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Theorem pmltpclem1 18771
Description: Lemma for pmltpc 18773. (Contributed by Mario Carneiro, 1-Jul-2014.)
Hypotheses
Ref Expression
pmltpclem1.1  |-  ( ph  ->  A  e.  S )
pmltpclem1.2  |-  ( ph  ->  B  e.  S )
pmltpclem1.3  |-  ( ph  ->  C  e.  S )
pmltpclem1.4  |-  ( ph  ->  A  <  B )
pmltpclem1.5  |-  ( ph  ->  B  <  C )
pmltpclem1.6  |-  ( ph  ->  ( ( ( F `
 A )  < 
( F `  B
)  /\  ( F `  C )  <  ( F `  B )
)  \/  ( ( F `  B )  <  ( F `  A )  /\  ( F `  B )  <  ( F `  C
) ) ) )
Assertion
Ref Expression
pmltpclem1  |-  ( ph  ->  E. a  e.  S  E. b  e.  S  E. c  e.  S  ( a  <  b  /\  b  <  c  /\  ( ( ( F `
 a )  < 
( F `  b
)  /\  ( F `  c )  <  ( F `  b )
)  \/  ( ( F `  b )  <  ( F `  a )  /\  ( F `  b )  <  ( F `  c
) ) ) ) )
Distinct variable groups:    a, b,
c, A    B, b,
c    C, c    F, a, b, c    S, a, b, c
Allowed substitution hints:    ph( a, b, c)    B( a)    C( a, b)

Proof of Theorem pmltpclem1
StepHypRef Expression
1 pmltpclem1.1 . 2  |-  ( ph  ->  A  e.  S )
2 pmltpclem1.2 . 2  |-  ( ph  ->  B  e.  S )
3 pmltpclem1.3 . 2  |-  ( ph  ->  C  e.  S )
4 pmltpclem1.4 . 2  |-  ( ph  ->  A  <  B )
5 pmltpclem1.5 . 2  |-  ( ph  ->  B  <  C )
6 pmltpclem1.6 . 2  |-  ( ph  ->  ( ( ( F `
 A )  < 
( F `  B
)  /\  ( F `  C )  <  ( F `  B )
)  \/  ( ( F `  B )  <  ( F `  A )  /\  ( F `  B )  <  ( F `  C
) ) ) )
7 breq1 4000 . . . 4  |-  ( a  =  A  ->  (
a  <  b  <->  A  <  b ) )
8 fveq2 5458 . . . . . . 7  |-  ( a  =  A  ->  ( F `  a )  =  ( F `  A ) )
98breq1d 4007 . . . . . 6  |-  ( a  =  A  ->  (
( F `  a
)  <  ( F `  b )  <->  ( F `  A )  <  ( F `  b )
) )
109anbi1d 688 . . . . 5  |-  ( a  =  A  ->  (
( ( F `  a )  <  ( F `  b )  /\  ( F `  c
)  <  ( F `  b ) )  <->  ( ( F `  A )  <  ( F `  b
)  /\  ( F `  c )  <  ( F `  b )
) ) )
118breq2d 4009 . . . . . 6  |-  ( a  =  A  ->  (
( F `  b
)  <  ( F `  a )  <->  ( F `  b )  <  ( F `  A )
) )
1211anbi1d 688 . . . . 5  |-  ( a  =  A  ->  (
( ( F `  b )  <  ( F `  a )  /\  ( F `  b
)  <  ( F `  c ) )  <->  ( ( F `  b )  <  ( F `  A
)  /\  ( F `  b )  <  ( F `  c )
) ) )
1310, 12orbi12d 693 . . . 4  |-  ( a  =  A  ->  (
( ( ( F `
 a )  < 
( F `  b
)  /\  ( F `  c )  <  ( F `  b )
)  \/  ( ( F `  b )  <  ( F `  a )  /\  ( F `  b )  <  ( F `  c
) ) )  <->  ( (
( F `  A
)  <  ( F `  b )  /\  ( F `  c )  <  ( F `  b
) )  \/  (
( F `  b
)  <  ( F `  A )  /\  ( F `  b )  <  ( F `  c
) ) ) ) )
147, 133anbi13d 1259 . . 3  |-  ( a  =  A  ->  (
( a  <  b  /\  b  <  c  /\  ( ( ( F `
 a )  < 
( F `  b
)  /\  ( F `  c )  <  ( F `  b )
)  \/  ( ( F `  b )  <  ( F `  a )  /\  ( F `  b )  <  ( F `  c
) ) ) )  <-> 
( A  <  b  /\  b  <  c  /\  ( ( ( F `
 A )  < 
( F `  b
)  /\  ( F `  c )  <  ( F `  b )
)  \/  ( ( F `  b )  <  ( F `  A )  /\  ( F `  b )  <  ( F `  c
) ) ) ) ) )
15 breq2 4001 . . . 4  |-  ( b  =  B  ->  ( A  <  b  <->  A  <  B ) )
16 breq1 4000 . . . 4  |-  ( b  =  B  ->  (
b  <  c  <->  B  <  c ) )
17 fveq2 5458 . . . . . . 7  |-  ( b  =  B  ->  ( F `  b )  =  ( F `  B ) )
1817breq2d 4009 . . . . . 6  |-  ( b  =  B  ->  (
( F `  A
)  <  ( F `  b )  <->  ( F `  A )  <  ( F `  B )
) )
1917breq2d 4009 . . . . . 6  |-  ( b  =  B  ->  (
( F `  c
)  <  ( F `  b )  <->  ( F `  c )  <  ( F `  B )
) )
2018, 19anbi12d 694 . . . . 5  |-  ( b  =  B  ->  (
( ( F `  A )  <  ( F `  b )  /\  ( F `  c
)  <  ( F `  b ) )  <->  ( ( F `  A )  <  ( F `  B
)  /\  ( F `  c )  <  ( F `  B )
) ) )
2117breq1d 4007 . . . . . 6  |-  ( b  =  B  ->  (
( F `  b
)  <  ( F `  A )  <->  ( F `  B )  <  ( F `  A )
) )
2217breq1d 4007 . . . . . 6  |-  ( b  =  B  ->  (
( F `  b
)  <  ( F `  c )  <->  ( F `  B )  <  ( F `  c )
) )
2321, 22anbi12d 694 . . . . 5  |-  ( b  =  B  ->  (
( ( F `  b )  <  ( F `  A )  /\  ( F `  b
)  <  ( F `  c ) )  <->  ( ( F `  B )  <  ( F `  A
)  /\  ( F `  B )  <  ( F `  c )
) ) )
2420, 23orbi12d 693 . . . 4  |-  ( b  =  B  ->  (
( ( ( F `
 A )  < 
( F `  b
)  /\  ( F `  c )  <  ( F `  b )
)  \/  ( ( F `  b )  <  ( F `  A )  /\  ( F `  b )  <  ( F `  c
) ) )  <->  ( (
( F `  A
)  <  ( F `  B )  /\  ( F `  c )  <  ( F `  B
) )  \/  (
( F `  B
)  <  ( F `  A )  /\  ( F `  B )  <  ( F `  c
) ) ) ) )
2515, 16, 243anbi123d 1257 . . 3  |-  ( b  =  B  ->  (
( A  <  b  /\  b  <  c  /\  ( ( ( F `
 A )  < 
( F `  b
)  /\  ( F `  c )  <  ( F `  b )
)  \/  ( ( F `  b )  <  ( F `  A )  /\  ( F `  b )  <  ( F `  c
) ) ) )  <-> 
( A  <  B  /\  B  <  c  /\  ( ( ( F `
 A )  < 
( F `  B
)  /\  ( F `  c )  <  ( F `  B )
)  \/  ( ( F `  B )  <  ( F `  A )  /\  ( F `  B )  <  ( F `  c
) ) ) ) ) )
26 breq2 4001 . . . 4  |-  ( c  =  C  ->  ( B  <  c  <->  B  <  C ) )
27 fveq2 5458 . . . . . . 7  |-  ( c  =  C  ->  ( F `  c )  =  ( F `  C ) )
2827breq1d 4007 . . . . . 6  |-  ( c  =  C  ->  (
( F `  c
)  <  ( F `  B )  <->  ( F `  C )  <  ( F `  B )
) )
2928anbi2d 687 . . . . 5  |-  ( c  =  C  ->  (
( ( F `  A )  <  ( F `  B )  /\  ( F `  c
)  <  ( F `  B ) )  <->  ( ( F `  A )  <  ( F `  B
)  /\  ( F `  C )  <  ( F `  B )
) ) )
3027breq2d 4009 . . . . . 6  |-  ( c  =  C  ->  (
( F `  B
)  <  ( F `  c )  <->  ( F `  B )  <  ( F `  C )
) )
3130anbi2d 687 . . . . 5  |-  ( c  =  C  ->  (
( ( F `  B )  <  ( F `  A )  /\  ( F `  B
)  <  ( F `  c ) )  <->  ( ( F `  B )  <  ( F `  A
)  /\  ( F `  B )  <  ( F `  C )
) ) )
3229, 31orbi12d 693 . . . 4  |-  ( c  =  C  ->  (
( ( ( F `
 A )  < 
( F `  B
)  /\  ( F `  c )  <  ( F `  B )
)  \/  ( ( F `  B )  <  ( F `  A )  /\  ( F `  B )  <  ( F `  c
) ) )  <->  ( (
( F `  A
)  <  ( F `  B )  /\  ( F `  C )  <  ( F `  B
) )  \/  (
( F `  B
)  <  ( F `  A )  /\  ( F `  B )  <  ( F `  C
) ) ) ) )
3326, 323anbi23d 1260 . . 3  |-  ( c  =  C  ->  (
( A  <  B  /\  B  <  c  /\  ( ( ( F `
 A )  < 
( F `  B
)  /\  ( F `  c )  <  ( F `  B )
)  \/  ( ( F `  B )  <  ( F `  A )  /\  ( F `  B )  <  ( F `  c
) ) ) )  <-> 
( A  <  B  /\  B  <  C  /\  ( ( ( F `
 A )  < 
( F `  B
)  /\  ( F `  C )  <  ( F `  B )
)  \/  ( ( F `  B )  <  ( F `  A )  /\  ( F `  B )  <  ( F `  C
) ) ) ) ) )
3414, 25, 33rcla43ev 2869 . 2  |-  ( ( ( A  e.  S  /\  B  e.  S  /\  C  e.  S
)  /\  ( A  <  B  /\  B  < 
C  /\  ( (
( F `  A
)  <  ( F `  B )  /\  ( F `  C )  <  ( F `  B
) )  \/  (
( F `  B
)  <  ( F `  A )  /\  ( F `  B )  <  ( F `  C
) ) ) ) )  ->  E. a  e.  S  E. b  e.  S  E. c  e.  S  ( a  <  b  /\  b  < 
c  /\  ( (
( F `  a
)  <  ( F `  b )  /\  ( F `  c )  <  ( F `  b
) )  \/  (
( F `  b
)  <  ( F `  a )  /\  ( F `  b )  <  ( F `  c
) ) ) ) )
351, 2, 3, 4, 5, 6, 34syl33anc 1202 1  |-  ( ph  ->  E. a  e.  S  E. b  e.  S  E. c  e.  S  ( a  <  b  /\  b  <  c  /\  ( ( ( F `
 a )  < 
( F `  b
)  /\  ( F `  c )  <  ( F `  b )
)  \/  ( ( F `  b )  <  ( F `  a )  /\  ( F `  b )  <  ( F `  c
) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    \/ wo 359    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   E.wrex 2519   class class class wbr 3997   ` cfv 4673    < clt 8835
This theorem is referenced by:  pmltpclem2  18772
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-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-rex 2524  df-rab 2527  df-v 2765  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-xp 4675  df-cnv 4677  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fv 4689
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