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Theorem nninfwlporlemd 7163
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 6426 . . . . . . . . 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 5510 . . . . . . . . . . . . 13  |-  ( i  =  j  ->  ( X `  i )  =  ( X `  j ) )
12 fveq2 5510 . . . . . . . . . . . . 13  |-  ( i  =  j  ->  ( Y `  i )  =  ( Y `  j ) )
1311, 12eqeq12d 2192 . . . . . . . . . . . 12  |-  ( i  =  j  ->  (
( X `  i
)  =  ( Y `
 i )  <->  ( X `  j )  =  ( Y `  j ) ) )
1413ifbid 3555 . . . . . . . . . . 11  |-  ( i  =  j  ->  if ( ( X `  i )  =  ( Y `  i ) ,  1o ,  (/) )  =  if (
( X `  j
)  =  ( Y `
 j ) ,  1o ,  (/) ) )
1514cbvmptv 4096 . . . . . . . . . 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 5510 . . . . . . . . . . 11  |-  ( j  =  i  ->  ( X `  j )  =  ( X `  i ) )
18 fveq2 5510 . . . . . . . . . . 11  |-  ( j  =  i  ->  ( Y `  j )  =  ( Y `  i ) )
1917, 18eqeq12d 2192 . . . . . . . . . 10  |-  ( j  =  i  ->  (
( X `  j
)  =  ( Y `
 j )  <->  ( X `  i )  =  ( Y `  i ) ) )
2019ifbid 3555 . . . . . . . . 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 6436 . . . . . . . . . . 11  |-  1o  e.  2o
2322a1i 9 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  om )  ->  1o  e.  2o )
24 0lt2o 6435 . . . . . . . . . . 11  |-  (/)  e.  2o
2524a1i 9 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  om )  ->  (/)  e.  2o )
26 2ssom 6518 . . . . . . . . . . . 12  |-  2o  C_  om
27 nninfwlporlem.x . . . . . . . . . . . . 13  |-  ( ph  ->  X : om --> 2o )
2827ffvelcdmda 5646 . . . . . . . . . . . 12  |-  ( (
ph  /\  i  e.  om )  ->  ( X `  i )  e.  2o )
2926, 28sselid 3153 . . . . . . . . . . 11  |-  ( (
ph  /\  i  e.  om )  ->  ( X `  i )  e.  om )
30 nninfwlporlem.y . . . . . . . . . . . . 13  |-  ( ph  ->  Y : om --> 2o )
3130ffvelcdmda 5646 . . . . . . . . . . . 12  |-  ( (
ph  /\  i  e.  om )  ->  ( Y `  i )  e.  2o )
3226, 31sselid 3153 . . . . . . . . . . 11  |-  ( (
ph  /\  i  e.  om )  ->  ( Y `  i )  e.  om )
33 nndceq 6493 . . . . . . . . . . 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 3569 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  om )  ->  if (
( X `  i
)  =  ( Y `
 i ) ,  1o ,  (/) )  e.  2o )
3616, 20, 21, 35fvmptd3 5604 . . . . . . . 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 3568 . . . . . . . 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 5519 . . . 4  |-  ( i  =  j  ->  (
( D `  i
)  =  1o  <->  ( D `  j )  =  1o ) )
4544cbvralv 2703 . . 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 5361 . . 3  |-  ( ph  ->  X  Fn  om )
4830ffnd 5361 . . 3  |-  ( ph  ->  Y  Fn  om )
49 eqfnfv 5608 . . 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 5337 . . . 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 6514 . . . 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 5615 . 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 3422   ifcif 3534    |-> cmpt 4061   omcom 4585    Fn wfn 5206   -->wf 5207   ` cfv 5211   1oc1o 6403   2oc2o 6404
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 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-iinf 4583
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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4289  df-iord 4362  df-on 4364  df-suc 4367  df-iom 4586  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-fv 5219  df-1o 6410  df-2o 6411
This theorem is referenced by:  nninfwlporlem  7164
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