Theorem mod2xnegi 13227

Description: Version of mod2xi 13225 where A ^ B does not equal K (mod N) but instead the negative of K (mod N). (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 9605	. . . 4 |- ( ( L e. NN0 /\ K e. NN ) -> ( L + K ) e. NN )
5	2, 3, 4	mp2an 430	. . 3 |- ( L + K ) e. NN
6	1, 5	eqeltrri 2312	. 2 |- N e. NN
7		mod2xnegi.1	. 2 |- A e. NN
8		mod2xnegi.2	. 2 |- B e. NN0
9	6	nnzi 9676	. . . 4 |- N e. ZZ
10		mod2xnegi.3	. . . 4 |- D e. ZZ
11		zaddcl 9695	. . . 4 |- ( ( N e. ZZ /\ D e. ZZ ) -> ( N + D ) e. ZZ )
12	9, 10, 11	mp2an 430	. . 3 |- ( N + D ) e. ZZ
13	3	nnnn0i 9582	. . . . 5 |- K e. NN0
14	13, 13	nn0addcli 9611	. . . 4 |- ( K + K ) e. NN0
15	14	nn0zi 9677	. . 3 |- ( K + K ) e. ZZ
16		zsubcl 9696	. . 3 |- ( ( ( N + D ) e. ZZ /\ ( K + K ) e. ZZ ) -> ( ( N + D ) - ( K + K ) ) e. ZZ )
17	12, 15, 16	mp2an 430	. 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 9323	. . . . . 6 |- N e. CC
22		zcn 9660	. . . . . . 7 |- ( D e. ZZ -> D e. CC )
23	10, 22	ax-mp 5	. . . . . 6 |- D e. CC
24	21, 23	addcli 8336	. . . . 5 |- ( N + D ) e. CC
25	3	nncni 9323	. . . . . 6 |- K e. CC
26	25, 25	addcli 8336	. . . . 5 |- ( K + K ) e. CC
27	24, 26, 21	subdiri 8742	. . . 4 |- ( ( ( N + D ) - ( K + K ) ) x. N ) = ( ( ( N + D ) x. N ) - ( ( K + K ) x. N ) )
28	27	oveq1i 6098	. . 3 |- ( ( ( ( N + D ) - ( K + K ) ) x. N ) + M ) = ( ( ( ( N + D ) x. N ) - ( ( K + K ) x. N ) ) + M )
29	24, 21	mulcli 8337	. . . 4 |- ( ( N + D ) x. N ) e. CC
30	18	nn0cni 9586	. . . 4 |- M e. CC
31	26, 21	mulcli 8337	. . . 4 |- ( ( K + K ) x. N ) e. CC
32	29, 30, 31	addsubi 8625	. . 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 6099	. . . . . 6 |- ( ( N x. N ) + ( ( D x. N ) + M ) ) = ( ( N x. N ) + ( K x. K ) )
35	21, 25, 25	adddii 8342	. . . . . 6 |- ( N x. ( K + K ) ) = ( ( N x. K ) + ( N x. K ) )
36	34, 35	oveq12i 6100	. . . . 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 8343	. . . . . . . 8 |- ( ( N + D ) x. N ) = ( ( N x. N ) + ( D x. N ) )
38	37	oveq1i 6098	. . . . . . 7 |- ( ( ( N + D ) x. N ) + M ) = ( ( ( N x. N ) + ( D x. N ) ) + M )
39	21, 21	mulcli 8337	. . . . . . . 8 |- ( N x. N ) e. CC
40	23, 21	mulcli 8337	. . . . . . . 8 |- ( D x. N ) e. CC
41	39, 40, 30	addassi 8340	. . . . . . 7 |- ( ( ( N x. N ) + ( D x. N ) ) + M ) = ( ( N x. N ) + ( ( D x. N ) + M ) )
42	38, 41	eqtr2i 2260	. . . . . 6 |- ( ( N x. N ) + ( ( D x. N ) + M ) ) = ( ( ( N + D ) x. N ) + M )
43	21, 26	mulcomi 8338	. . . . . 6 |- ( N x. ( K + K ) ) = ( ( K + K ) x. N )
44	42, 43	oveq12i 6100	. . . . 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 2261	. . . 4 |- ( ( ( N x. N ) + ( K x. K ) ) - ( ( N x. K ) + ( N x. K ) ) ) = ( ( ( ( N + D ) x. N ) + M ) - ( ( K + K ) x. N ) )
46		mulsub 8735	. . . . . 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 431	. . . . 5 |- ( ( N - K ) x. ( N - K ) ) = ( ( ( N x. N ) + ( K x. K ) ) - ( ( N x. K ) + ( N x. K ) ) )
48	2	nn0cni 9586	. . . . . . . 8 |- L e. CC
49	21, 25, 48	subadd2i 8621	. . . . . . 7 |- ( ( N - K ) = L <-> ( L + K ) = N )
50	1, 49	mpbir 146	. . . . . 6 |- ( N - K ) = L
51	50, 50	oveq12i 6100	. . . . 5 |- ( ( N - K ) x. ( N - K ) ) = ( L x. L )
52	47, 51	eqtr3i 2261	. . . 4 |- ( ( ( N x. N ) + ( K x. K ) ) - ( ( N x. K ) + ( N x. K ) ) ) = ( L x. L )
53	45, 52	eqtr3i 2261	. . 3 |- ( ( ( ( N + D ) x. N ) + M ) - ( ( K + K ) x. N ) ) = ( L x. L )
54	28, 32, 53	3eqtr2i 2265	. 2 |- ( ( ( ( N + D ) - ( K + K ) ) x. N ) + M ) = ( L x. L )
55	6, 7, 8, 17, 2, 18, 19, 20, 54	mod2xi 13225	1 |- ( ( A ^ E ) mod N ) = ( M mod N )

This proof depends on syntax axioms:   = wceq 1402   e. wcel 2209  (class class class)co 6088   CCcc 8183   + caddc 8188   x. cmul 8190   - cmin 8504   NNcn 9313   2c2 9364   NN0cn0 9574   ZZcz 9655   mod cmo 10780   ^cexp 10996

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-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304

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-rmo 2536  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-if 3639  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-po 4443  df-iso 4444  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-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-n0 9575  df-z 9656  df-uz 9938  df-q 10036  df-rp 10072  df-fl 10722  df-mod 10781  df-seqfrec 10906  df-exp 10997

This theorem is used by:  1259lem4  13274