Theorem dvdsaddre2b 12367

Description: Adding a multiple of the base does not affect divisibility. Variant of dvdsadd2b 12366 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 12318	. . . 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 1076	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> C e. ZZ )
5	4	zcnd 9581	. . . . 5 |- ( ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> C e. CC )
6		simpl2 1025	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> B e. RR )
7	6	recnd 8186	. . . . 5 |- ( ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> B e. CC )
8	5, 7	pncan2d 8470	. . . 4 |- ( ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> ( ( C + B ) - C ) = B )
9		dvdszrcl 12318	. . . . . . 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 9585	. . . 4 |- ( ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> ( ( C + B ) - C ) e. ZZ )
13	8, 12	eqeltrrd 2307	. . 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 12366	. . . . . 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 1226	. . . 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 1217	. 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 710	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 1002   e. wcel 2200   class class class wbr 4083  (class class class)co 6007   RRcr 8009   + caddc 8013   - cmin 8328   ZZcz 9457   || cdvds 12313

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-0id 8118  ax-rnegex 8119  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-ltadd 8126

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-inn 9122  df-n0 9381  df-z 9458  df-dvds 12314

This theorem is referenced by:  2lgsoddprmlem2  15800