2o )
2827ffvelcdmda 5652 . . . . . . . . . . . 12  |-  ( (
ph  /\  i  e.  om )  ->  ( X `  i )  e.  2o )
2926, 28sselid 3154 . . . . . . . . . . 11  |-  ( (
ph  /\  i  e.  om )  ->  ( X `  i )  e.  om )
30 nninfwlporlem.y . . . . . . . . . . . . 13  |-  ( ph  ->  Y : om --> 2o )
3130ffvelcdmda 5652 . . . . . . . . . . . 12  |-  ( (
ph  /\  i  e.  om )  ->  ( Y `  i )  e.  2o )
3226, 31sselid 3154 . . . . . . . . . . 11  |-  ( (
ph  /\  i  e.  om )  ->  ( Y `  i )  e.  om )
33 nndceq 6500 . . . . . . . . . . 11  |-  ( ( ( X `  i
)  e.  om  /\  ( Y `  i )  e.  om )  -> DECID  ( X `  i )  =  ( Y `  i ) )
3429, 32, 33syl2anc 411 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  om )  -> DECID  ( X `  i
)  =  ( Y `
 i ) )
3523, 25, 34ifcldcd 3571 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  om )  ->  if (
( X `  i
)  =  ( Y `
 i ) ,  1o ,  (/) )  e.  2o )
3616, 20, 21, 35fvmptd3 5610 . . . . . . . 8  |-  ( (
ph  /\  i  e.  om )  ->  ( D `  i )  =  if ( ( X `  i )  =  ( Y `  i ) ,  1o ,  (/) ) )
3736eqeq2d 2189 . . . . . . 7  |-  ( (
ph  /\  i  e.  om )  ->  ( 1o  =  ( D `  i )  <->  1o  =  if ( ( X `  i )  =  ( Y `  i ) ,  1o ,  (/) ) ) )
38 eqifdc 3570 . . . . . . . 8  |-  (DECID  ( X `
 i )  =  ( Y `  i
)  ->  ( 1o  =  if ( ( X `
 i )  =  ( Y `  i
) ,  1o ,  (/) )  <->  ( ( ( X `  i )  =  ( Y `  i )  /\  1o  =  1o )  \/  ( -.  ( X `  i
)  =  ( Y `
 i )  /\  1o  =  (/) ) ) ) )
3934, 38syl 14 . . . . . . 7  |-  ( (
ph  /\  i  e.  om )  ->  ( 1o  =  if ( ( X `
 i )  =  ( Y `  i
) ,  1o ,  (/) )  <->  ( ( ( X `  i )  =  ( Y `  i )  /\  1o  =  1o )  \/  ( -.  ( X `  i
)  =  ( Y `
 i )  /\  1o  =  (/) ) ) ) )
4037, 39bitrd 188 . . . . . 6  |-  ( (
ph  /\  i  e.  om )  ->  ( 1o  =  ( D `  i )  <->  ( (
( X `  i
)  =  ( Y `
 i )  /\  1o  =  1o )  \/  ( -.  ( X `
 i )  =  ( Y `  i
)  /\  1o  =  (/) ) ) ) )
419, 40bitr3id 194 . . . . 5  |-  ( (
ph  /\  i  e.  om )  ->  ( ( D `  i )  =  1o  <->  ( ( ( X `  i )  =  ( Y `  i )  /\  1o  =  1o )  \/  ( -.  ( X `  i
)  =  ( Y `
 i )  /\  1o  =  (/) ) ) ) )
428, 41bitr4id 199 . . . 4  |-  ( (
ph  /\  i  e.  om )  ->  ( ( X `  i )  =  ( Y `  i )  <->  ( D `  i )  =  1o ) )
4342ralbidva 2473 . . 3  |-  ( ph  ->  ( A. i  e. 
om  ( X `  i )  =  ( Y `  i )  <->  A. i  e.  om  ( D `  i )  =  1o ) )
44 fveqeq2 5525 . . . 4  |-  ( i  =  j  ->  (
( D `  i
)  =  1o  <->  ( D `  j )  =  1o ) )
4544cbvralv 2704 . . 3  |-  ( A. i  e.  om  ( D `  i )  =  1o  <->  A. j  e.  om  ( D `  j )  =  1o )
4643, 45bitrdi 196 . 2  |-  ( ph  ->  ( A. i  e. 
om  ( X `  i )  =  ( Y `  i )  <->  A. j  e.  om  ( D `  j )  =  1o ) )
4727ffnd 5367 . . 3  |-  ( ph  ->  X  Fn  om )
4830ffnd 5367 . . 3  |-  ( ph  ->  Y  Fn  om )
49 eqfnfv 5614 . . 3  |-  ( ( X  Fn  om  /\  Y  Fn  om )  ->  ( X  =  Y  <->  A. i  e.  om  ( X `  i )  =  ( Y `  i ) ) )
5047, 48, 49syl2anc 411 . 2  |-  ( ph  ->  ( X  =  Y  <->  A. i  e.  om  ( X `  i )  =  ( Y `  i ) ) )
5135ralrimiva 2550 . . . 4  |-  ( ph  ->  A. i  e.  om  if ( ( X `  i )  =  ( Y `  i ) ,  1o ,  (/) )  e.  2o )
5210fnmpt 5343 . . . 4  |-  ( A. i  e.  om  if ( ( X `  i
)  =  ( Y `
 i ) ,  1o ,  (/) )  e.  2o  ->  D  Fn  om )
5351, 52syl 14 . . 3  |-  ( ph  ->  D  Fn  om )
54 eqidd 2178 . . 3  |-  ( j  =  i  ->  1o  =  1o )
55 1onn 6521 . . . 4  |-  1o  e.  om
5655a1i 9 . . 3  |-  ( (
ph  /\  j  e.  om )  ->  1o  e.  om )
5755a1i 9 . . 3  |-  ( (
ph  /\  i  e.  om )  ->  1o  e.  om )
5853, 54, 56, 57fnmptfvd 5621 . 2  |-  ( ph  ->  ( D  =  ( i  e.  om  |->  1o )  <->  A. j  e.  om  ( D `  j )  =  1o ) )
5946, 50, 583bitr4d 220 1  |-  ( ph  ->  ( X  =  Y  <-> 
D  =  ( i  e.  om  |->  1o ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 708  DECID wdc 834    = wceq 1353    e. wcel 2148   A.wral 2455   (/)c0 3423   ifcif 3535    |-> cmpt 4065   omcom 4590    Fn wfn 5212   -->wf 5213   ` cfv 5217   1oc1o 6410   2oc2o 6411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-fv 5225  df-1o 6417  df-2o 6418
This theorem is referenced by:  nninfwlporlem  7171
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