Theorem 3dvds2decOLD 15057

Description: Old version of 3dvds2dec 15056. Obsolete as of 1-Aug-2021. (Contributed by AV, 14-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
3dvdsdec.a	|- A e. NN0
3dvdsdec.b	|- B e. NN0
3dvds2dec.c	|- C e. NN0

Assertion

Ref	Expression
3dvds2decOLD	|- ( 3 || ;; A B C <-> 3 || ( ( A + B ) + C ) )

Proof of Theorem 3dvds2decOLD

Step	Hyp	Ref	Expression
1		3dvdsdec.a	. . . . 5 |- A e. NN0
2		3dvdsdec.b	. . . . 5 |- B e. NN0
3	1, 2	3decOLD 13053	. . . 4 |- ;; A B C = ( ( ( ( 10 ^ 2 ) x. A ) + ( 10 x. B ) ) + C )
4		sq10e99m1OLD 13052	. . . . . . . 8 |- ( 10 ^ 2 ) = (; 9 9 + 1 )
5	4	oveq1i 6660	. . . . . . 7 |- ( ( 10 ^ 2 ) x. A ) = ( (; 9 9 + 1 ) x. A )
6		9nn0 11316	. . . . . . . . . 10 |- 9 e. NN0
7	6, 6	deccl 11512	. . . . . . . . 9 |- ; 9 9 e. NN0
8	7	nn0cni 11304	. . . . . . . 8 |- ; 9 9 e. CC
9		ax-1cn 9994	. . . . . . . 8 |- 1 e. CC
10	1	nn0cni 11304	. . . . . . . 8 |- A e. CC
11	8, 9, 10	adddiri 10051	. . . . . . 7 |- ( (; 9 9 + 1 ) x. A ) = ( (; 9 9 x. A ) + ( 1 x. A ) )
12	10	mulid2i 10043	. . . . . . . 8 |- ( 1 x. A ) = A
13	12	oveq2i 6661	. . . . . . 7 |- ( (; 9 9 x. A ) + ( 1 x. A ) ) = ( (; 9 9 x. A ) + A )
14	5, 11, 13	3eqtri 2648	. . . . . 6 |- ( ( 10 ^ 2 ) x. A ) = ( (; 9 9 x. A ) + A )
15		df-10OLD 11087	. . . . . . . 8 |- 10 = ( 9 + 1 )
16	15	oveq1i 6660	. . . . . . 7 |- ( 10 x. B ) = ( ( 9 + 1 ) x. B )
17		9cn 11108	. . . . . . . 8 |- 9 e. CC
18	2	nn0cni 11304	. . . . . . . 8 |- B e. CC
19	17, 9, 18	adddiri 10051	. . . . . . 7 |- ( ( 9 + 1 ) x. B ) = ( ( 9 x. B ) + ( 1 x. B ) )
20	18	mulid2i 10043	. . . . . . . 8 |- ( 1 x. B ) = B
21	20	oveq2i 6661	. . . . . . 7 |- ( ( 9 x. B ) + ( 1 x. B ) ) = ( ( 9 x. B ) + B )
22	16, 19, 21	3eqtri 2648	. . . . . 6 |- ( 10 x. B ) = ( ( 9 x. B ) + B )
23	14, 22	oveq12i 6662	. . . . 5 |- ( ( ( 10 ^ 2 ) x. A ) + ( 10 x. B ) ) = ( ( (; 9 9 x. A ) + A ) + ( ( 9 x. B ) + B ) )
24	23	oveq1i 6660	. . . 4 |- ( ( ( ( 10 ^ 2 ) x. A ) + ( 10 x. B ) ) + C ) = ( ( ( (; 9 9 x. A ) + A ) + ( ( 9 x. B ) + B ) ) + C )
25	8, 10	mulcli 10045	. . . . . 6 |- (; 9 9 x. A ) e. CC
26	17, 18	mulcli 10045	. . . . . 6 |- ( 9 x. B ) e. CC
27		add4 10256	. . . . . . 7 |- ( ( ( (; 9 9 x. A ) e. CC /\ A e. CC ) /\ ( ( 9 x. B ) e. CC /\ B e. CC ) ) -> ( ( (; 9 9 x. A ) + A ) + ( ( 9 x. B ) + B ) ) = ( ( (; 9 9 x. A ) + ( 9 x. B ) ) + ( A + B ) ) )
28	27	oveq1d 6665	. . . . . 6 |- ( ( ( (; 9 9 x. A ) e. CC /\ A e. CC ) /\ ( ( 9 x. B ) e. CC /\ B e. CC ) ) -> ( ( ( (; 9 9 x. A ) + A ) + ( ( 9 x. B ) + B ) ) + C ) = ( ( ( (; 9 9 x. A ) + ( 9 x. B ) ) + ( A + B ) ) + C ) )
29	25, 10, 26, 18, 28	mp4an 709	. . . . 5 |- ( ( ( (; 9 9 x. A ) + A ) + ( ( 9 x. B ) + B ) ) + C ) = ( ( ( (; 9 9 x. A ) + ( 9 x. B ) ) + ( A + B ) ) + C )
30	25, 26	addcli 10044	. . . . . 6 |- ( (; 9 9 x. A ) + ( 9 x. B ) ) e. CC
31	10, 18	addcli 10044	. . . . . 6 |- ( A + B ) e. CC
32		3dvds2dec.c	. . . . . . 7 |- C e. NN0
33	32	nn0cni 11304	. . . . . 6 |- C e. CC
34	30, 31, 33	addassi 10048	. . . . 5 |- ( ( ( (; 9 9 x. A ) + ( 9 x. B ) ) + ( A + B ) ) + C ) = ( ( (; 9 9 x. A ) + ( 9 x. B ) ) + ( ( A + B ) + C ) )
35		9t11e99 11671	. . . . . . . . . . 11 |- ( 9 x. ; 1 1 ) = ; 9 9
36	35	eqcomi 2631	. . . . . . . . . 10 |- ; 9 9 = ( 9 x. ; 1 1 )
37	36	oveq1i 6660	. . . . . . . . 9 |- (; 9 9 x. A ) = ( ( 9 x. ; 1 1 ) x. A )
38		1nn0 11308	. . . . . . . . . . . 12 |- 1 e. NN0
39	38, 38	deccl 11512	. . . . . . . . . . 11 |- ; 1 1 e. NN0
40	39	nn0cni 11304	. . . . . . . . . 10 |- ; 1 1 e. CC
41	17, 40, 10	mulassi 10049	. . . . . . . . 9 |- ( ( 9 x. ; 1 1 ) x. A ) = ( 9 x. (; 1 1 x. A ) )
42	37, 41	eqtri 2644	. . . . . . . 8 |- (; 9 9 x. A ) = ( 9 x. (; 1 1 x. A ) )
43	42	oveq1i 6660	. . . . . . 7 |- ( (; 9 9 x. A ) + ( 9 x. B ) ) = ( ( 9 x. (; 1 1 x. A ) ) + ( 9 x. B ) )
44	40, 10	mulcli 10045	. . . . . . . . 9 |- (; 1 1 x. A ) e. CC
45	17, 44, 18	adddii 10050	. . . . . . . 8 |- ( 9 x. ( (; 1 1 x. A ) + B ) ) = ( ( 9 x. (; 1 1 x. A ) ) + ( 9 x. B ) )
46	45	eqcomi 2631	. . . . . . 7 |- ( ( 9 x. (; 1 1 x. A ) ) + ( 9 x. B ) ) = ( 9 x. ( (; 1 1 x. A ) + B ) )
47		3t3e9 11180	. . . . . . . . . 10 |- ( 3 x. 3 ) = 9
48	47	eqcomi 2631	. . . . . . . . 9 |- 9 = ( 3 x. 3 )
49	48	oveq1i 6660	. . . . . . . 8 |- ( 9 x. ( (; 1 1 x. A ) + B ) ) = ( ( 3 x. 3 ) x. ( (; 1 1 x. A ) + B ) )
50		3cn 11095	. . . . . . . . 9 |- 3 e. CC
51	44, 18	addcli 10044	. . . . . . . . 9 |- ( (; 1 1 x. A ) + B ) e. CC
52	50, 50, 51	mulassi 10049	. . . . . . . 8 |- ( ( 3 x. 3 ) x. ( (; 1 1 x. A ) + B ) ) = ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) )
53	49, 52	eqtri 2644	. . . . . . 7 |- ( 9 x. ( (; 1 1 x. A ) + B ) ) = ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) )
54	43, 46, 53	3eqtri 2648	. . . . . 6 |- ( (; 9 9 x. A ) + ( 9 x. B ) ) = ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) )
55	54	oveq1i 6660	. . . . 5 |- ( ( (; 9 9 x. A ) + ( 9 x. B ) ) + ( ( A + B ) + C ) ) = ( ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) ) + ( ( A + B ) + C ) )
56	29, 34, 55	3eqtri 2648	. . . 4 |- ( ( ( (; 9 9 x. A ) + A ) + ( ( 9 x. B ) + B ) ) + C ) = ( ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) ) + ( ( A + B ) + C ) )
57	3, 24, 56	3eqtri 2648	. . 3 |- ;; A B C = ( ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) ) + ( ( A + B ) + C ) )
58	57	breq2i 4661	. 2 |- ( 3 || ;; A B C <-> 3 || ( ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) ) + ( ( A + B ) + C ) ) )
59		3z 11410	. . 3 |- 3 e. ZZ
60	1	nn0zi 11402	. . . . 5 |- A e. ZZ
61	2	nn0zi 11402	. . . . 5 |- B e. ZZ
62		zaddcl 11417	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A + B ) e. ZZ )
63	60, 61, 62	mp2an 708	. . . 4 |- ( A + B ) e. ZZ
64	32	nn0zi 11402	. . . 4 |- C e. ZZ
65		zaddcl 11417	. . . 4 |- ( ( ( A + B ) e. ZZ /\ C e. ZZ ) -> ( ( A + B ) + C ) e. ZZ )
66	63, 64, 65	mp2an 708	. . 3 |- ( ( A + B ) + C ) e. ZZ
67	39	nn0zi 11402	. . . . . . . 8 |- ; 1 1 e. ZZ
68		zmulcl 11426	. . . . . . . 8 |- ( (; 1 1 e. ZZ /\ A e. ZZ ) -> (; 1 1 x. A ) e. ZZ )
69	67, 60, 68	mp2an 708	. . . . . . 7 |- (; 1 1 x. A ) e. ZZ
70		zaddcl 11417	. . . . . . 7 |- ( ( (; 1 1 x. A ) e. ZZ /\ B e. ZZ ) -> ( (; 1 1 x. A ) + B ) e. ZZ )
71	69, 61, 70	mp2an 708	. . . . . 6 |- ( (; 1 1 x. A ) + B ) e. ZZ
72		zmulcl 11426	. . . . . 6 |- ( ( 3 e. ZZ /\ ( (; 1 1 x. A ) + B ) e. ZZ ) -> ( 3 x. ( (; 1 1 x. A ) + B ) ) e. ZZ )
73	59, 71, 72	mp2an 708	. . . . 5 |- ( 3 x. ( (; 1 1 x. A ) + B ) ) e. ZZ
74		zmulcl 11426	. . . . 5 |- ( ( 3 e. ZZ /\ ( 3 x. ( (; 1 1 x. A ) + B ) ) e. ZZ ) -> ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) ) e. ZZ )
75	59, 73, 74	mp2an 708	. . . 4 |- ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) ) e. ZZ
76		dvdsmul1 15003	. . . . 5 |- ( ( 3 e. ZZ /\ ( 3 x. ( (; 1 1 x. A ) + B ) ) e. ZZ ) -> 3 || ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) ) )
77	59, 73, 76	mp2an 708	. . . 4 |- 3 || ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) )
78	75, 77	pm3.2i 471	. . 3 |- ( ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) ) e. ZZ /\ 3 || ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) ) )
79		dvdsadd2b 15028	. . 3 |- ( ( 3 e. ZZ /\ ( ( A + B ) + C ) e. ZZ /\ ( ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) ) e. ZZ /\ 3 || ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) ) ) ) -> ( 3 || ( ( A + B ) + C ) <-> 3 || ( ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) ) + ( ( A + B ) + C ) ) ) )
80	59, 66, 78, 79	mp3an 1424	. 2 |- ( 3 || ( ( A + B ) + C ) <-> 3 || ( ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) ) + ( ( A + B ) + C ) ) )
81	58, 80	bitr4i 267	1 |- ( 3 || ;; A B C <-> 3 || ( ( A + B ) + C ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   class class class wbr 4653  (class class class)co 6650   CCcc 9934   1c1 9937   + caddc 9939   x. cmul 9941   2c2 11070   3c3 11071   9c9 11077   10c10 11078   NN0cn0 11292   ZZcz 11377  ;cdc 11493   ^cexp 12860   || cdvds 14983

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

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-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-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-10OLD 11087  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-seq 12802  df-exp 12861  df-dvds 14984

This theorem is referenced by: (None)