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Mirrors > Home > MPE Home > Th. List > dvdsr | Structured version Unicode version |
Description: Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.) |
Ref | Expression |
---|---|
dvdsr.1 |
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dvdsr.2 |
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dvdsr.3 |
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Ref | Expression |
---|---|
dvdsr |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvdsr.2 |
. . . 4
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2 | 1 | reldvdsr 16862 |
. . 3
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3 | brrelex12 4987 |
. . 3
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4 | 2, 3 | mpan 670 |
. 2
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5 | elex 3087 |
. . 3
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6 | id 22 |
. . . . 5
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7 | ovex 6228 |
. . . . 5
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8 | 6, 7 | syl6eqelr 2551 |
. . . 4
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9 | 8 | rexlimivw 2943 |
. . 3
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10 | 5, 9 | anim12i 566 |
. 2
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11 | simpl 457 |
. . . . 5
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12 | 11 | eleq1d 2523 |
. . . 4
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13 | 11 | oveq2d 6219 |
. . . . . 6
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14 | simpr 461 |
. . . . . 6
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15 | 13, 14 | eqeq12d 2476 |
. . . . 5
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16 | 15 | rexbidv 2868 |
. . . 4
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17 | 12, 16 | anbi12d 710 |
. . 3
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18 | dvdsr.1 |
. . . 4
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19 | dvdsr.3 |
. . . 4
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20 | 18, 1, 19 | dvdsrval 16863 |
. . 3
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21 | 17, 20 | brabga 4714 |
. 2
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22 | 4, 10, 21 | pm5.21nii 353 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-rep 4514 ax-sep 4524 ax-nul 4532 ax-pow 4581 ax-pr 4642 ax-un 6485 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2650 df-ral 2804 df-rex 2805 df-reu 2806 df-rab 2808 df-v 3080 df-sbc 3295 df-csb 3399 df-dif 3442 df-un 3444 df-in 3446 df-ss 3453 df-nul 3749 df-if 3903 df-pw 3973 df-sn 3989 df-pr 3991 df-op 3995 df-uni 4203 df-iun 4284 df-br 4404 df-opab 4462 df-mpt 4463 df-id 4747 df-xp 4957 df-rel 4958 df-cnv 4959 df-co 4960 df-dm 4961 df-rn 4962 df-res 4963 df-ima 4964 df-iota 5492 df-fun 5531 df-fn 5532 df-f 5533 df-f1 5534 df-fo 5535 df-f1o 5536 df-fv 5537 df-ov 6206 df-dvdsr 16859 |
This theorem is referenced by: dvdsr2 16865 dvdsrmul 16866 dvdsrcl 16867 dvdsrcl2 16868 dvdsrtr 16870 dvdsrmul1 16871 opprunit 16879 crngunit 16880 subrgdvds 17005 rhmdvdsr 26451 |
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