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Theorem fzsdom2 12584
Description: Condition for finite ranges to have a strict dominance relation. (Contributed by Stefan O'Rear, 12-Sep-2014.) (Revised by Mario Carneiro, 15-Apr-2015.)
Assertion
Ref Expression
fzsdom2  |-  ( ( ( B  e.  (
ZZ>= `  A )  /\  C  e.  ZZ )  /\  B  <  C )  ->  ( A ... B )  ~<  ( A ... C ) )

Proof of Theorem fzsdom2
StepHypRef Expression
1 eluzelz 11157 . . . . . . 7  |-  ( B  e.  ( ZZ>= `  A
)  ->  B  e.  ZZ )
21ad2antrr 730 . . . . . 6  |-  ( ( ( B  e.  (
ZZ>= `  A )  /\  C  e.  ZZ )  /\  B  <  C )  ->  B  e.  ZZ )
32zred 11029 . . . . 5  |-  ( ( ( B  e.  (
ZZ>= `  A )  /\  C  e.  ZZ )  /\  B  <  C )  ->  B  e.  RR )
4 eluzel2 11153 . . . . . . 7  |-  ( B  e.  ( ZZ>= `  A
)  ->  A  e.  ZZ )
54ad2antrr 730 . . . . . 6  |-  ( ( ( B  e.  (
ZZ>= `  A )  /\  C  e.  ZZ )  /\  B  <  C )  ->  A  e.  ZZ )
65zred 11029 . . . . 5  |-  ( ( ( B  e.  (
ZZ>= `  A )  /\  C  e.  ZZ )  /\  B  <  C )  ->  A  e.  RR )
73, 6resubcld 10036 . . . 4  |-  ( ( ( B  e.  (
ZZ>= `  A )  /\  C  e.  ZZ )  /\  B  <  C )  ->  ( B  -  A )  e.  RR )
8 simplr 760 . . . . . 6  |-  ( ( ( B  e.  (
ZZ>= `  A )  /\  C  e.  ZZ )  /\  B  <  C )  ->  C  e.  ZZ )
98zred 11029 . . . . 5  |-  ( ( ( B  e.  (
ZZ>= `  A )  /\  C  e.  ZZ )  /\  B  <  C )  ->  C  e.  RR )
109, 6resubcld 10036 . . . 4  |-  ( ( ( B  e.  (
ZZ>= `  A )  /\  C  e.  ZZ )  /\  B  <  C )  ->  ( C  -  A )  e.  RR )
11 1red 9647 . . . 4  |-  ( ( ( B  e.  (
ZZ>= `  A )  /\  C  e.  ZZ )  /\  B  <  C )  ->  1  e.  RR )
12 simpr 462 . . . . 5  |-  ( ( ( B  e.  (
ZZ>= `  A )  /\  C  e.  ZZ )  /\  B  <  C )  ->  B  <  C
)
133, 9, 6, 12ltsub1dd 10214 . . . 4  |-  ( ( ( B  e.  (
ZZ>= `  A )  /\  C  e.  ZZ )  /\  B  <  C )  ->  ( B  -  A )  <  ( C  -  A )
)
147, 10, 11, 13ltadd1dd 10213 . . 3  |-  ( ( ( B  e.  (
ZZ>= `  A )  /\  C  e.  ZZ )  /\  B  <  C )  ->  ( ( B  -  A )  +  1 )  <  (
( C  -  A
)  +  1 ) )
15 hashfz 12583 . . . 4  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( # `  ( A ... B ) )  =  ( ( B  -  A )  +  1 ) )
1615ad2antrr 730 . . 3  |-  ( ( ( B  e.  (
ZZ>= `  A )  /\  C  e.  ZZ )  /\  B  <  C )  ->  ( # `  ( A ... B ) )  =  ( ( B  -  A )  +  1 ) )
173, 9, 12ltled 9772 . . . . . 6  |-  ( ( ( B  e.  (
ZZ>= `  A )  /\  C  e.  ZZ )  /\  B  <  C )  ->  B  <_  C
)
18 eluz2 11154 . . . . . 6  |-  ( C  e.  ( ZZ>= `  B
)  <->  ( B  e.  ZZ  /\  C  e.  ZZ  /\  B  <_  C ) )
192, 8, 17, 18syl3anbrc 1189 . . . . 5  |-  ( ( ( B  e.  (
ZZ>= `  A )  /\  C  e.  ZZ )  /\  B  <  C )  ->  C  e.  (
ZZ>= `  B ) )
20 simpll 758 . . . . 5  |-  ( ( ( B  e.  (
ZZ>= `  A )  /\  C  e.  ZZ )  /\  B  <  C )  ->  B  e.  (
ZZ>= `  A ) )
21 uztrn 11164 . . . . 5  |-  ( ( C  e.  ( ZZ>= `  B )  /\  B  e.  ( ZZ>= `  A )
)  ->  C  e.  ( ZZ>= `  A )
)
2219, 20, 21syl2anc 665 . . . 4  |-  ( ( ( B  e.  (
ZZ>= `  A )  /\  C  e.  ZZ )  /\  B  <  C )  ->  C  e.  (
ZZ>= `  A ) )
23 hashfz 12583 . . . 4  |-  ( C  e.  ( ZZ>= `  A
)  ->  ( # `  ( A ... C ) )  =  ( ( C  -  A )  +  1 ) )
2422, 23syl 17 . . 3  |-  ( ( ( B  e.  (
ZZ>= `  A )  /\  C  e.  ZZ )  /\  B  <  C )  ->  ( # `  ( A ... C ) )  =  ( ( C  -  A )  +  1 ) )
2514, 16, 243brtr4d 4447 . 2  |-  ( ( ( B  e.  (
ZZ>= `  A )  /\  C  e.  ZZ )  /\  B  <  C )  ->  ( # `  ( A ... B ) )  <  ( # `  ( A ... C ) ) )
26 fzfi 12171 . . 3  |-  ( A ... B )  e. 
Fin
27 fzfi 12171 . . 3  |-  ( A ... C )  e. 
Fin
28 hashsdom 12546 . . 3  |-  ( ( ( A ... B
)  e.  Fin  /\  ( A ... C )  e.  Fin )  -> 
( ( # `  ( A ... B ) )  <  ( # `  ( A ... C ) )  <-> 
( A ... B
)  ~<  ( A ... C ) ) )
2926, 27, 28mp2an 676 . 2  |-  ( (
# `  ( A ... B ) )  < 
( # `  ( A ... C ) )  <-> 
( A ... B
)  ~<  ( A ... C ) )
3025, 29sylib 199 1  |-  ( ( ( B  e.  (
ZZ>= `  A )  /\  C  e.  ZZ )  /\  B  <  C )  ->  ( A ... B )  ~<  ( A ... C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1867   class class class wbr 4417   ` cfv 5592  (class class class)co 6296    ~< csdm 7567   Fincfn 7568   1c1 9529    + caddc 9531    < clt 9664    <_ cle 9665    - cmin 9849   ZZcz 10926   ZZ>=cuz 11148   ...cfz 11771   #chash 12501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-oadd 7185  df-er 7362  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-card 8363  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-nn 10599  df-n0 10859  df-z 10927  df-uz 11149  df-fz 11772  df-hash 12502
This theorem is referenced by:  irrapxlem1  35420
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