Theorem dvdsaddre2b 15029

Description: Adding a multiple of the base does not affect divisibility. Variant of dvdsadd2b 15028 only requiring B to be a real number (not necessarily an integer). (Contributed by AV, 19-Jul-2021.)

Assertion

Ref	Expression
dvdsaddre2b	|- ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) -> ( A || B <-> A || ( C + B ) ) )

Proof of Theorem dvdsaddre2b

Step	Hyp	Ref	Expression
1		dvdsadd2b 15028	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) -> ( A || B <-> A || ( C + B ) ) )
2	1	a1d 25	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) -> ( B e. RR -> ( A || B <-> A || ( C + B ) ) ) )
3	2	3exp 1264	. . . . 5 |- ( A e. ZZ -> ( B e. ZZ -> ( ( C e. ZZ /\ A || C ) -> ( B e. RR -> ( A || B <-> A || ( C + B ) ) ) ) ) )
4	3	com24 95	. . . 4 |- ( A e. ZZ -> ( B e. RR -> ( ( C e. ZZ /\ A || C ) -> ( B e. ZZ -> ( A || B <-> A || ( C + B ) ) ) ) ) )
5	4	3imp 1256	. . 3 |- ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) -> ( B e. ZZ -> ( A || B <-> A || ( C + B ) ) ) )
6	5	com12 32	. 2 |- ( B e. ZZ -> ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) -> ( A || B <-> A || ( C + B ) ) ) )
7		dvdszrcl 14988	. . . . . . 7 |- ( A || B -> ( A e. ZZ /\ B e. ZZ ) )
8		pm2.24 121	. . . . . . 7 |- ( B e. ZZ -> ( -. B e. ZZ -> A || ( C + B ) ) )
9	7, 8	simpl2im 658	. . . . . 6 |- ( A || B -> ( -. B e. ZZ -> A || ( C + B ) ) )
10	9	com12 32	. . . . 5 |- ( -. B e. ZZ -> ( A || B -> A || ( C + B ) ) )
11	10	adantr 481	. . . 4 |- ( ( -. B e. ZZ /\ ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) ) -> ( A || B -> A || ( C + B ) ) )
12		dvdszrcl 14988	. . . . . 6 |- ( A || ( C + B ) -> ( A e. ZZ /\ ( C + B ) e. ZZ ) )
13		zcn 11382	. . . . . . . . . . . . . . . . 17 |- ( C e. ZZ -> C e. CC )
14	13	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( C e. ZZ /\ ( B e. RR /\ -. B e. ZZ ) ) -> C e. CC )
15		recn 10026	. . . . . . . . . . . . . . . . 17 |- ( B e. RR -> B e. CC )
16	15	ad2antrl 764	. . . . . . . . . . . . . . . 16 |- ( ( C e. ZZ /\ ( B e. RR /\ -. B e. ZZ ) ) -> B e. CC )
17	14, 16	addcomd 10238	. . . . . . . . . . . . . . 15 |- ( ( C e. ZZ /\ ( B e. RR /\ -. B e. ZZ ) ) -> ( C + B ) = ( B + C ) )
18		eldif 3584	. . . . . . . . . . . . . . . . 17 |- ( B e. ( RR \ ZZ ) <-> ( B e. RR /\ -. B e. ZZ ) )
19		nzadd 11425	. . . . . . . . . . . . . . . . . . 19 |- ( ( B e. ( RR \ ZZ ) /\ C e. ZZ ) -> ( B + C ) e. ( RR \ ZZ ) )
20	19	eldifbd 3587	. . . . . . . . . . . . . . . . . 18 |- ( ( B e. ( RR \ ZZ ) /\ C e. ZZ ) -> -. ( B + C ) e. ZZ )
21	20	expcom 451	. . . . . . . . . . . . . . . . 17 |- ( C e. ZZ -> ( B e. ( RR \ ZZ ) -> -. ( B + C ) e. ZZ ) )
22	18, 21	syl5bir 233	. . . . . . . . . . . . . . . 16 |- ( C e. ZZ -> ( ( B e. RR /\ -. B e. ZZ ) -> -. ( B + C ) e. ZZ ) )
23	22	imp 445	. . . . . . . . . . . . . . 15 |- ( ( C e. ZZ /\ ( B e. RR /\ -. B e. ZZ ) ) -> -. ( B + C ) e. ZZ )
24	17, 23	eqneltrd 2720	. . . . . . . . . . . . . 14 |- ( ( C e. ZZ /\ ( B e. RR /\ -. B e. ZZ ) ) -> -. ( C + B ) e. ZZ )
25	24	exp32 631	. . . . . . . . . . . . 13 |- ( C e. ZZ -> ( B e. RR -> ( -. B e. ZZ -> -. ( C + B ) e. ZZ ) ) )
26		pm2.21 120	. . . . . . . . . . . . 13 |- ( -. ( C + B ) e. ZZ -> ( ( C + B ) e. ZZ -> A || B ) )
27	25, 26	syl8 76	. . . . . . . . . . . 12 |- ( C e. ZZ -> ( B e. RR -> ( -. B e. ZZ -> ( ( C + B ) e. ZZ -> A || B ) ) ) )
28	27	adantr 481	. . . . . . . . . . 11 |- ( ( C e. ZZ /\ A || C ) -> ( B e. RR -> ( -. B e. ZZ -> ( ( C + B ) e. ZZ -> A || B ) ) ) )
29	28	com12 32	. . . . . . . . . 10 |- ( B e. RR -> ( ( C e. ZZ /\ A || C ) -> ( -. B e. ZZ -> ( ( C + B ) e. ZZ -> A || B ) ) ) )
30	29	a1i 11	. . . . . . . . 9 |- ( A e. ZZ -> ( B e. RR -> ( ( C e. ZZ /\ A || C ) -> ( -. B e. ZZ -> ( ( C + B ) e. ZZ -> A || B ) ) ) ) )
31	30	3imp 1256	. . . . . . . 8 |- ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) -> ( -. B e. ZZ -> ( ( C + B ) e. ZZ -> A || B ) ) )
32	31	impcom 446	. . . . . . 7 |- ( ( -. B e. ZZ /\ ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) ) -> ( ( C + B ) e. ZZ -> A || B ) )
33	32	com12 32	. . . . . 6 |- ( ( C + B ) e. ZZ -> ( ( -. B e. ZZ /\ ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) ) -> A || B ) )
34	12, 33	simpl2im 658	. . . . 5 |- ( A || ( C + B ) -> ( ( -. B e. ZZ /\ ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) ) -> A || B ) )
35	34	com12 32	. . . 4 |- ( ( -. B e. ZZ /\ ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) ) -> ( A || ( C + B ) -> A || B ) )
36	11, 35	impbid 202	. . 3 |- ( ( -. B e. ZZ /\ ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) ) -> ( A || B <-> A || ( C + B ) ) )
37	36	ex 450	. 2 |- ( -. B e. ZZ -> ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) -> ( A || B <-> A || ( C + B ) ) ) )
38	6, 37	pm2.61i 176	1 |- ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) -> ( A || B <-> A || ( C + B ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   e. wcel 1990   \ cdif 3571   class class class wbr 4653  (class class class)co 6650   CCcc 9934   RRcr 9935   + caddc 9939   ZZcz 11377   || 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-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-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-n0 11293  df-z 11378  df-dvds 14984

This theorem is referenced by:  2lgsoddprmlem2  25134