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Theorem konigth 8207
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
Dummy variables  a 
e  f  j  b are mutually distinct and distinct from all other variables.
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 5541 . . . . 5  |-  ( b  =  a  ->  (
f `  b )  =  ( f `  a ) )
54fveq1d 5543 . . . 4  |-  ( b  =  a  ->  (
( f `  b
) `  i )  =  ( ( f `
 a ) `  i ) )
65cbvmptv 4127 . . 3  |-  ( b  e.  ( M `  i )  |->  ( ( f `  b ) `
 i ) )  =  ( a  e.  ( M `  i
)  |->  ( ( f `
 a ) `  i ) )
76mpteq2i 4119 . 2  |-  ( i  e.  A  |->  ( b  e.  ( M `  i )  |->  ( ( f `  b ) `
 i ) ) )  =  ( i  e.  A  |->  ( a  e.  ( M `  i )  |->  ( ( f `  a ) `
 i ) ) )
8 fveq2 5541 . . 3  |-  ( j  =  i  ->  (
e `  j )  =  ( e `  i ) )
98cbvmptv 4127 . 2  |-  ( j  e.  A  |->  ( e `
 j ) )  =  ( i  e.  A  |->  ( e `  i ) )
101, 2, 3, 7, 9konigthlem 8206 1  |-  ( A. i  e.  A  ( M `  i )  ~<  ( N `  i
)  ->  S  ~<  P )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801   U_ciun 3921   class class class wbr 4039    e. cmpt 4093   ` cfv 5271   X_cixp 6833    ~< csdm 6878
This theorem is referenced by:  pwcfsdom  8221
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-ac2 8105
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-er 6676  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-card 7588  df-acn 7591  df-ac 7759
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