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Mirrors > Home > ILE Home > Th. List > divconjdvds | Unicode version |
Description: If a nonzero integer divides another integer , the other integer divided by the nonzero integer (i.e. the divisor conjugate of to ) divides the other integer . Theorem 1.1(k) in [ApostolNT] p. 14. (Contributed by AV, 7-Aug-2021.) |
Ref | Expression |
---|---|
divconjdvds |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvdszrcl 11498 | . . 3 | |
2 | simpll 518 | . . . . . . . 8 | |
3 | oveq1 5781 | . . . . . . . . . 10 | |
4 | 3 | eqeq1d 2148 | . . . . . . . . 9 |
5 | 4 | adantl 275 | . . . . . . . 8 |
6 | zcn 9059 | . . . . . . . . . . 11 | |
7 | 6 | adantl 275 | . . . . . . . . . 10 |
8 | 7 | adantr 274 | . . . . . . . . 9 |
9 | zcn 9059 | . . . . . . . . . . 11 | |
10 | 9 | adantr 274 | . . . . . . . . . 10 |
11 | 10 | adantr 274 | . . . . . . . . 9 |
12 | 0z 9065 | . . . . . . . . . . . 12 | |
13 | zapne 9125 | . . . . . . . . . . . 12 # | |
14 | 12, 13 | mpan2 421 | . . . . . . . . . . 11 # |
15 | 14 | adantr 274 | . . . . . . . . . 10 # |
16 | 15 | biimpar 295 | . . . . . . . . 9 # |
17 | 8, 11, 16 | divcanap2d 8552 | . . . . . . . 8 |
18 | 2, 5, 17 | rspcedvd 2795 | . . . . . . 7 |
19 | 18 | adantr 274 | . . . . . 6 |
20 | simpr 109 | . . . . . . . 8 | |
21 | simpr 109 | . . . . . . . . . . 11 | |
22 | simpr 109 | . . . . . . . . . . . 12 | |
23 | 22 | adantr 274 | . . . . . . . . . . 11 |
24 | 2, 21, 23 | 3jca 1161 | . . . . . . . . . 10 |
25 | 24 | adantr 274 | . . . . . . . . 9 |
26 | dvdsval2 11496 | . . . . . . . . 9 | |
27 | 25, 26 | syl 14 | . . . . . . . 8 |
28 | 20, 27 | mpbid 146 | . . . . . . 7 |
29 | 23 | adantr 274 | . . . . . . 7 |
30 | divides 11495 | . . . . . . 7 | |
31 | 28, 29, 30 | syl2anc 408 | . . . . . 6 |
32 | 19, 31 | mpbird 166 | . . . . 5 |
33 | 32 | exp31 361 | . . . 4 |
34 | 33 | com3r 79 | . . 3 |
35 | 1, 34 | mpd 13 | . 2 |
36 | 35 | imp 123 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wceq 1331 wcel 1480 wne 2308 wrex 2417 class class class wbr 3929 (class class class)co 5774 cc 7618 cc0 7620 cmul 7625 # cap 8343 cdiv 8432 cz 9054 cdvds 11493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-id 4215 df-po 4218 df-iso 4219 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-n0 8978 df-z 9055 df-dvds 11494 |
This theorem is referenced by: dvdsdivcl 11548 |
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