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Mirrors > Home > MPE Home > Th. List > dvdsrval | Structured version Visualization version Unicode version |
Description: Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 6-Jan-2015.) |
Ref | Expression |
---|---|
dvdsr.1 |
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dvdsr.2 |
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dvdsr.3 |
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Ref | Expression |
---|---|
dvdsrval |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvdsr.2 |
. . 3
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2 | fveq2 5847 |
. . . . . . . . 9
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3 | dvdsr.1 |
. . . . . . . . 9
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4 | 2, 3 | syl6eqr 2503 |
. . . . . . . 8
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5 | 4 | eleq2d 2514 |
. . . . . . 7
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6 | 4 | rexeqdv 2961 |
. . . . . . 7
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7 | 5, 6 | anbi12d 722 |
. . . . . 6
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8 | fveq2 5847 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
9 | dvdsr.3 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
10 | 8, 9 | syl6eqr 2503 |
. . . . . . . . . 10
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11 | 10 | oveqd 6292 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
12 | 11 | eqeq1d 2453 |
. . . . . . . 8
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13 | 12 | rexbidv 2872 |
. . . . . . 7
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14 | 13 | anbi2d 715 |
. . . . . 6
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15 | 7, 14 | bitrd 261 |
. . . . 5
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16 | 15 | opabbidv 4437 |
. . . 4
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17 | df-dvdsr 17879 |
. . . 4
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18 | fvex 5857 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
19 | 3, 18 | eqeltri 2525 |
. . . . 5
![]() ![]() ![]() ![]() |
20 | eqcom 2458 |
. . . . . . . . 9
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21 | 20 | rexbii 2861 |
. . . . . . . 8
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22 | 21 | abbii 2567 |
. . . . . . 7
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23 | 19 | abrexex 6754 |
. . . . . . 7
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24 | 22, 23 | eqeltri 2525 |
. . . . . 6
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25 | 24 | a1i 11 |
. . . . 5
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26 | 19, 25 | opabex3 6759 |
. . . 4
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27 | 16, 17, 26 | fvmpt 5931 |
. . 3
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28 | 1, 27 | syl5eq 2497 |
. 2
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29 | fvprc 5841 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
30 | 1, 29 | syl5eq 2497 |
. . 3
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31 | opabn0 4704 |
. . . . 5
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32 | n0i 3703 |
. . . . . . . 8
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33 | fvprc 5841 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
34 | 3, 33 | syl5eq 2497 |
. . . . . . . 8
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35 | 32, 34 | nsyl2 132 |
. . . . . . 7
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36 | 35 | adantr 471 |
. . . . . 6
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37 | 36 | exlimivv 1781 |
. . . . 5
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38 | 31, 37 | sylbi 200 |
. . . 4
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39 | 38 | necon1bi 2651 |
. . 3
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40 | 30, 39 | eqtr4d 2488 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
41 | 28, 40 | pm2.61i 169 |
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 1672 ax-4 1685 ax-5 1761 ax-6 1808 ax-7 1854 ax-8 1892 ax-9 1899 ax-10 1918 ax-11 1923 ax-12 1936 ax-13 2091 ax-ext 2431 ax-rep 4486 ax-sep 4496 ax-nul 4505 ax-pow 4553 ax-pr 4611 ax-un 6570 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3an 988 df-tru 1450 df-ex 1667 df-nf 1671 df-sb 1801 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2623 df-ral 2741 df-rex 2742 df-reu 2743 df-rab 2745 df-v 3014 df-sbc 3235 df-csb 3331 df-dif 3374 df-un 3376 df-in 3378 df-ss 3385 df-nul 3699 df-if 3849 df-pw 3920 df-sn 3936 df-pr 3938 df-op 3942 df-uni 4168 df-iun 4249 df-br 4374 df-opab 4433 df-mpt 4434 df-id 4726 df-xp 4817 df-rel 4818 df-cnv 4819 df-co 4820 df-dm 4821 df-rn 4822 df-res 4823 df-ima 4824 df-iota 5524 df-fun 5562 df-fn 5563 df-f 5564 df-f1 5565 df-fo 5566 df-f1o 5567 df-fv 5568 df-ov 6278 df-dvdsr 17879 |
This theorem is referenced by: dvdsr 17884 dvdsrpropd 17934 dvdsrzring 19062 |
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