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Theorem hashfz 11284
Description: Value of the numeric cardinality of a nonempty integer range. (Contributed by Stefan O'Rear, 12-Sep-2014.) (Proof shortened by Mario Carneiro, 15-Apr-2015.)
Assertion
Ref Expression
hashfz |- ( B e. ( ZZ>= ` A ) -> (♯ ` ( A ... B ) ) = ( ( B - A ) + 1 ) )
Proof of Theorem hashfz
StepHypRef Expression
1 eluzel2 9942 . . . . 5 |- ( B e. ( ZZ>= ` A ) -> A e. ZZ )
2 eluzelz 9947 . . . . 5 |- ( B e. ( ZZ>= ` A ) -> B e. ZZ )
3 1z 9681 . . . . . 6 |- 1 e. ZZ
4 zsubcl 9696 . . . . . 6 |- ( ( 1 e. ZZ /\ A e. ZZ ) -> ( 1 - A ) e. ZZ )
53, 1, 4sylancr 418 . . . . 5 |- ( B e. ( ZZ>= ` A ) -> ( 1 - A ) e. ZZ )
6 fzen 10464 . . . . 5 |- ( ( A e. ZZ /\ B e. ZZ /\ ( 1 - A ) e. ZZ ) -> ( A ... B ) ~~ ( ( A + ( 1 - A ) ) ... ( B + ( 1 - A ) ) ) )
71, 2, 5, 6syl3anc 1278 . . . 4 |- ( B e. ( ZZ>= ` A ) -> ( A ... B ) ~~ ( ( A + ( 1 - A ) ) ... ( B + ( 1 - A ) ) ) )
81zcnd 9780 . . . . . 6 |- ( B e. ( ZZ>= ` A ) -> A e. CC )
9 ax-1cn 8278 . . . . . 6 |- 1 e. CC
10 pncan3 8541 . . . . . 6 |- ( ( A e. CC /\ 1 e. CC ) -> ( A + ( 1 - A ) ) = 1 )
118, 9, 10sylancl 417 . . . . 5 |- ( B e. ( ZZ>= ` A ) -> ( A + ( 1 - A ) ) = 1 )
12 1cnd 8348 . . . . . 6 |- ( B e. ( ZZ>= ` A ) -> 1 e. CC )
132zcnd 9780 . . . . . . 7 |- ( B e. ( ZZ>= ` A ) -> B e. CC )
1413, 8subcld 8644 . . . . . 6 |- ( B e. ( ZZ>= ` A ) -> ( B - A ) e. CC )
1513, 12, 8addsub12d 8667 . . . . . 6 |- ( B e. ( ZZ>= ` A ) -> ( B + ( 1 - A ) ) = ( 1 + ( B - A ) ) )
1612, 14, 15comraddd 8490 . . . . 5 |- ( B e. ( ZZ>= ` A ) -> ( B + ( 1 - A ) ) = ( ( B - A ) + 1 ) )
1711, 16oveq12d 6106 . . . 4 |- ( B e. ( ZZ>= ` A ) -> ( ( A + ( 1 - A ) ) ... ( B + ( 1 - A ) ) ) = ( 1 ... ( ( B - A ) + 1 ) ) )
187, 17breqtrd 4157 . . 3 |- ( B e. ( ZZ>= ` A ) -> ( A ... B ) ~~ ( 1 ... ( ( B - A ) + 1 ) ) )
191, 2fzfigd 10889 . . . 4 |- ( B e. ( ZZ>= ` A ) -> ( A ... B ) e. Fin )
20 1zzd 9682 . . . . 5 |- ( B e. ( ZZ>= ` A ) -> 1 e. ZZ )
212, 1zsubcld 9784 . . . . . 6 |- ( B e. ( ZZ>= ` A ) -> ( B - A ) e. ZZ )
2221peano2zd 9782 . . . . 5 |- ( B e. ( ZZ>= ` A ) -> ( ( B - A ) + 1 ) e. ZZ )
2320, 22fzfigd 10889 . . . 4 |- ( B e. ( ZZ>= ` A ) -> ( 1 ... ( ( B - A ) + 1 ) ) e. Fin )
24 hashen 11245 . . . 4 |- ( ( ( A ... B ) e. Fin /\ ( 1 ... ( ( B - A ) + 1 ) ) e. Fin ) -> ( (♯ ` ( A ... B ) ) = (♯ ` ( 1 ... ( ( B - A ) + 1 ) ) ) <-> ( A ... B ) ~~ ( 1 ... ( ( B - A ) + 1 ) ) ) )
2519, 23, 24syl2anc 415 . . 3 |- ( B e. ( ZZ>= ` A ) -> ( (♯ ` ( A ... B ) ) = (♯ ` ( 1 ... ( ( B - A ) + 1 ) ) ) <-> ( A ... B ) ~~ ( 1 ... ( ( B - A ) + 1 ) ) ) )
2618, 25mpbird 167 . 2 |- ( B e. ( ZZ>= ` A ) -> (♯ ` ( A ... B ) ) = (♯ ` ( 1 ... ( ( B - A ) + 1 ) ) ) )
27 uznn0sub 9970 . . 3 |- ( B e. ( ZZ>= ` A ) -> ( B - A ) e. NN0 )
28 peano2nn0 9614 . . 3 |- ( ( B - A ) e. NN0 -> ( ( B - A ) + 1 ) e. NN0 )
29 hashfz1 11244 . . 3 |- ( ( ( B - A ) + 1 ) e. NN0 -> (♯ ` ( 1 ... ( ( B - A ) + 1 ) ) ) = ( ( B - A ) + 1 ) )
3027, 28, 293syl 17 . 2 |- ( B e. ( ZZ>= ` A ) -> (♯ ` ( 1 ... ( ( B - A ) + 1 ) ) ) = ( ( B - A ) + 1 ) )
3126, 30eqtrd 2271 1 |- ( B e. ( ZZ>= ` A ) -> (♯ ` ( A ... B ) ) = ( ( B - A ) + 1 ) )
Colors of variables:    wff set class
This proof depends on syntax axioms:   -> wi 4   <-> wb 105   = wceq 1402   e. wcel 2209   class class class wbr 4131   ` cfv 5380  (class class class)co 6088   ~~ cen 7024   Fincfn 7026   CCcc 8183   1c1 8186   + caddc 8188   - cmin 8504   NN0cn0 9574   ZZcz 9655   ZZ>=cuz 9937   ...cfz 10428  ♯chash 11236
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-addass 8287  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-0id 8293  ax-rnegex 8294  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301
This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-1o 6691  df-er 6811  df-en 7027  df-dom 7028  df-fin 7029  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-inn 9314  df-n0 9575  df-z 9656  df-uz 9938  df-fz 10429  df-ihash 11237
This theorem is used by:  hashfzo  11285  hashfzp1  11287  hashfz0  11288  ballotfilem2  13286  gzsumgsum  14245  0sgmppw  16287  gausslemma2dlem5  16390  lgseisenlem4  16397
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