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Theorem fzass4 10417
Description: Two ways to express a nondecreasing sequence of four integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)
Assertion
Ref Expression
fzass4 |- ( ( B e. ( A ... D ) /\ C e. ( B ... D ) ) <-> ( B e. ( A ... C ) /\ C e. ( A ... D ) ) )
Proof of Theorem fzass4
StepHypRef Expression
1 simpll 527 . . . . 5 |- ( ( ( B e. ( ZZ>= ` A ) /\ D e. ( ZZ>= ` B ) ) /\ ( C e. ( ZZ>= ` B ) /\ D e. ( ZZ>= ` C ) ) ) -> B e. ( ZZ>= ` A ) )
2 simprl 531 . . . . 5 |- ( ( ( B e. ( ZZ>= ` A ) /\ D e. ( ZZ>= ` B ) ) /\ ( C e. ( ZZ>= ` B ) /\ D e. ( ZZ>= ` C ) ) ) -> C e. ( ZZ>= ` B ) )
31, 2jca 306 . . . 4 |- ( ( ( B e. ( ZZ>= ` A ) /\ D e. ( ZZ>= ` B ) ) /\ ( C e. ( ZZ>= ` B ) /\ D e. ( ZZ>= ` C ) ) ) -> ( B e. ( ZZ>= ` A ) /\ C e. ( ZZ>= ` B ) ) )
4 uztrn 9889 . . . . . 6 |- ( ( C e. ( ZZ>= ` B ) /\ B e. ( ZZ>= ` A ) ) -> C e. ( ZZ>= ` A ) )
54ancoms 268 . . . . 5 |- ( ( B e. ( ZZ>= ` A ) /\ C e. ( ZZ>= ` B ) ) -> C e. ( ZZ>= ` A ) )
65ad2ant2r 509 . . . 4 |- ( ( ( B e. ( ZZ>= ` A ) /\ D e. ( ZZ>= ` B ) ) /\ ( C e. ( ZZ>= ` B ) /\ D e. ( ZZ>= ` C ) ) ) -> C e. ( ZZ>= ` A ) )
7 simprr 533 . . . 4 |- ( ( ( B e. ( ZZ>= ` A ) /\ D e. ( ZZ>= ` B ) ) /\ ( C e. ( ZZ>= ` B ) /\ D e. ( ZZ>= ` C ) ) ) -> D e. ( ZZ>= ` C ) )
83, 6, 7jca32 310 . . 3 |- ( ( ( B e. ( ZZ>= ` A ) /\ D e. ( ZZ>= ` B ) ) /\ ( C e. ( ZZ>= ` B ) /\ D e. ( ZZ>= ` C ) ) ) -> ( ( B e. ( ZZ>= ` A ) /\ C e. ( ZZ>= ` B ) ) /\ ( C e. ( ZZ>= ` A ) /\ D e. ( ZZ>= ` C ) ) ) )
9 simpll 527 . . . . 5 |- ( ( ( B e. ( ZZ>= ` A ) /\ C e. ( ZZ>= ` B ) ) /\ ( C e. ( ZZ>= ` A ) /\ D e. ( ZZ>= ` C ) ) ) -> B e. ( ZZ>= ` A ) )
10 uztrn 9889 . . . . . . 7 |- ( ( D e. ( ZZ>= ` C ) /\ C e. ( ZZ>= ` B ) ) -> D e. ( ZZ>= ` B ) )
1110ancoms 268 . . . . . 6 |- ( ( C e. ( ZZ>= ` B ) /\ D e. ( ZZ>= ` C ) ) -> D e. ( ZZ>= ` B ) )
1211ad2ant2l 508 . . . . 5 |- ( ( ( B e. ( ZZ>= ` A ) /\ C e. ( ZZ>= ` B ) ) /\ ( C e. ( ZZ>= ` A ) /\ D e. ( ZZ>= ` C ) ) ) -> D e. ( ZZ>= ` B ) )
139, 12jca 306 . . . 4 |- ( ( ( B e. ( ZZ>= ` A ) /\ C e. ( ZZ>= ` B ) ) /\ ( C e. ( ZZ>= ` A ) /\ D e. ( ZZ>= ` C ) ) ) -> ( B e. ( ZZ>= ` A ) /\ D e. ( ZZ>= ` B ) ) )
14 simplr 529 . . . 4 |- ( ( ( B e. ( ZZ>= ` A ) /\ C e. ( ZZ>= ` B ) ) /\ ( C e. ( ZZ>= ` A ) /\ D e. ( ZZ>= ` C ) ) ) -> C e. ( ZZ>= ` B ) )
15 simprr 533 . . . 4 |- ( ( ( B e. ( ZZ>= ` A ) /\ C e. ( ZZ>= ` B ) ) /\ ( C e. ( ZZ>= ` A ) /\ D e. ( ZZ>= ` C ) ) ) -> D e. ( ZZ>= ` C ) )
1613, 14, 15jca32 310 . . 3 |- ( ( ( B e. ( ZZ>= ` A ) /\ C e. ( ZZ>= ` B ) ) /\ ( C e. ( ZZ>= ` A ) /\ D e. ( ZZ>= ` C ) ) ) -> ( ( B e. ( ZZ>= ` A ) /\ D e. ( ZZ>= ` B ) ) /\ ( C e. ( ZZ>= ` B ) /\ D e. ( ZZ>= ` C ) ) ) )
178, 16impbii 126 . 2 |- ( ( ( B e. ( ZZ>= ` A ) /\ D e. ( ZZ>= ` B ) ) /\ ( C e. ( ZZ>= ` B ) /\ D e. ( ZZ>= ` C ) ) ) <-> ( ( B e. ( ZZ>= ` A ) /\ C e. ( ZZ>= ` B ) ) /\ ( C e. ( ZZ>= ` A ) /\ D e. ( ZZ>= ` C ) ) ) )
18 elfzuzb 10372 . . 3 |- ( B e. ( A ... D ) <-> ( B e. ( ZZ>= ` A ) /\ D e. ( ZZ>= ` B ) ) )
19 elfzuzb 10372 . . 3 |- ( C e. ( B ... D ) <-> ( C e. ( ZZ>= ` B ) /\ D e. ( ZZ>= ` C ) ) )
2018, 19anbi12i 460 . 2 |- ( ( B e. ( A ... D ) /\ C e. ( B ... D ) ) <-> ( ( B e. ( ZZ>= ` A ) /\ D e. ( ZZ>= ` B ) ) /\ ( C e. ( ZZ>= ` B ) /\ D e. ( ZZ>= ` C ) ) ) )
21 elfzuzb 10372 . . 3 |- ( B e. ( A ... C ) <-> ( B e. ( ZZ>= ` A ) /\ C e. ( ZZ>= ` B ) ) )
22 elfzuzb 10372 . . 3 |- ( C e. ( A ... D ) <-> ( C e. ( ZZ>= ` A ) /\ D e. ( ZZ>= ` C ) ) )
2321, 22anbi12i 460 . 2 |- ( ( B e. ( A ... C ) /\ C e. ( A ... D ) ) <-> ( ( B e. ( ZZ>= ` A ) /\ C e. ( ZZ>= ` B ) ) /\ ( C e. ( ZZ>= ` A ) /\ D e. ( ZZ>= ` C ) ) ) )
2417, 20, 233bitr4i 212 1 |- ( ( B e. ( A ... D ) /\ C e. ( B ... D ) ) <-> ( B e. ( A ... C ) /\ C e. ( A ... D ) ) )
Colors of variables: wff set class
Syntax hints:   /\ wa 104   <-> wb 105   e. wcel 2205   ` cfv 5357  (class class class)co 6058   ZZ>=cuz 9871   ...cfz 10361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-pre-ltwlin 8256
This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-neg 8463  df-z 9595  df-uz 9872  df-fz 10362
This theorem is referenced by:  ccatswrd  11387  ccatpfx  11418
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