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Theorem pmltpclem1 23217
Description: Lemma for pmltpc 23219. (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 4656 . . . 4 |- ( a = A -> ( a < b <-> A < b ) )
8 fveq2 6191 . . . . . . 7 |- ( a = A -> ( F ` a ) = ( F ` A ) )
98breq1d 4663 . . . . . 6 |- ( a = A -> ( ( F ` a ) < ( F ` b ) <-> ( F ` A ) < ( F ` b ) ) )
109anbi1d 741 . . . . 5 |- ( a = A -> ( ( ( F ` a ) < ( F ` b ) /\ ( F ` c ) < ( F ` b ) ) <-> ( ( F ` A ) < ( F ` b ) /\ ( F ` c ) < ( F ` b ) ) ) )
118breq2d 4665 . . . . . 6 |- ( a = A -> ( ( F ` b ) < ( F ` a ) <-> ( F ` b ) < ( F ` A ) ) )
1211anbi1d 741 . . . . 5 |- ( a = A -> ( ( ( F ` b ) < ( F ` a ) /\ ( F ` b ) < ( F ` c ) ) <-> ( ( F ` b ) < ( F ` A ) /\ ( F ` b ) < ( F ` c ) ) ) )
1310, 12orbi12d 746 . . . 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 1401 . . 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 4657 . . . 4 |- ( b = B -> ( A < b <-> A < B ) )
16 breq1 4656 . . . 4 |- ( b = B -> ( b < c <-> B < c ) )
17 fveq2 6191 . . . . . . 7 |- ( b = B -> ( F ` b ) = ( F ` B ) )
1817breq2d 4665 . . . . . 6 |- ( b = B -> ( ( F ` A ) < ( F ` b ) <-> ( F ` A ) < ( F ` B ) ) )
1917breq2d 4665 . . . . . 6 |- ( b = B -> ( ( F ` c ) < ( F ` b ) <-> ( F ` c ) < ( F ` B ) ) )
2018, 19anbi12d 747 . . . . 5 |- ( b = B -> ( ( ( F ` A ) < ( F ` b ) /\ ( F ` c ) < ( F ` b ) ) <-> ( ( F ` A ) < ( F ` B ) /\ ( F ` c ) < ( F ` B ) ) ) )
2117breq1d 4663 . . . . . 6 |- ( b = B -> ( ( F ` b ) < ( F ` A ) <-> ( F ` B ) < ( F ` A ) ) )
2217breq1d 4663 . . . . . 6 |- ( b = B -> ( ( F ` b ) < ( F ` c ) <-> ( F ` B ) < ( F ` c ) ) )
2321, 22anbi12d 747 . . . . 5 |- ( b = B -> ( ( ( F ` b ) < ( F ` A ) /\ ( F ` b ) < ( F ` c ) ) <-> ( ( F ` B ) < ( F ` A ) /\ ( F ` B ) < ( F ` c ) ) ) )
2420, 23orbi12d 746 . . . 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 1399 . . 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 4657 . . . 4 |- ( c = C -> ( B < c <-> B < C ) )
27 fveq2 6191 . . . . . . 7 |- ( c = C -> ( F ` c ) = ( F ` C ) )
2827breq1d 4663 . . . . . 6 |- ( c = C -> ( ( F ` c ) < ( F ` B ) <-> ( F ` C ) < ( F ` B ) ) )
2928anbi2d 740 . . . . 5 |- ( c = C -> ( ( ( F ` A ) < ( F ` B ) /\ ( F ` c ) < ( F ` B ) ) <-> ( ( F ` A ) < ( F ` B ) /\ ( F ` C ) < ( F ` B ) ) ) )
3027breq2d 4665 . . . . . 6 |- ( c = C -> ( ( F ` B ) < ( F ` c ) <-> ( F ` B ) < ( F ` C ) ) )
3130anbi2d 740 . . . . 5 |- ( c = C -> ( ( ( F ` B ) < ( F ` A ) /\ ( F ` B ) < ( F ` c ) ) <-> ( ( F ` B ) < ( F ` A ) /\ ( F ` B ) < ( F ` C ) ) ) )
3229, 31orbi12d 746 . . . 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 1402 . . 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, 33rspc3ev 3326 . 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 1341 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 setvar class
Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   class class class wbr 4653   ` cfv 5888   < clt 10074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896
This theorem is referenced by:  pmltpclem2  23218
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