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Theorem konigth 8428
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 5714 . . . . 5  |-  ( b  =  a  ->  (
f `  b )  =  ( f `  a ) )
54fveq1d 5716 . . . 4  |-  ( b  =  a  ->  (
( f `  b
) `  i )  =  ( ( f `
 a ) `  i ) )
65cbvmptv 4287 . . 3  |-  ( b  e.  ( M `  i )  |->  ( ( f `  b ) `
 i ) )  =  ( a  e.  ( M `  i
)  |->  ( ( f `
 a ) `  i ) )
76mpteq2i 4279 . 2  |-  ( i  e.  A  |->  ( b  e.  ( M `  i )  |->  ( ( f `  b ) `
 i ) ) )  =  ( i  e.  A  |->  ( a  e.  ( M `  i )  |->  ( ( f `  a ) `
 i ) ) )
8 fveq2 5714 . . 3  |-  ( j  =  i  ->  (
e `  j )  =  ( e `  i ) )
98cbvmptv 4287 . 2  |-  ( j  e.  A  |->  ( e `
 j ) )  =  ( i  e.  A  |->  ( e `  i ) )
101, 2, 3, 7, 9konigthlem 8427 1  |-  ( A. i  e.  A  ( M `  i )  ~<  ( N `  i
)  ->  S  ~<  P )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   A.wral 2692   _Vcvv 2943   U_ciun 4080   class class class wbr 4199    e. cmpt 4253   ` cfv 5440   X_cixp 7049    ~< csdm 7094
This theorem is referenced by:  pwcfsdom  8442
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-rep 4307  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687  ax-ac2 8327
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-ral 2697  df-rex 2698  df-reu 2699  df-rmo 2700  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-pss 3323  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-tp 3809  df-op 3810  df-uni 4003  df-int 4038  df-iun 4082  df-br 4200  df-opab 4254  df-mpt 4255  df-tr 4290  df-eprel 4481  df-id 4485  df-po 4490  df-so 4491  df-fr 4528  df-se 4529  df-we 4530  df-ord 4571  df-on 4572  df-suc 4574  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-isom 5449  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-1st 6335  df-2nd 6336  df-riota 6535  df-recs 6619  df-er 6891  df-map 7006  df-ixp 7050  df-en 7096  df-dom 7097  df-sdom 7098  df-card 7810  df-acn 7813  df-ac 7981
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