Theorem prmdivdiv 12872

Description: The (modular) inverse of the inverse of a number is itself. (Contributed by Mario Carneiro, 24-Jan-2015.)

Hypothesis

Ref	Expression
prmdiv.1	|- R = ( ( A ^ ( P - 2 ) ) mod P )

Assertion

Ref	Expression
prmdivdiv	|- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> A = ( ( R ^ ( P - 2 ) ) mod P ) )

Proof of Theorem prmdivdiv

Step	Hyp	Ref	Expression
1		fz1ssfz0 10397	. . 3 |- ( 1 ... ( P - 1 ) ) C_ ( 0 ... ( P - 1 ) )
2		simpr 110	. . 3 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> A e. ( 1 ... ( P - 1 ) ) )
3	1, 2	sselid 3226	. 2 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> A e. ( 0 ... ( P - 1 ) ) )
4		simpl 109	. . . . 5 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> P e. Prime )
5		elfznn 10334	. . . . . . 7 |- ( A e. ( 1 ... ( P - 1 ) ) -> A e. NN )
6	5	adantl 277	. . . . . 6 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> A e. NN )
7	6	nnzd 9645	. . . . 5 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> A e. ZZ )
8		prmnn 12745	. . . . . 6 |- ( P e. Prime -> P e. NN )
9		fzm1ndvds 12480	. . . . . 6 |- ( ( P e. NN /\ A e. ( 1 ... ( P - 1 ) ) ) -> -. P || A )
10	8, 9	sylan 283	. . . . 5 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> -. P || A )
11		prmdiv.1	. . . . . 6 |- R = ( ( A ^ ( P - 2 ) ) mod P )
12	11	prmdiv 12870	. . . . 5 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( R e. ( 1 ... ( P - 1 ) ) /\ P || ( ( A x. R ) - 1 ) ) )
13	4, 7, 10, 12	syl3anc 1274	. . . 4 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> ( R e. ( 1 ... ( P - 1 ) ) /\ P || ( ( A x. R ) - 1 ) ) )
14	13	simprd 114	. . 3 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> P || ( ( A x. R ) - 1 ) )
15	6	nncnd 9199	. . . . 5 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> A e. CC )
16	13	simpld 112	. . . . . . 7 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> R e. ( 1 ... ( P - 1 ) ) )
17		elfznn 10334	. . . . . . 7 |- ( R e. ( 1 ... ( P - 1 ) ) -> R e. NN )
18	16, 17	syl 14	. . . . . 6 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> R e. NN )
19	18	nncnd 9199	. . . . 5 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> R e. CC )
20	15, 19	mulcomd 8243	. . . 4 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> ( A x. R ) = ( R x. A ) )
21	20	oveq1d 6043	. . 3 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> ( ( A x. R ) - 1 ) = ( ( R x. A ) - 1 ) )
22	14, 21	breqtrd 4119	. 2 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> P || ( ( R x. A ) - 1 ) )
23		elfzelz 10305	. . . 4 |- ( R e. ( 1 ... ( P - 1 ) ) -> R e. ZZ )
24	16, 23	syl 14	. . 3 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> R e. ZZ )
25		fzm1ndvds 12480	. . . 4 |- ( ( P e. NN /\ R e. ( 1 ... ( P - 1 ) ) ) -> -. P || R )
26	8, 16, 25	syl2an2r 599	. . 3 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> -. P || R )
27		eqid 2231	. . . 4 |- ( ( R ^ ( P - 2 ) ) mod P ) = ( ( R ^ ( P - 2 ) ) mod P )
28	27	prmdiveq 12871	. . 3 |- ( ( P e. Prime /\ R e. ZZ /\ -. P || R ) -> ( ( A e. ( 0 ... ( P - 1 ) ) /\ P || ( ( R x. A ) - 1 ) ) <-> A = ( ( R ^ ( P - 2 ) ) mod P ) ) )
29	4, 24, 26, 28	syl3anc 1274	. 2 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> ( ( A e. ( 0 ... ( P - 1 ) ) /\ P || ( ( R x. A ) - 1 ) ) <-> A = ( ( R ^ ( P - 2 ) ) mod P ) ) )
30	3, 22, 29	mpbi2and 952	1 |- ( ( P e. Prime /\ A e. ( 1 ... ( P - 1 ) ) ) -> A = ( ( R ^ ( P - 2 ) ) mod P ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2202   class class class wbr 4093  (class class class)co 6028   0cc0 8075   1c1 8076   x. cmul 8080   - cmin 8392   NNcn 9185   2c2 9236   ZZcz 9523   ...cfz 10288   mod cmo 10630   ^cexp 10846   || cdvds 12411   Primecprime 12742

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193  ax-arch 8194  ax-caucvg 8195

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-isom 5342  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-frec 6600  df-1o 6625  df-2o 6626  df-oadd 6629  df-er 6745  df-en 6953  df-dom 6954  df-fin 6955  df-sup 7226  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-reap 8797  df-ap 8804  df-div 8895  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-n0 9445  df-z 9524  df-uz 9800  df-q 9898  df-rp 9933  df-fz 10289  df-fzo 10423  df-fl 10576  df-mod 10631  df-seqfrec 10756  df-exp 10847  df-ihash 11084  df-cj 11465  df-re 11466  df-im 11467  df-rsqrt 11621  df-abs 11622  df-clim 11902  df-proddc 12175  df-dvds 12412  df-gcd 12588  df-prm 12743  df-phi 12846

This theorem is referenced by: (None)