Theorem dvdsaddre2b 12527

Description: Adding a multiple of the base does not affect divisibility. Variant of dvdsadd2b 12526 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		dvdszrcl 12478	. . . 4 |- ( A || B -> ( A e. ZZ /\ B e. ZZ ) )
2	1	simprd 114	. . 3 |- ( A || B -> B e. ZZ )
3	2	a1i 9	. 2 |- ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) -> ( A || B -> B e. ZZ ) )
4		simpl3l 1079	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> C e. ZZ )
5	4	zcnd 9701	. . . . 5 |- ( ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> C e. CC )
6		simpl2 1028	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> B e. RR )
7	6	recnd 8302	. . . . 5 |- ( ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> B e. CC )
8	5, 7	pncan2d 8586	. . . 4 |- ( ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> ( ( C + B ) - C ) = B )
9		dvdszrcl 12478	. . . . . . 7 |- ( A || ( C + B ) -> ( A e. ZZ /\ ( C + B ) e. ZZ ) )
10	9	simprd 114	. . . . . 6 |- ( A || ( C + B ) -> ( C + B ) e. ZZ )
11	10	adantl 277	. . . . 5 |- ( ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> ( C + B ) e. ZZ )
12	11, 4	zsubcld 9705	. . . 4 |- ( ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> ( ( C + B ) - C ) e. ZZ )
13	8, 12	eqeltrrd 2310	. . 3 |- ( ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> B e. ZZ )
14	13	ex 115	. 2 |- ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) -> ( A || ( C + B ) -> B e. ZZ ) )
15		dvdsadd2b 12526	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) -> ( A || B <-> A || ( C + B ) ) )
16	15	a1d 22	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) -> ( B e. RR -> ( A || B <-> A || ( C + B ) ) ) )
17	16	3exp 1229	. . . 4 |- ( A e. ZZ -> ( B e. ZZ -> ( ( C e. ZZ /\ A || C ) -> ( B e. RR -> ( A || B <-> A || ( C + B ) ) ) ) ) )
18	17	com24 87	. . 3 |- ( A e. ZZ -> ( B e. RR -> ( ( C e. ZZ /\ A || C ) -> ( B e. ZZ -> ( A || B <-> A || ( C + B ) ) ) ) ) )
19	18	3imp 1220	. 2 |- ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) -> ( B e. ZZ -> ( A || B <-> A || ( C + B ) ) ) )
20	3, 14, 19	pm5.21ndd 713	1 |- ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) -> ( A || B <-> A || ( C + B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   e. wcel 2203   class class class wbr 4109  (class class class)co 6050   RRcr 8126   + caddc 8130   - cmin 8444   ZZcz 9577   || cdvds 12473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-n0 9497  df-z 9578  df-dvds 12474

This theorem is referenced by:  2lgsoddprmlem2  15979