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Theorem konigth 8159
Description: Konig's Theorem. If  m ( i )  ~<  n
( i ) for all  i  e.  A, then  sum_ i  e.  A m ( i )  ~<  prod_ i  e.  A n ( i ), where the sums and products stand in for disjoint union and infinite cartesian product. The version here is proven with regular unions rather than disjoint unions for convenience, but the version with disjoint unions is clearly a special case of this version. The Axiom of Choice is needed for this proof, but it contains AC as a simple corollary (letting  m ( i )  =  (/), this theorem says that an infinite cartesian product of nonempty sets is nonempty), so this is an AC equivalent. Theorem 11.26 of [TakeutiZaring] p. 107. (Contributed by Mario Carneiro, 22-Feb-2013.)
Hypotheses
Ref Expression
konigth.1  |-  A  e. 
_V
konigth.2  |-  S  = 
U_ i  e.  A  ( M `  i )
konigth.3  |-  P  = 
X_ i  e.  A  ( N `  i )
Assertion
Ref Expression
konigth  |-  ( A. i  e.  A  ( M `  i )  ~<  ( N `  i
)  ->  S  ~<  P )
Distinct variable group:    A, i
Allowed substitution hints:    P( i)    S( i)    M( i)    N( i)

Proof of Theorem konigth
StepHypRef Expression
1 konigth.1 . 2  |-  A  e. 
_V
2 konigth.2 . 2  |-  S  = 
U_ i  e.  A  ( M `  i )
3 konigth.3 . 2  |-  P  = 
X_ i  e.  A  ( N `  i )
4 fveq2 5458 . . . . 5  |-  ( b  =  a  ->  (
f `  b )  =  ( f `  a ) )
54fveq1d 5460 . . . 4  |-  ( b  =  a  ->  (
( f `  b
) `  i )  =  ( ( f `
 a ) `  i ) )
65cbvmptv 4085 . . 3  |-  ( b  e.  ( M `  i )  |->  ( ( f `  b ) `
 i ) )  =  ( a  e.  ( M `  i
)  |->  ( ( f `
 a ) `  i ) )
76mpteq2i 4077 . 2  |-  ( i  e.  A  |->  ( b  e.  ( M `  i )  |->  ( ( f `  b ) `
 i ) ) )  =  ( i  e.  A  |->  ( a  e.  ( M `  i )  |->  ( ( f `  a ) `
 i ) ) )
8 fveq2 5458 . . 3  |-  ( j  =  i  ->  (
e `  j )  =  ( e `  i ) )
98cbvmptv 4085 . 2  |-  ( j  e.  A  |->  ( e `
 j ) )  =  ( i  e.  A  |->  ( e `  i ) )
101, 2, 3, 7, 9konigthlem 8158 1  |-  ( A. i  e.  A  ( M `  i )  ~<  ( N `  i
)  ->  S  ~<  P )
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1619    e. wcel 1621   A.wral 2518   _Vcvv 2763   U_ciun 3879   class class class wbr 3997    e. cmpt 4051   ` cfv 4673   X_cixp 6785    ~< csdm 6830
This theorem is referenced by:  pwcfsdom  8173
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-ac2 8057
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-se 4325  df-we 4326  df-ord 4367  df-on 4368  df-suc 4370  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-isom 4690  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-er 6628  df-map 6742  df-ixp 6786  df-en 6832  df-dom 6833  df-sdom 6834  df-card 7540  df-acn 7543  df-ac 7711
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