Theorem modaddabs 13612

Description: Absorbtion law for modulo. (Contributed by Paul Chapman, 22-Jun-2011.)

Assertion

Ref	Expression
modaddabs	|- ((A e. RR /\ B e. RR /\ C e. RR+) -> (((A mod C) + (B mod C)) mod C) = ((A + B) mod C))

Proof of Theorem modaddabs

Step	Hyp	Ref	Expression
1		modcl 7502	. . . . . 6 |- ((A e. RR /\ C e. RR+) -> (A mod C) e. RR)
2		recn 6466	. . . . . 6 |- ((A mod C) e. RR -> (A mod C) e. CC)
3	1, 2	syl 12	. . . . 5 |- ((A e. RR /\ C e. RR+) -> (A mod C) e. CC)
4	3	3adant2 895	. . . 4 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> (A mod C) e. CC)
5		modcl 7502	. . . . . 6 |- ((B e. RR /\ C e. RR+) -> (B mod C) e. RR)
6		recn 6466	. . . . . 6 |- ((B mod C) e. RR -> (B mod C) e. CC)
7	5, 6	syl 12	. . . . 5 |- ((B e. RR /\ C e. RR+) -> (B mod C) e. CC)
8	7	3adant1 894	. . . 4 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> (B mod C) e. CC)
9		addcom 6458	. . . 4 |- (((A mod C) e. CC /\ (B mod C) e. CC) -> ((A mod C) + (B mod C)) = ((B mod C) + (A mod C)))
10	4, 8, 9	syl11anc 524	. . 3 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> ((A mod C) + (B mod C)) = ((B mod C) + (A mod C)))
11	10	opreq1d 4897	. 2 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> (((A mod C) + (B mod C)) mod C) = (((B mod C) + (A mod C)) mod C))
12		simpl 346	. . . . . 6 |- ((B e. RR /\ C e. RR+) -> B e. RR)
13	5, 12	jca 310	. . . . 5 |- ((B e. RR /\ C e. RR+) -> ((B mod C) e. RR /\ B e. RR))
14	13	3adant1 894	. . . 4 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> ((B mod C) e. RR /\ B e. RR))
15		simpr 350	. . . . . 6 |- ((A e. RR /\ C e. RR+) -> C e. RR+)
16	1, 15	jca 310	. . . . 5 |- ((A e. RR /\ C e. RR+) -> ((A mod C) e. RR /\ C e. RR+))
17	16	3adant2 895	. . . 4 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> ((A mod C) e. RR /\ C e. RR+))
18		modabs2 7515	. . . . 5 |- ((B e. RR /\ C e. RR+) -> ((B mod C) mod C) = (B mod C))
19	18	3adant1 894	. . . 4 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> ((B mod C) mod C) = (B mod C))
20		modadd1 7518	. . . 4 |- ((((B mod C) e. RR /\ B e. RR) /\ ((A mod C) e. RR /\ C e. RR+) /\ ((B mod C) mod C) = (B mod C)) -> (((B mod C) + (A mod C)) mod C) = ((B + (A mod C)) mod C))
21	14, 17, 19, 20	syl111anc 1100	. . 3 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> (((B mod C) + (A mod C)) mod C) = ((B + (A mod C)) mod C))
22		recn 6466	. . . . . 6 |- (B e. RR -> B e. CC)
23	22	3ad2ant2 898	. . . . 5 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> B e. CC)
24		addcom 6458	. . . . 5 |- (((A mod C) e. CC /\ B e. CC) -> ((A mod C) + B) = (B + (A mod C)))
25	4, 23, 24	syl11anc 524	. . . 4 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> ((A mod C) + B) = (B + (A mod C)))
26	25	opreq1d 4897	. . 3 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> (((A mod C) + B) mod C) = ((B + (A mod C)) mod C))
27		simpl 346	. . . . . 6 |- ((A e. RR /\ C e. RR+) -> A e. RR)
28	1, 27	jca 310	. . . . 5 |- ((A e. RR /\ C e. RR+) -> ((A mod C) e. RR /\ A e. RR))
29	28	3adant2 895	. . . 4 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> ((A mod C) e. RR /\ A e. RR))
30		3simpc 874	. . . 4 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> (B e. RR /\ C e. RR+))
31		modabs2 7515	. . . . 5 |- ((A e. RR /\ C e. RR+) -> ((A mod C) mod C) = (A mod C))
32	31	3adant2 895	. . . 4 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> ((A mod C) mod C) = (A mod C))
33		modadd1 7518	. . . 4 |- ((((A mod C) e. RR /\ A e. RR) /\ (B e. RR /\ C e. RR+) /\ ((A mod C) mod C) = (A mod C)) -> (((A mod C) + B) mod C) = ((A + B) mod C))
34	29, 30, 32, 33	syl111anc 1100	. . 3 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> (((A mod C) + B) mod C) = ((A + B) mod C))
35	21, 26, 34	3eqtr2rd 1933	. 2 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> ((A + B) mod C) = (((B mod C) + (A mod C)) mod C))
36	11, 35	eqtr4d 1928	1 |- ((A e. RR /\ B e. RR /\ C e. RR+) -> (((A mod C) + (B mod C)) mod C) = ((A + B) mod C))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  (class class class)co 4884  CCcc 6384  RRcr 6385   + caddc 6389  RR+crp 6453   mod cmo 7499

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

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