MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax5seglem3a Structured version   Unicode version

Theorem ax5seglem3a 24360
Description: Lemma for ax5seg 24368. (Contributed by Scott Fenton, 7-May-2015.)
Assertion
Ref Expression
ax5seglem3a  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  j  e.  ( 1 ... N
) )  ->  (
( ( A `  j )  -  ( C `  j )
)  e.  RR  /\  ( ( D `  j )  -  ( F `  j )
)  e.  RR ) )

Proof of Theorem ax5seglem3a
StepHypRef Expression
1 simpl21 1074 . . . 4  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  j  e.  ( 1 ... N
) )  ->  A  e.  ( EE `  N
) )
2 fveere 24331 . . . 4  |-  ( ( A  e.  ( EE
`  N )  /\  j  e.  ( 1 ... N ) )  ->  ( A `  j )  e.  RR )
31, 2sylancom 667 . . 3  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  j  e.  ( 1 ... N
) )  ->  ( A `  j )  e.  RR )
4 simpl23 1076 . . . 4  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  j  e.  ( 1 ... N
) )  ->  C  e.  ( EE `  N
) )
5 fveere 24331 . . . 4  |-  ( ( C  e.  ( EE
`  N )  /\  j  e.  ( 1 ... N ) )  ->  ( C `  j )  e.  RR )
64, 5sylancom 667 . . 3  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  j  e.  ( 1 ... N
) )  ->  ( C `  j )  e.  RR )
73, 6resubcld 10008 . 2  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  j  e.  ( 1 ... N
) )  ->  (
( A `  j
)  -  ( C `
 j ) )  e.  RR )
8 simpl31 1077 . . . 4  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  j  e.  ( 1 ... N
) )  ->  D  e.  ( EE `  N
) )
9 fveere 24331 . . . 4  |-  ( ( D  e.  ( EE
`  N )  /\  j  e.  ( 1 ... N ) )  ->  ( D `  j )  e.  RR )
108, 9sylancom 667 . . 3  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  j  e.  ( 1 ... N
) )  ->  ( D `  j )  e.  RR )
11 simpl33 1079 . . . 4  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  j  e.  ( 1 ... N
) )  ->  F  e.  ( EE `  N
) )
12 fveere 24331 . . . 4  |-  ( ( F  e.  ( EE
`  N )  /\  j  e.  ( 1 ... N ) )  ->  ( F `  j )  e.  RR )
1311, 12sylancom 667 . . 3  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  j  e.  ( 1 ... N
) )  ->  ( F `  j )  e.  RR )
1410, 13resubcld 10008 . 2  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  j  e.  ( 1 ... N
) )  ->  (
( D `  j
)  -  ( F `
 j ) )  e.  RR )
157, 14jca 532 1  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  j  e.  ( 1 ... N
) )  ->  (
( ( A `  j )  -  ( C `  j )
)  e.  RR  /\  ( ( D `  j )  -  ( F `  j )
)  e.  RR ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    e. wcel 1819   ` cfv 5594  (class class class)co 6296   RRcr 9508   1c1 9510    - cmin 9824   NNcn 10556   ...cfz 11697   EEcee 24318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-ltxr 9650  df-sub 9826  df-neg 9827  df-ee 24321
This theorem is referenced by:  ax5seglem3  24361
  Copyright terms: Public domain W3C validator