Theorem mod2xnegi 15775

Description: Version of mod2xi 15773 with a negative mod value. (Contributed by Mario Carneiro, 21-Feb-2014.)

Hypotheses

Ref	Expression
mod2xnegi.1	|- A e. NN
mod2xnegi.2	|- B e. NN0
mod2xnegi.3	|- D e. ZZ
mod2xnegi.4	|- K e. NN
mod2xnegi.5	|- M e. NN0
mod2xnegi.6	|- L e. NN0
mod2xnegi.10	|- ( ( A ^ B ) mod N ) = ( L mod N )
mod2xnegi.7	|- ( 2 x. B ) = E
mod2xnegi.8	|- ( L + K ) = N
mod2xnegi.9	|- ( ( D x. N ) + M ) = ( K x. K )

Assertion

Ref	Expression
mod2xnegi	|- ( ( A ^ E ) mod N ) = ( M mod N )

Proof of Theorem mod2xnegi

Step	Hyp	Ref	Expression
1		mod2xnegi.8	. . 3 |- ( L + K ) = N
2		mod2xnegi.6	. . . 4 |- L e. NN0
3		mod2xnegi.4	. . . 4 |- K e. NN
4		nn0nnaddcl 11324	. . . 4 |- ( ( L e. NN0 /\ K e. NN ) -> ( L + K ) e. NN )
5	2, 3, 4	mp2an 708	. . 3 |- ( L + K ) e. NN
6	1, 5	eqeltrri 2698	. 2 |- N e. NN
7		mod2xnegi.1	. 2 |- A e. NN
8		mod2xnegi.2	. 2 |- B e. NN0
9	6	nnzi 11401	. . . 4 |- N e. ZZ
10		mod2xnegi.3	. . . 4 |- D e. ZZ
11		zaddcl 11417	. . . 4 |- ( ( N e. ZZ /\ D e. ZZ ) -> ( N + D ) e. ZZ )
12	9, 10, 11	mp2an 708	. . 3 |- ( N + D ) e. ZZ
13	3	nnnn0i 11300	. . . . 5 |- K e. NN0
14	13, 13	nn0addcli 11330	. . . 4 |- ( K + K ) e. NN0
15	14	nn0zi 11402	. . 3 |- ( K + K ) e. ZZ
16		zsubcl 11419	. . 3 |- ( ( ( N + D ) e. ZZ /\ ( K + K ) e. ZZ ) -> ( ( N + D ) - ( K + K ) ) e. ZZ )
17	12, 15, 16	mp2an 708	. 2 |- ( ( N + D ) - ( K + K ) ) e. ZZ
18		mod2xnegi.5	. 2 |- M e. NN0
19		mod2xnegi.10	. 2 |- ( ( A ^ B ) mod N ) = ( L mod N )
20		mod2xnegi.7	. 2 |- ( 2 x. B ) = E
21	6	nncni 11030	. . . . . 6 |- N e. CC
22		zcn 11382	. . . . . . 7 |- ( D e. ZZ -> D e. CC )
23	10, 22	ax-mp 5	. . . . . 6 |- D e. CC
24	21, 23	addcli 10044	. . . . 5 |- ( N + D ) e. CC
25	3	nncni 11030	. . . . . 6 |- K e. CC
26	25, 25	addcli 10044	. . . . 5 |- ( K + K ) e. CC
27	24, 26, 21	subdiri 10480	. . . 4 |- ( ( ( N + D ) - ( K + K ) ) x. N ) = ( ( ( N + D ) x. N ) - ( ( K + K ) x. N ) )
28	27	oveq1i 6660	. . 3 |- ( ( ( ( N + D ) - ( K + K ) ) x. N ) + M ) = ( ( ( ( N + D ) x. N ) - ( ( K + K ) x. N ) ) + M )
29	24, 21	mulcli 10045	. . . 4 |- ( ( N + D ) x. N ) e. CC
30	18	nn0cni 11304	. . . 4 |- M e. CC
31	26, 21	mulcli 10045	. . . 4 |- ( ( K + K ) x. N ) e. CC
32	29, 30, 31	addsubi 10373	. . 3 |- ( ( ( ( N + D ) x. N ) + M ) - ( ( K + K ) x. N ) ) = ( ( ( ( N + D ) x. N ) - ( ( K + K ) x. N ) ) + M )
33		mod2xnegi.9	. . . . . . 7 |- ( ( D x. N ) + M ) = ( K x. K )
34	33	oveq2i 6661	. . . . . 6 |- ( ( N x. N ) + ( ( D x. N ) + M ) ) = ( ( N x. N ) + ( K x. K ) )
35	21, 25, 25	adddii 10050	. . . . . 6 |- ( N x. ( K + K ) ) = ( ( N x. K ) + ( N x. K ) )
36	34, 35	oveq12i 6662	. . . . 5 |- ( ( ( N x. N ) + ( ( D x. N ) + M ) ) - ( N x. ( K + K ) ) ) = ( ( ( N x. N ) + ( K x. K ) ) - ( ( N x. K ) + ( N x. K ) ) )
37	21, 23, 21	adddiri 10051	. . . . . . . 8 |- ( ( N + D ) x. N ) = ( ( N x. N ) + ( D x. N ) )
38	37	oveq1i 6660	. . . . . . 7 |- ( ( ( N + D ) x. N ) + M ) = ( ( ( N x. N ) + ( D x. N ) ) + M )
39	21, 21	mulcli 10045	. . . . . . . 8 |- ( N x. N ) e. CC
40	23, 21	mulcli 10045	. . . . . . . 8 |- ( D x. N ) e. CC
41	39, 40, 30	addassi 10048	. . . . . . 7 |- ( ( ( N x. N ) + ( D x. N ) ) + M ) = ( ( N x. N ) + ( ( D x. N ) + M ) )
42	38, 41	eqtr2i 2645	. . . . . 6 |- ( ( N x. N ) + ( ( D x. N ) + M ) ) = ( ( ( N + D ) x. N ) + M )
43	21, 26	mulcomi 10046	. . . . . 6 |- ( N x. ( K + K ) ) = ( ( K + K ) x. N )
44	42, 43	oveq12i 6662	. . . . 5 |- ( ( ( N x. N ) + ( ( D x. N ) + M ) ) - ( N x. ( K + K ) ) ) = ( ( ( ( N + D ) x. N ) + M ) - ( ( K + K ) x. N ) )
45	36, 44	eqtr3i 2646	. . . 4 |- ( ( ( N x. N ) + ( K x. K ) ) - ( ( N x. K ) + ( N x. K ) ) ) = ( ( ( ( N + D ) x. N ) + M ) - ( ( K + K ) x. N ) )
46		mulsub 10473	. . . . . 6 |- ( ( ( N e. CC /\ K e. CC ) /\ ( N e. CC /\ K e. CC ) ) -> ( ( N - K ) x. ( N - K ) ) = ( ( ( N x. N ) + ( K x. K ) ) - ( ( N x. K ) + ( N x. K ) ) ) )
47	21, 25, 21, 25, 46	mp4an 709	. . . . 5 |- ( ( N - K ) x. ( N - K ) ) = ( ( ( N x. N ) + ( K x. K ) ) - ( ( N x. K ) + ( N x. K ) ) )
48	2	nn0cni 11304	. . . . . . . 8 |- L e. CC
49	21, 25, 48	subadd2i 10369	. . . . . . 7 |- ( ( N - K ) = L <-> ( L + K ) = N )
50	1, 49	mpbir 221	. . . . . 6 |- ( N - K ) = L
51	50, 50	oveq12i 6662	. . . . 5 |- ( ( N - K ) x. ( N - K ) ) = ( L x. L )
52	47, 51	eqtr3i 2646	. . . 4 |- ( ( ( N x. N ) + ( K x. K ) ) - ( ( N x. K ) + ( N x. K ) ) ) = ( L x. L )
53	45, 52	eqtr3i 2646	. . 3 |- ( ( ( ( N + D ) x. N ) + M ) - ( ( K + K ) x. N ) ) = ( L x. L )
54	28, 32, 53	3eqtr2i 2650	. 2 |- ( ( ( ( N + D ) - ( K + K ) ) x. N ) + M ) = ( L x. L )
55	6, 7, 8, 17, 2, 18, 19, 20, 54	mod2xi 15773	1 |- ( ( A ^ E ) mod N ) = ( M mod N )

Syntax hints:   = wceq 1483   e. wcel 1990  (class class class)co 6650   CCcc 9934   + caddc 9939   x. cmul 9941   - cmin 10266   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   mod cmo 12668   ^cexp 12860

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  ax-pre-sup 10014

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-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  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-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861

This theorem is referenced by:  1259lem4  15841  2503lem2  15845