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Mirrors > Home > MPE Home > Th. List > dvdsr | Structured version Visualization version Unicode version |
Description: Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.) |
Ref | Expression |
---|---|
dvdsr.1 | |
dvdsr.2 | r |
dvdsr.3 |
Ref | Expression |
---|---|
dvdsr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvdsr.2 | . . . 4 r | |
2 | 1 | reldvdsr 18644 | . . 3 |
3 | brrelex12 5155 | . . 3 | |
4 | 2, 3 | mpan 706 | . 2 |
5 | elex 3212 | . . 3 | |
6 | id 22 | . . . . 5 | |
7 | ovex 6678 | . . . . 5 | |
8 | 6, 7 | syl6eqelr 2710 | . . . 4 |
9 | 8 | rexlimivw 3029 | . . 3 |
10 | 5, 9 | anim12i 590 | . 2 |
11 | simpl 473 | . . . . 5 | |
12 | 11 | eleq1d 2686 | . . . 4 |
13 | 11 | oveq2d 6666 | . . . . . 6 |
14 | simpr 477 | . . . . . 6 | |
15 | 13, 14 | eqeq12d 2637 | . . . . 5 |
16 | 15 | rexbidv 3052 | . . . 4 |
17 | 12, 16 | anbi12d 747 | . . 3 |
18 | dvdsr.1 | . . . 4 | |
19 | dvdsr.3 | . . . 4 | |
20 | 18, 1, 19 | dvdsrval 18645 | . . 3 |
21 | 17, 20 | brabga 4989 | . 2 |
22 | 4, 10, 21 | pm5.21nii 368 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wb 196 wa 384 wceq 1483 wcel 1990 wrex 2913 cvv 3200 class class class wbr 4653 wrel 5119 cfv 5888 (class class class)co 6650 cbs 15857 cmulr 15942 rcdsr 18638 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-dvdsr 18641 |
This theorem is referenced by: dvdsr2 18647 dvdsrmul 18648 dvdsrcl 18649 dvdsrcl2 18650 dvdsrtr 18652 dvdsrmul1 18653 opprunit 18661 crngunit 18662 subrgdvds 18794 rhmdvdsr 29818 |
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