Theorem dvdsaddre2b 12267

Description: Adding a multiple of the base does not affect divisibility. Variant of dvdsadd2b 12266 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 12218	. . . 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 1055	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> C e. ZZ )
5	4	zcnd 9531	. . . . 5 |- ( ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> C e. CC )
6		simpl2 1004	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> B e. RR )
7	6	recnd 8136	. . . . 5 |- ( ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> B e. CC )
8	5, 7	pncan2d 8420	. . . 4 |- ( ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> ( ( C + B ) - C ) = B )
9		dvdszrcl 12218	. . . . . . 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 9535	. . . 4 |- ( ( ( A e. ZZ /\ B e. RR /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> ( ( C + B ) - C ) e. ZZ )
13	8, 12	eqeltrrd 2285	. . 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 12266	. . . . . 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 1205	. . . 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 1196	. 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 707	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 981   e. wcel 2178   class class class wbr 4059  (class class class)co 5967   RRcr 7959   + caddc 7963   - cmin 8278   ZZcz 9407   || cdvds 12213

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-iota 5251  df-fun 5292  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-inn 9072  df-n0 9331  df-z 9408  df-dvds 12214

This theorem is referenced by:  2lgsoddprmlem2  15698