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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rhmdvdsr | Structured version Visualization version Unicode version |
Description: A ring homomorphism preserves the divisibility relation. (Contributed by Thierry Arnoux, 22-Oct-2017.) |
Ref | Expression |
---|---|
rhmdvdsr.x |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
rhmdvdsr.m |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
rhmdvdsr.n |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Ref | Expression |
---|---|
rhmdvdsr |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 1012 |
. . 3
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2 | simpl2 1013 |
. . 3
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3 | rhmdvdsr.x |
. . . . 5
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4 | eqid 2453 |
. . . . 5
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5 | 3, 4 | rhmf 17966 |
. . . 4
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6 | 5 | ffvelrnda 6027 |
. . 3
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7 | 1, 2, 6 | syl2anc 667 |
. 2
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8 | simpll1 1048 |
. . . . . 6
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9 | simpr 463 |
. . . . . 6
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10 | 5 | ffvelrnda 6027 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
11 | 8, 9, 10 | syl2anc 667 |
. . . . 5
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12 | 11 | ralrimiva 2804 |
. . . 4
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13 | 2 | adantr 467 |
. . . . . . 7
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14 | eqid 2453 |
. . . . . . . 8
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15 | eqid 2453 |
. . . . . . . 8
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16 | 3, 14, 15 | rhmmul 17967 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
17 | 8, 9, 13, 16 | syl3anc 1269 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
18 | 17 | ralrimiva 2804 |
. . . . 5
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19 | simpr 463 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
20 | rhmdvdsr.m |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
21 | 3, 20, 14 | dvdsr2 17887 |
. . . . . . 7
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22 | 21 | biimpac 489 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 19, 2, 22 | syl2anc 667 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | r19.29 2927 |
. . . . . 6
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25 | simpl 459 |
. . . . . . . 8
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26 | simpr 463 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
27 | 26 | fveq2d 5874 |
. . . . . . . 8
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28 | 25, 27 | eqtr3d 2489 |
. . . . . . 7
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29 | 28 | reximi 2857 |
. . . . . 6
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30 | 24, 29 | syl 17 |
. . . . 5
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31 | 18, 23, 30 | syl2anc 667 |
. . . 4
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32 | r19.29 2927 |
. . . 4
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33 | 12, 31, 32 | syl2anc 667 |
. . 3
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34 | oveq1 6302 |
. . . . . 6
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35 | 34 | eqeq1d 2455 |
. . . . 5
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36 | 35 | rspcev 3152 |
. . . 4
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37 | 36 | rexlimivw 2878 |
. . 3
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38 | 33, 37 | syl 17 |
. 2
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39 | rhmdvdsr.n |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
40 | 4, 39, 15 | dvdsr 17886 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
41 | 7, 38, 40 | sylanbrc 671 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 1671 ax-4 1684 ax-5 1760 ax-6 1807 ax-7 1853 ax-8 1891 ax-9 1898 ax-10 1917 ax-11 1922 ax-12 1935 ax-13 2093 ax-ext 2433 ax-rep 4518 ax-sep 4528 ax-nul 4537 ax-pow 4584 ax-pr 4642 ax-un 6588 ax-cnex 9600 ax-resscn 9601 ax-1cn 9602 ax-icn 9603 ax-addcl 9604 ax-addrcl 9605 ax-mulcl 9606 ax-mulrcl 9607 ax-mulcom 9608 ax-addass 9609 ax-mulass 9610 ax-distr 9611 ax-i2m1 9612 ax-1ne0 9613 ax-1rid 9614 ax-rnegex 9615 ax-rrecex 9616 ax-cnre 9617 ax-pre-lttri 9618 ax-pre-lttrn 9619 ax-pre-ltadd 9620 ax-pre-mulgt0 9621 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 987 df-3an 988 df-tru 1449 df-ex 1666 df-nf 1670 df-sb 1800 df-eu 2305 df-mo 2306 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2583 df-ne 2626 df-nel 2627 df-ral 2744 df-rex 2745 df-reu 2746 df-rab 2748 df-v 3049 df-sbc 3270 df-csb 3366 df-dif 3409 df-un 3411 df-in 3413 df-ss 3420 df-pss 3422 df-nul 3734 df-if 3884 df-pw 3955 df-sn 3971 df-pr 3973 df-tp 3975 df-op 3977 df-uni 4202 df-iun 4283 df-br 4406 df-opab 4465 df-mpt 4466 df-tr 4501 df-eprel 4748 df-id 4752 df-po 4758 df-so 4759 df-fr 4796 df-we 4798 df-xp 4843 df-rel 4844 df-cnv 4845 df-co 4846 df-dm 4847 df-rn 4848 df-res 4849 df-ima 4850 df-pred 5383 df-ord 5429 df-on 5430 df-lim 5431 df-suc 5432 df-iota 5549 df-fun 5587 df-fn 5588 df-f 5589 df-f1 5590 df-fo 5591 df-f1o 5592 df-fv 5593 df-riota 6257 df-ov 6298 df-oprab 6299 df-mpt2 6300 df-om 6698 df-wrecs 7033 df-recs 7095 df-rdg 7133 df-er 7368 df-map 7479 df-en 7575 df-dom 7576 df-sdom 7577 df-pnf 9682 df-mnf 9683 df-xr 9684 df-ltxr 9685 df-le 9686 df-sub 9867 df-neg 9868 df-nn 10617 df-2 10675 df-ndx 15136 df-slot 15137 df-base 15138 df-sets 15139 df-plusg 15215 df-0g 15352 df-mhm 16594 df-ghm 16893 df-mgp 17736 df-ur 17748 df-ring 17794 df-dvdsr 17881 df-rnghom 17955 |
This theorem is referenced by: elrhmunit 28595 |
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