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Theorem hashfzp1 13218
Description: Value of the numeric cardinality of a (possibly empty) integer range. (Contributed by AV, 19-Jun-2021.)
Assertion
Ref Expression
hashfzp1 |- ( B e. ( ZZ>= ` A ) -> ( # ` ( ( A + 1 ) ... B ) ) = ( B - A ) )
Proof of Theorem hashfzp1
StepHypRef Expression
1 hash0 13158 . . . 4 |- ( # ` (/) ) = 0
2 eluzelre 11698 . . . . . . 7 |- ( B e. ( ZZ>= ` A ) -> B e. RR )
32ltp1d 10954 . . . . . 6 |- ( B e. ( ZZ>= ` A ) -> B < ( B + 1 ) )
4 eluzelz 11697 . . . . . . 7 |- ( B e. ( ZZ>= ` A ) -> B e. ZZ )
5 peano2z 11418 . . . . . . . 8 |- ( B e. ZZ -> ( B + 1 ) e. ZZ )
65ancri 575 . . . . . . 7 |- ( B e. ZZ -> ( ( B + 1 ) e. ZZ /\ B e. ZZ ) )
7 fzn 12357 . . . . . . 7 |- ( ( ( B + 1 ) e. ZZ /\ B e. ZZ ) -> ( B < ( B + 1 ) <-> ( ( B + 1 ) ... B ) = (/) ) )
84, 6, 73syl 18 . . . . . 6 |- ( B e. ( ZZ>= ` A ) -> ( B < ( B + 1 ) <-> ( ( B + 1 ) ... B ) = (/) ) )
93, 8mpbid 222 . . . . 5 |- ( B e. ( ZZ>= ` A ) -> ( ( B + 1 ) ... B ) = (/) )
109fveq2d 6195 . . . 4 |- ( B e. ( ZZ>= ` A ) -> ( # ` ( ( B + 1 ) ... B ) ) = ( # ` (/) ) )
114zcnd 11483 . . . . 5 |- ( B e. ( ZZ>= ` A ) -> B e. CC )
1211subidd 10380 . . . 4 |- ( B e. ( ZZ>= ` A ) -> ( B - B ) = 0 )
131, 10, 123eqtr4a 2682 . . 3 |- ( B e. ( ZZ>= ` A ) -> ( # ` ( ( B + 1 ) ... B ) ) = ( B - B ) )
14 oveq1 6657 . . . . . 6 |- ( A = B -> ( A + 1 ) = ( B + 1 ) )
1514oveq1d 6665 . . . . 5 |- ( A = B -> ( ( A + 1 ) ... B ) = ( ( B + 1 ) ... B ) )
1615fveq2d 6195 . . . 4 |- ( A = B -> ( # ` ( ( A + 1 ) ... B ) ) = ( # ` ( ( B + 1 ) ... B ) ) )
17 oveq2 6658 . . . 4 |- ( A = B -> ( B - A ) = ( B - B ) )
1816, 17eqeq12d 2637 . . 3 |- ( A = B -> ( ( # ` ( ( A + 1 ) ... B ) ) = ( B - A ) <-> ( # ` ( ( B + 1 ) ... B ) ) = ( B - B ) ) )
1913, 18syl5ibr 236 . 2 |- ( A = B -> ( B e. ( ZZ>= ` A ) -> ( # ` ( ( A + 1 ) ... B ) ) = ( B - A ) ) )
20 uzp1 11721 . . . . . . 7 |- ( B e. ( ZZ>= ` A ) -> ( B = A \/ B e. ( ZZ>= ` ( A + 1 ) ) ) )
21 pm2.24 121 . . . . . . . . 9 |- ( A = B -> ( -. A = B -> B e. ( ZZ>= ` ( A + 1 ) ) ) )
2221eqcoms 2630 . . . . . . . 8 |- ( B = A -> ( -. A = B -> B e. ( ZZ>= ` ( A + 1 ) ) ) )
23 ax-1 6 . . . . . . . 8 |- ( B e. ( ZZ>= ` ( A + 1 ) ) -> ( -. A = B -> B e. ( ZZ>= ` ( A + 1 ) ) ) )
2422, 23jaoi 394 . . . . . . 7 |- ( ( B = A \/ B e. ( ZZ>= ` ( A + 1 ) ) ) -> ( -. A = B -> B e. ( ZZ>= ` ( A + 1 ) ) ) )
2520, 24syl 17 . . . . . 6 |- ( B e. ( ZZ>= ` A ) -> ( -. A = B -> B e. ( ZZ>= ` ( A + 1 ) ) ) )
2625impcom 446 . . . . 5 |- ( ( -. A = B /\ B e. ( ZZ>= ` A ) ) -> B e. ( ZZ>= ` ( A + 1 ) ) )
27 hashfz 13214 . . . . 5 |- ( B e. ( ZZ>= ` ( A + 1 ) ) -> ( # ` ( ( A + 1 ) ... B ) ) = ( ( B - ( A + 1 ) ) + 1 ) )
2826, 27syl 17 . . . 4 |- ( ( -. A = B /\ B e. ( ZZ>= ` A ) ) -> ( # ` ( ( A + 1 ) ... B ) ) = ( ( B - ( A + 1 ) ) + 1 ) )
29 eluzel2 11692 . . . . . . 7 |- ( B e. ( ZZ>= ` A ) -> A e. ZZ )
3029zcnd 11483 . . . . . 6 |- ( B e. ( ZZ>= ` A ) -> A e. CC )
31 1cnd 10056 . . . . . 6 |- ( B e. ( ZZ>= ` A ) -> 1 e. CC )
3211, 30, 31nppcan2d 10418 . . . . 5 |- ( B e. ( ZZ>= ` A ) -> ( ( B - ( A + 1 ) ) + 1 ) = ( B - A ) )
3332adantl 482 . . . 4 |- ( ( -. A = B /\ B e. ( ZZ>= ` A ) ) -> ( ( B - ( A + 1 ) ) + 1 ) = ( B - A ) )
3428, 33eqtrd 2656 . . 3 |- ( ( -. A = B /\ B e. ( ZZ>= ` A ) ) -> ( # ` ( ( A + 1 ) ... B ) ) = ( B - A ) )
3534ex 450 . 2 |- ( -. A = B -> ( B e. ( ZZ>= ` A ) -> ( # ` ( ( A + 1 ) ... B ) ) = ( B - A ) ) )
3619, 35pm2.61i 176 1 |- ( B e. ( ZZ>= ` A ) -> ( # ` ( ( A + 1 ) ... B ) ) = ( B - A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   (/)c0 3915   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   - cmin 10266   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   #chash 13117
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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118
This theorem is referenced by:  2lgslem1  25119
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