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Mirrors > Home > MPE Home > Th. List > dvdsr2 | Structured version Visualization version GIF version |
Description: Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.) |
Ref | Expression |
---|---|
dvdsr.1 | โข ๐ต = (Baseโ๐ ) |
dvdsr.2 | โข โฅ = (โฅrโ๐ ) |
dvdsr.3 | โข ยท = (.rโ๐ ) |
Ref | Expression |
---|---|
dvdsr2 | โข (๐ โ ๐ต โ (๐ โฅ ๐ โ โ๐ง โ ๐ต (๐ง ยท ๐) = ๐)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvdsr.1 | . . 3 โข ๐ต = (Baseโ๐ ) | |
2 | dvdsr.2 | . . 3 โข โฅ = (โฅrโ๐ ) | |
3 | dvdsr.3 | . . 3 โข ยท = (.rโ๐ ) | |
4 | 1, 2, 3 | dvdsr 20083 | . 2 โข (๐ โฅ ๐ โ (๐ โ ๐ต โง โ๐ง โ ๐ต (๐ง ยท ๐) = ๐)) |
5 | 4 | baib 537 | 1 โข (๐ โ ๐ต โ (๐ โฅ ๐ โ โ๐ง โ ๐ต (๐ง ยท ๐) = ๐)) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โ wb 205 = wceq 1542 โ wcel 2107 โwrex 3070 class class class wbr 5109 โcfv 6500 (class class class)co 7361 Basecbs 17091 .rcmulr 17142 โฅrcdsr 20075 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fv 6508 df-ov 7364 df-dvdsr 20078 |
This theorem is referenced by: dvdsr01 20092 dvdsr02 20093 unitgrp 20104 rhmdvdsr 20191 rspsn 20769 znunit 20993 dvdsq1p 25548 |
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