Theorem dvdsaddre2b 12552

Description: Adding a multiple of the base does not affect divisibility. Variant of dvdsadd2b 12551 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 12503	. . . 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 9719	. . . . 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 8318	. . . . 5 |- ( ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> B e. CC )
8	5, 7	pncan2d 8602	. . . 4 |- ( ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> ( ( C + B ) - C ) = B )
9		dvdszrcl 12503	. . . . . . 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 9723	. . . 4 |- ( ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> ( ( C + B ) - C ) e. ZZ )
13	8, 12	eqeltrrd 2312	. . 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 12551	. . . . . 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 2205   class class class wbr 4114  (class class class)co 6058   RRcr 8142   + caddc 8146   - cmin 8460   ZZcz 9594   || cdvds 12498

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-ltadd 8259

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-iota 5317  df-fun 5359  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-n0 9514  df-z 9595  df-dvds 12499

This theorem is referenced by:  2lgsoddprmlem2  16105