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Theorem etransclem10 40461
Description: The given if term is an integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
Hypotheses
Ref Expression
etransclem10.n |- ( ph -> P e. NN )
etransclem10.m |- ( ph -> M e. NN0 )
etransclem10.c |- ( ph -> C : ( 0 ... M ) --> ( 0 ... N ) )
etransclem10.j |- ( ph -> J e. ZZ )
Assertion
Ref Expression
etransclem10 |- ( ph -> if ( ( P - 1 ) < ( C ` 0 ) , 0 , ( ( ( ! ` ( P - 1 ) ) / ( ! ` ( ( P - 1 ) - ( C ` 0 ) ) ) ) x. ( J ^ ( ( P - 1 ) - ( C ` 0 ) ) ) ) ) e. ZZ )
Proof of Theorem etransclem10
StepHypRef Expression
1 0zd 11389 . 2 |- ( ( ph /\ ( P - 1 ) < ( C ` 0 ) ) -> 0 e. ZZ )
2 0zd 11389 . . . . . . . . 9 |- ( ph -> 0 e. ZZ )
3 etransclem10.n . . . . . . . . . . 11 |- ( ph -> P e. NN )
4 nnm1nn0 11334 . . . . . . . . . . 11 |- ( P e. NN -> ( P - 1 ) e. NN0 )
53, 4syl 17 . . . . . . . . . 10 |- ( ph -> ( P - 1 ) e. NN0 )
65nn0zd 11480 . . . . . . . . 9 |- ( ph -> ( P - 1 ) e. ZZ )
7 etransclem10.c . . . . . . . . . . . 12 |- ( ph -> C : ( 0 ... M ) --> ( 0 ... N ) )
8 etransclem10.m . . . . . . . . . . . . . 14 |- ( ph -> M e. NN0 )
9 nn0uz 11722 . . . . . . . . . . . . . 14 |- NN0 = ( ZZ>= ` 0 )
108, 9syl6eleq 2711 . . . . . . . . . . . . 13 |- ( ph -> M e. ( ZZ>= ` 0 ) )
11 eluzfz1 12348 . . . . . . . . . . . . 13 |- ( M e. ( ZZ>= ` 0 ) -> 0 e. ( 0 ... M ) )
1210, 11syl 17 . . . . . . . . . . . 12 |- ( ph -> 0 e. ( 0 ... M ) )
137, 12ffvelrnd 6360 . . . . . . . . . . 11 |- ( ph -> ( C ` 0 ) e. ( 0 ... N ) )
1413elfzelzd 39530 . . . . . . . . . 10 |- ( ph -> ( C ` 0 ) e. ZZ )
156, 14zsubcld 11487 . . . . . . . . 9 |- ( ph -> ( ( P - 1 ) - ( C ` 0 ) ) e. ZZ )
162, 6, 153jca 1242 . . . . . . . 8 |- ( ph -> ( 0 e. ZZ /\ ( P - 1 ) e. ZZ /\ ( ( P - 1 ) - ( C ` 0 ) ) e. ZZ ) )
1716adantr 481 . . . . . . 7 |- ( ( ph /\ -. ( P - 1 ) < ( C ` 0 ) ) -> ( 0 e. ZZ /\ ( P - 1 ) e. ZZ /\ ( ( P - 1 ) - ( C ` 0 ) ) e. ZZ ) )
1814zred 11482 . . . . . . . . . 10 |- ( ph -> ( C ` 0 ) e. RR )
1918adantr 481 . . . . . . . . 9 |- ( ( ph /\ -. ( P - 1 ) < ( C ` 0 ) ) -> ( C ` 0 ) e. RR )
205nn0red 11352 . . . . . . . . . 10 |- ( ph -> ( P - 1 ) e. RR )
2120adantr 481 . . . . . . . . 9 |- ( ( ph /\ -. ( P - 1 ) < ( C ` 0 ) ) -> ( P - 1 ) e. RR )
22 simpr 477 . . . . . . . . 9 |- ( ( ph /\ -. ( P - 1 ) < ( C ` 0 ) ) -> -. ( P - 1 ) < ( C ` 0 ) )
2319, 21, 22nltled 10187 . . . . . . . 8 |- ( ( ph /\ -. ( P - 1 ) < ( C ` 0 ) ) -> ( C ` 0 ) <_ ( P - 1 ) )
2421, 19subge0d 10617 . . . . . . . 8 |- ( ( ph /\ -. ( P - 1 ) < ( C ` 0 ) ) -> ( 0 <_ ( ( P - 1 ) - ( C ` 0 ) ) <-> ( C ` 0 ) <_ ( P - 1 ) ) )
2523, 24mpbird 247 . . . . . . 7 |- ( ( ph /\ -. ( P - 1 ) < ( C ` 0 ) ) -> 0 <_ ( ( P - 1 ) - ( C ` 0 ) ) )
26 elfzle1 12344 . . . . . . . . . 10 |- ( ( C ` 0 ) e. ( 0 ... N ) -> 0 <_ ( C ` 0 ) )
2713, 26syl 17 . . . . . . . . 9 |- ( ph -> 0 <_ ( C ` 0 ) )
2827adantr 481 . . . . . . . 8 |- ( ( ph /\ -. ( P - 1 ) < ( C ` 0 ) ) -> 0 <_ ( C ` 0 ) )
2921, 19subge02d 10619 . . . . . . . 8 |- ( ( ph /\ -. ( P - 1 ) < ( C ` 0 ) ) -> ( 0 <_ ( C ` 0 ) <-> ( ( P - 1 ) - ( C ` 0 ) ) <_ ( P - 1 ) ) )
3028, 29mpbid 222 . . . . . . 7 |- ( ( ph /\ -. ( P - 1 ) < ( C ` 0 ) ) -> ( ( P - 1 ) - ( C ` 0 ) ) <_ ( P - 1 ) )
3117, 25, 30jca32 558 . . . . . 6 |- ( ( ph /\ -. ( P - 1 ) < ( C ` 0 ) ) -> ( ( 0 e. ZZ /\ ( P - 1 ) e. ZZ /\ ( ( P - 1 ) - ( C ` 0 ) ) e. ZZ ) /\ ( 0 <_ ( ( P - 1 ) - ( C ` 0 ) ) /\ ( ( P - 1 ) - ( C ` 0 ) ) <_ ( P - 1 ) ) ) )
32 elfz2 12333 . . . . . 6 |- ( ( ( P - 1 ) - ( C ` 0 ) ) e. ( 0 ... ( P - 1 ) ) <-> ( ( 0 e. ZZ /\ ( P - 1 ) e. ZZ /\ ( ( P - 1 ) - ( C ` 0 ) ) e. ZZ ) /\ ( 0 <_ ( ( P - 1 ) - ( C ` 0 ) ) /\ ( ( P - 1 ) - ( C ` 0 ) ) <_ ( P - 1 ) ) ) )
3331, 32sylibr 224 . . . . 5 |- ( ( ph /\ -. ( P - 1 ) < ( C ` 0 ) ) -> ( ( P - 1 ) - ( C ` 0 ) ) e. ( 0 ... ( P - 1 ) ) )
34 permnn 13113 . . . . 5 |- ( ( ( P - 1 ) - ( C ` 0 ) ) e. ( 0 ... ( P - 1 ) ) -> ( ( ! ` ( P - 1 ) ) / ( ! ` ( ( P - 1 ) - ( C ` 0 ) ) ) ) e. NN )
3533, 34syl 17 . . . 4 |- ( ( ph /\ -. ( P - 1 ) < ( C ` 0 ) ) -> ( ( ! ` ( P - 1 ) ) / ( ! ` ( ( P - 1 ) - ( C ` 0 ) ) ) ) e. NN )
3635nnzd 11481 . . 3 |- ( ( ph /\ -. ( P - 1 ) < ( C ` 0 ) ) -> ( ( ! ` ( P - 1 ) ) / ( ! ` ( ( P - 1 ) - ( C ` 0 ) ) ) ) e. ZZ )
37 etransclem10.j . . . . 5 |- ( ph -> J e. ZZ )
3837adantr 481 . . . 4 |- ( ( ph /\ -. ( P - 1 ) < ( C ` 0 ) ) -> J e. ZZ )
3915adantr 481 . . . . 5 |- ( ( ph /\ -. ( P - 1 ) < ( C ` 0 ) ) -> ( ( P - 1 ) - ( C ` 0 ) ) e. ZZ )
40 elnn0z 11390 . . . . 5 |- ( ( ( P - 1 ) - ( C ` 0 ) ) e. NN0 <-> ( ( ( P - 1 ) - ( C ` 0 ) ) e. ZZ /\ 0 <_ ( ( P - 1 ) - ( C ` 0 ) ) ) )
4139, 25, 40sylanbrc 698 . . . 4 |- ( ( ph /\ -. ( P - 1 ) < ( C ` 0 ) ) -> ( ( P - 1 ) - ( C ` 0 ) ) e. NN0 )
42 zexpcl 12875 . . . 4 |- ( ( J e. ZZ /\ ( ( P - 1 ) - ( C ` 0 ) ) e. NN0 ) -> ( J ^ ( ( P - 1 ) - ( C ` 0 ) ) ) e. ZZ )
4338, 41, 42syl2anc 693 . . 3 |- ( ( ph /\ -. ( P - 1 ) < ( C ` 0 ) ) -> ( J ^ ( ( P - 1 ) - ( C ` 0 ) ) ) e. ZZ )
4436, 43zmulcld 11488 . 2 |- ( ( ph /\ -. ( P - 1 ) < ( C ` 0 ) ) -> ( ( ( ! ` ( P - 1 ) ) / ( ! ` ( ( P - 1 ) - ( C ` 0 ) ) ) ) x. ( J ^ ( ( P - 1 ) - ( C ` 0 ) ) ) ) e. ZZ )
451, 44ifclda 4120 1 |- ( ph -> if ( ( P - 1 ) < ( C ` 0 ) , 0 , ( ( ( ! ` ( P - 1 ) ) / ( ! ` ( ( P - 1 ) - ( C ` 0 ) ) ) ) x. ( J ^ ( ( P - 1 ) - ( C ` 0 ) ) ) ) ) e. ZZ )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   e. wcel 1990   ifcif 4086   class class class wbr 4653   -->wf 5884   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   NNcn 11020   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   ^cexp 12860   !cfa 13060
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-rmo 2920  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-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-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090
This theorem is referenced by:  etransclem25  40476  etransclem26  40477  etransclem35  40486  etransclem37  40488
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