Theorem gxmodid 9402

Description: Casting out powers of the identity element leaves the group power unchanged. (Contributed by Paul Chapman, 17-Apr-2009.)

Hypotheses

Ref	Expression
gxmodid.1	|- X = ran G
gxmodid.2	|- U = (Id` G)
gxmodid.3	|- P = (^g` G)

Assertion

Ref	Expression
gxmodid	|- ((G e. Grp /\ (K e. ZZ /\ M e. NN) /\ (A e. X /\ (APM) = U)) -> (AP(K mod M)) = (APK))

Proof of Theorem gxmodid

Step	Hyp	Ref	Expression
1		modval 7501	. . . . . 6 |- ((K e. RR /\ M e. RR+) -> (K mod M) = (K - (M x. (|_` (K / M)))))
2		zre 7348	. . . . . 6 |- (K e. ZZ -> K e. RR)
3		nnrp 7238	. . . . . 6 |- (M e. NN -> M e. RR+)
4	1, 2, 3	syl2an 503	. . . . 5 |- ((K e. ZZ /\ M e. NN) -> (K mod M) = (K - (M x. (|_` (K / M)))))
5	4	3ad2ant2 898	. . . 4 |- ((G e. Grp /\ (K e. ZZ /\ M e. NN) /\ (A e. X /\ (APM) = U)) -> (K mod M) = (K - (M x. (|_` (K / M)))))
6	5	opreq2d 4898	. . 3 |- ((G e. Grp /\ (K e. ZZ /\ M e. NN) /\ (A e. X /\ (APM) = U)) -> (AP(K mod M)) = (AP(K - (M x. (|_` (K / M))))))
7		simpl 346	. . . . . . 7 |- ((K e. ZZ /\ M e. NN) -> K e. ZZ)
8		zcn 7349	. . . . . . 7 |- (K e. ZZ -> K e. CC)
9	7, 8	syl 12	. . . . . 6 |- ((K e. ZZ /\ M e. NN) -> K e. CC)
10		nnz 7362	. . . . . . . . 9 |- (M e. NN -> M e. ZZ)
11	10	adantl 424	. . . . . . . 8 |- ((K e. ZZ /\ M e. NN) -> M e. ZZ)
12		redivcl 6978	. . . . . . . . . . 11 |- ((K e. RR /\ M e. RR /\ M =/= 0) -> (K / M) e. RR)
13		nnre 7112	. . . . . . . . . . 11 |- (M e. NN -> M e. RR)
14		nnne0 7132	. . . . . . . . . . 11 |- (M e. NN -> M =/= 0)
15	12, 2, 13, 14	syl3an 1139	. . . . . . . . . 10 |- ((K e. ZZ /\ M e. NN /\ M e. NN) -> (K / M) e. RR)
16	15	3anidm23 1156	. . . . . . . . 9 |- ((K e. ZZ /\ M e. NN) -> (K / M) e. RR)
17		flcl 7465	. . . . . . . . 9 |- ((K / M) e. RR -> (|_` (K / M)) e. ZZ)
18	16, 17	syl 12	. . . . . . . 8 |- ((K e. ZZ /\ M e. NN) -> (|_` (K / M)) e. ZZ)
19		zmulcl 7389	. . . . . . . 8 |- ((M e. ZZ /\ (|_` (K / M)) e. ZZ) -> (M x. (|_` (K / M))) e. ZZ)
20	11, 18, 19	syl11anc 524	. . . . . . 7 |- ((K e. ZZ /\ M e. NN) -> (M x. (|_` (K / M))) e. ZZ)
21		zcn 7349	. . . . . . 7 |- ((M x. (|_` (K / M))) e. ZZ -> (M x. (|_` (K / M))) e. CC)
22	20, 21	syl 12	. . . . . 6 |- ((K e. ZZ /\ M e. NN) -> (M x. (|_` (K / M))) e. CC)
23		negsub 6540	. . . . . 6 |- ((K e. CC /\ (M x. (|_` (K / M))) e. CC) -> (K + -u(M x. (|_` (K / M)))) = (K - (M x. (|_` (K / M)))))
24	9, 22, 23	syl11anc 524	. . . . 5 |- ((K e. ZZ /\ M e. NN) -> (K + -u(M x. (|_` (K / M)))) = (K - (M x. (|_` (K / M)))))
25	24	3ad2ant2 898	. . . 4 |- ((G e. Grp /\ (K e. ZZ /\ M e. NN) /\ (A e. X /\ (APM) = U)) -> (K + -u(M x. (|_` (K / M)))) = (K - (M x. (|_` (K / M)))))
26	25	opreq2d 4898	. . 3 |- ((G e. Grp /\ (K e. ZZ /\ M e. NN) /\ (A e. X /\ (APM) = U)) -> (AP(K + -u(M x. (|_` (K / M))))) = (AP(K - (M x. (|_` (K / M))))))
27		simp1 876	. . . 4 |- ((G e. Grp /\ (K e. ZZ /\ M e. NN) /\ (A e. X /\ (APM) = U)) -> G e. Grp)
28		simp3l 904	. . . 4 |- ((G e. Grp /\ (K e. ZZ /\ M e. NN) /\ (A e. X /\ (APM) = U)) -> A e. X)
29		znegcl 7372	. . . . . . 7 |- ((M x. (|_` (K / M))) e. ZZ -> -u(M x. (|_` (K / M))) e. ZZ)
30	20, 29	syl 12	. . . . . 6 |- ((K e. ZZ /\ M e. NN) -> -u(M x. (|_` (K / M))) e. ZZ)
31	7, 30	jca 310	. . . . 5 |- ((K e. ZZ /\ M e. NN) -> (K e. ZZ /\ -u(M x. (|_` (K / M))) e. ZZ))
32	31	3ad2ant2 898	. . . 4 |- ((G e. Grp /\ (K e. ZZ /\ M e. NN) /\ (A e. X /\ (APM) = U)) -> (K e. ZZ /\ -u(M x. (|_` (K / M))) e. ZZ))
33		gxmodid.1	. . . . 5 |- X = ran G
34		gxmodid.3	. . . . 5 |- P = (^g` G)
35	33, 34	gxadd 9398	. . . 4 |- ((G e. Grp /\ A e. X /\ (K e. ZZ /\ -u(M x. (|_` (K / M))) e. ZZ)) -> (AP(K + -u(M x. (|_` (K / M))))) = ((APK)G(AP-u(M x. (|_` (K / M))))))
36	27, 28, 32, 35	syl111anc 1100	. . 3 |- ((G e. Grp /\ (K e. ZZ /\ M e. NN) /\ (A e. X /\ (APM) = U)) -> (AP(K + -u(M x. (|_` (K / M))))) = ((APK)G(AP-u(M x. (|_` (K / M))))))
37	6, 26, 36	3eqtr2d 1932	. 2 |- ((G e. Grp /\ (K e. ZZ /\ M e. NN) /\ (A e. X /\ (APM) = U)) -> (AP(K mod M)) = ((APK)G(AP-u(M x. (|_` (K / M))))))
38		zcn 7349	. . . . . . . 8 |- (M e. ZZ -> M e. CC)
39	11, 38	syl 12	. . . . . . 7 |- ((K e. ZZ /\ M e. NN) -> M e. CC)
40		zcn 7349	. . . . . . . 8 |- ((|_` (K / M)) e. ZZ -> (|_` (K / M)) e. CC)
41	18, 40	syl 12	. . . . . . 7 |- ((K e. ZZ /\ M e. NN) -> (|_` (K / M)) e. CC)
42		mulneg2 6616	. . . . . . 7 |- ((M e. CC /\ (|_` (K / M)) e. CC) -> (M x. -u(|_` (K / M))) = -u(M x. (|_` (K / M))))
43	39, 41, 42	syl11anc 524	. . . . . 6 |- ((K e. ZZ /\ M e. NN) -> (M x. -u(|_` (K / M))) = -u(M x. (|_` (K / M))))
44	43	3ad2ant2 898	. . . . 5 |- ((G e. Grp /\ (K e. ZZ /\ M e. NN) /\ (A e. X /\ (APM) = U)) -> (M x. -u(|_` (K / M))) = -u(M x. (|_` (K / M))))
45	44	opreq2d 4898	. . . 4 |- ((G e. Grp /\ (K e. ZZ /\ M e. NN) /\ (A e. X /\ (APM) = U)) -> (AP(M x. -u(|_` (K / M)))) = (AP-u(M x. (|_` (K / M)))))
46	11	3ad2ant2 898	. . . . . 6 |- ((G e. Grp /\ (K e. ZZ /\ M e. NN) /\ (A e. X /\ (APM) = U)) -> M e. ZZ)
47		znegcl 7372	. . . . . . . 8 |- ((|_` (K / M)) e. ZZ -> -u(|_` (K / M)) e. ZZ)
48	18, 47	syl 12	. . . . . . 7 |- ((K e. ZZ /\ M e. NN) -> -u(|_` (K / M)) e. ZZ)
49	48	3ad2ant2 898	. . . . . 6 |- ((G e. Grp /\ (K e. ZZ /\ M e. NN) /\ (A e. X /\ (APM) = U)) -> -u(|_` (K / M)) e. ZZ)
50	33, 34	gxmul 9401	. . . . . 6 |- ((G e. Grp /\ A e. X /\ (M e. ZZ /\ -u(|_` (K / M)) e. ZZ)) -> (AP(M x. -u(|_` (K / M)))) = ((APM)P-u(|_` (K / M))))
51	27, 28, 46, 49, 50	syl112anc 1104	. . . . 5 |- ((G e. Grp /\ (K e. ZZ /\ M e. NN) /\ (A e. X /\ (APM) = U)) -> (AP(M x. -u(|_` (K / M)))) = ((APM)P-u(|_` (K / M))))
52		opreq1 4889	. . . . . . 7 |- ((APM) = U -> ((APM)P-u(|_` (K / M))) = (UP-u(|_` (K / M))))
53	52	adantl 424	. . . . . 6 |- ((A e. X /\ (APM) = U) -> ((APM)P-u(|_` (K / M))) = (UP-u(|_` (K / M))))
54	53	3ad2ant3 899	. . . . 5 |- ((G e. Grp /\ (K e. ZZ /\ M e. NN) /\ (A e. X /\ (APM) = U)) -> ((APM)P-u(|_` (K / M))) = (UP-u(|_` (K / M))))
55		gxmodid.2	. . . . . . 7 |- U = (Id` G)
56	55, 34	gxid 9396	. . . . . 6 |- ((G e. Grp /\ -u(|_` (K / M)) e. ZZ) -> (UP-u(|_` (K / M))) = U)
57	27, 49, 56	syl11anc 524	. . . . 5 |- ((G e. Grp /\ (K e. ZZ /\ M e. NN) /\ (A e. X /\ (APM) = U)) -> (UP-u(|_` (K / M))) = U)
58	51, 54, 57	3eqtrd 1929	. . . 4 |- ((G e. Grp /\ (K e. ZZ /\ M e. NN) /\ (A e. X /\ (APM) = U)) -> (AP(M x. -u(|_` (K / M)))) = U)
59	45, 58	eqtr3d 1927	. . 3 |- ((G e. Grp /\ (K e. ZZ /\ M e. NN) /\ (A e. X /\ (APM) = U)) -> (AP-u(M x. (|_` (K / M)))) = U)
60	59	opreq2d 4898	. 2 |- ((G e. Grp /\ (K e. ZZ /\ M e. NN) /\ (A e. X /\ (APM) = U)) -> ((APK)G(AP-u(M x. (|_` (K / M))))) = ((APK)GU))
61		simp2l 902	. . . 4 |- ((G e. Grp /\ (K e. ZZ /\ M e. NN) /\ (A e. X /\ (APM) = U)) -> K e. ZZ)
62	33, 34	gxcl 9388	. . . 4 |- ((G e. Grp /\ A e. X /\ K e. ZZ) -> (APK) e. X)
63	27, 28, 61, 62	syl111anc 1100	. . 3 |- ((G e. Grp /\ (K e. ZZ /\ M e. NN) /\ (A e. X /\ (APM) = U)) -> (APK) e. X)
64	33, 55	grprid 9346	. . 3 |- ((G e. Grp /\ (APK) e. X) -> ((APK)GU) = (APK))
65	27, 63, 64	syl11anc 524	. 2 |- ((G e. Grp /\ (K e. ZZ /\ M e. NN) /\ (A e. X /\ (APM) = U)) -> ((APK)GU) = (APK))
66	37, 60, 65	3eqtrd 1929	1 |- ((G e. Grp /\ (K e. ZZ /\ M e. NN) /\ (A e. X /\ (APM) = U)) -> (AP(K mod M)) = (APK))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  ran crn 3987  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386   + caddc 6389   x. cmul 6391   - cmin 6445  -ucneg 6446   / cdiv 6447  NNcn 6449  ZZcz 6451  RR+crp 6453  |_cfl 7462   mod cmo 7499  Grpcgr 9311  Idcgi 9312  ^gcgx 9315

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-rp 7232  df-n0 7309  df-z 7345  df-fl 7463  df-mod 7500  df-seq1 7721  df-grp 9316  df-gid 9317  df-ginv 9318  df-gx 9320

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