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Theorem konigth 8186
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
Dummy variables  a 
e  f  j  b are mutually distinct and distinct from all other variables.
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 5485 . . . . 5  |-  ( b  =  a  ->  (
f `  b )  =  ( f `  a ) )
54fveq1d 5487 . . . 4  |-  ( b  =  a  ->  (
( f `  b
) `  i )  =  ( ( f `
 a ) `  i ) )
65cbvmptv 4112 . . 3  |-  ( b  e.  ( M `  i )  |->  ( ( f `  b ) `
 i ) )  =  ( a  e.  ( M `  i
)  |->  ( ( f `
 a ) `  i ) )
76mpteq2i 4104 . 2  |-  ( i  e.  A  |->  ( b  e.  ( M `  i )  |->  ( ( f `  b ) `
 i ) ) )  =  ( i  e.  A  |->  ( a  e.  ( M `  i )  |->  ( ( f `  a ) `
 i ) ) )
8 fveq2 5485 . . 3  |-  ( j  =  i  ->  (
e `  j )  =  ( e `  i ) )
98cbvmptv 4112 . 2  |-  ( j  e.  A  |->  ( e `
 j ) )  =  ( i  e.  A  |->  ( e `  i ) )
101, 2, 3, 7, 9konigthlem 8185 1  |-  ( A. i  e.  A  ( M `  i )  ~<  ( N `  i
)  ->  S  ~<  P )
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1624    e. wcel 1685   A.wral 2544   _Vcvv 2789   U_ciun 3906   class class class wbr 4024    e. cmpt 4078   ` cfv 5221   X_cixp 6812    ~< csdm 6857
This theorem is referenced by:  pwcfsdom  8200
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-ac2 8084
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-suc 4397  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-er 6655  df-map 6769  df-ixp 6813  df-en 6859  df-dom 6860  df-sdom 6861  df-card 7567  df-acn 7570  df-ac 7738
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