Theorem dvdsaddre2b 12586

Description: Adding a multiple of the base does not affect divisibility. Variant of dvdsadd2b 12585 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 12537	. . . 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 1083	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> C e. ZZ )
5	4	zcnd 9748	. . . . 5 |- ( ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> C e. CC )
6		simpl2 1032	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> B e. RR )
7	6	recnd 8344	. . . . 5 |- ( ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> B e. CC )
8	5, 7	pncan2d 8629	. . . 4 |- ( ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> ( ( C + B ) - C ) = B )
9		dvdszrcl 12537	. . . . . . 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 9752	. . . 4 |- ( ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> ( ( C + B ) - C ) e. ZZ )
13	8, 12	eqeltrrd 2316	. . 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 12585	. . . . . 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 1233	. . . 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 1224	. 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 717	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 1009   e. wcel 2209   class class class wbr 4125  (class class class)co 6075   RRcr 8168   + caddc 8172   - cmin 8487   ZZcz 9623   || cdvds 12532

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-n0 9543  df-z 9624  df-dvds 12533

This theorem is referenced by:  2lgsoddprmlem2  16139