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Mirrors > Home > MPE Home > Th. List > zrhpsgnmhm | Structured version Visualization version Unicode version |
Description: Embedding of permutation signs into an arbitrary ring is a homomorphism. (Contributed by SO, 9-Jul-2018.) |
Ref | Expression |
---|---|
zrhpsgnmhm |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2462 |
. . . 4
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2 | 1 | zrhrhm 19132 |
. . 3
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3 | eqid 2462 |
. . . 4
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4 | eqid 2462 |
. . . 4
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5 | 3, 4 | rhmmhm 17999 |
. . 3
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6 | 2, 5 | syl 17 |
. 2
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7 | eqid 2462 |
. . . . 5
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8 | eqid 2462 |
. . . . 5
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9 | eqid 2462 |
. . . . 5
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10 | 7, 8, 9 | psgnghm2 19198 |
. . . 4
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11 | ghmmhm 16942 |
. . . 4
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12 | 10, 11 | syl 17 |
. . 3
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13 | eqid 2462 |
. . . . . . . 8
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14 | 13 | cnmsgnsubg 19194 |
. . . . . . 7
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15 | subgsubm 16888 |
. . . . . . 7
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16 | 14, 15 | ax-mp 5 |
. . . . . 6
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17 | cnring 19039 |
. . . . . . 7
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18 | cnfldbas 19023 |
. . . . . . . . 9
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19 | cnfld0 19041 |
. . . . . . . . 9
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20 | cndrng 19046 |
. . . . . . . . 9
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21 | 18, 19, 20 | drngui 18030 |
. . . . . . . 8
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22 | eqid 2462 |
. . . . . . . 8
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23 | 21, 22 | unitsubm 17947 |
. . . . . . 7
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24 | 13 | subsubm 16653 |
. . . . . . 7
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25 | 17, 23, 24 | mp2b 10 |
. . . . . 6
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26 | 16, 25 | mpbi 213 |
. . . . 5
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27 | 26 | simpli 464 |
. . . 4
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28 | 1z 10996 |
. . . . 5
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29 | neg1z 11002 |
. . . . 5
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30 | prssi 4141 |
. . . . 5
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31 | 28, 29, 30 | mp2an 683 |
. . . 4
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32 | zsubrg 19070 |
. . . . 5
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33 | 22 | subrgsubm 18070 |
. . . . 5
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34 | zringmpg 19112 |
. . . . . . 7
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35 | 34 | eqcomi 2471 |
. . . . . 6
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36 | 35 | subsubm 16653 |
. . . . 5
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37 | 32, 33, 36 | mp2b 10 |
. . . 4
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38 | 27, 31, 37 | mpbir2an 936 |
. . 3
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39 | zex 10975 |
. . . . . 6
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40 | ressabs 15237 |
. . . . . 6
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41 | 39, 31, 40 | mp2an 683 |
. . . . 5
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42 | 34 | oveq1i 6325 |
. . . . 5
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43 | 41, 42 | eqtr3i 2486 |
. . . 4
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44 | 43 | resmhm2 16656 |
. . 3
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45 | 12, 38, 44 | sylancl 673 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
46 | mhmco 16658 |
. 2
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47 | 6, 45, 46 | syl2an 484 |
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 1680 ax-4 1693 ax-5 1769 ax-6 1816 ax-7 1862 ax-8 1900 ax-9 1907 ax-10 1926 ax-11 1931 ax-12 1944 ax-13 2102 ax-ext 2442 ax-rep 4529 ax-sep 4539 ax-nul 4548 ax-pow 4595 ax-pr 4653 ax-un 6610 ax-inf2 8172 ax-cnex 9621 ax-resscn 9622 ax-1cn 9623 ax-icn 9624 ax-addcl 9625 ax-addrcl 9626 ax-mulcl 9627 ax-mulrcl 9628 ax-mulcom 9629 ax-addass 9630 ax-mulass 9631 ax-distr 9632 ax-i2m1 9633 ax-1ne0 9634 ax-1rid 9635 ax-rnegex 9636 ax-rrecex 9637 ax-cnre 9638 ax-pre-lttri 9639 ax-pre-lttrn 9640 ax-pre-ltadd 9641 ax-pre-mulgt0 9642 ax-addf 9644 ax-mulf 9645 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3or 992 df-3an 993 df-xor 1416 df-tru 1458 df-ex 1675 df-nf 1679 df-sb 1809 df-eu 2314 df-mo 2315 df-clab 2449 df-cleq 2455 df-clel 2458 df-nfc 2592 df-ne 2635 df-nel 2636 df-ral 2754 df-rex 2755 df-reu 2756 df-rmo 2757 df-rab 2758 df-v 3059 df-sbc 3280 df-csb 3376 df-dif 3419 df-un 3421 df-in 3423 df-ss 3430 df-pss 3432 df-nul 3744 df-if 3894 df-pw 3965 df-sn 3981 df-pr 3983 df-tp 3985 df-op 3987 df-ot 3989 df-uni 4213 df-int 4249 df-iun 4294 df-iin 4295 df-br 4417 df-opab 4476 df-mpt 4477 df-tr 4512 df-eprel 4764 df-id 4768 df-po 4774 df-so 4775 df-fr 4812 df-se 4813 df-we 4814 df-xp 4859 df-rel 4860 df-cnv 4861 df-co 4862 df-dm 4863 df-rn 4864 df-res 4865 df-ima 4866 df-pred 5399 df-ord 5445 df-on 5446 df-lim 5447 df-suc 5448 df-iota 5565 df-fun 5603 df-fn 5604 df-f 5605 df-f1 5606 df-fo 5607 df-f1o 5608 df-fv 5609 df-isom 5610 df-riota 6277 df-ov 6318 df-oprab 6319 df-mpt2 6320 df-om 6720 df-1st 6820 df-2nd 6821 df-tpos 6999 df-wrecs 7054 df-recs 7116 df-rdg 7154 df-1o 7208 df-2o 7209 df-oadd 7212 df-er 7389 df-map 7500 df-en 7596 df-dom 7597 df-sdom 7598 df-fin 7599 df-card 8399 df-cda 8624 df-pnf 9703 df-mnf 9704 df-xr 9705 df-ltxr 9706 df-le 9707 df-sub 9888 df-neg 9889 df-div 10298 df-nn 10638 df-2 10696 df-3 10697 df-4 10698 df-5 10699 df-6 10700 df-7 10701 df-8 10702 df-9 10703 df-10 10704 df-n0 10899 df-z 10967 df-dec 11081 df-uz 11189 df-rp 11332 df-fz 11814 df-fzo 11947 df-seq 12246 df-exp 12305 df-hash 12548 df-word 12697 df-lsw 12698 df-concat 12699 df-s1 12700 df-substr 12701 df-splice 12702 df-reverse 12703 df-s2 12981 df-struct 15172 df-ndx 15173 df-slot 15174 df-base 15175 df-sets 15176 df-ress 15177 df-plusg 15252 df-mulr 15253 df-starv 15254 df-tset 15258 df-ple 15259 df-ds 15261 df-unif 15262 df-0g 15389 df-gsum 15390 df-mre 15541 df-mrc 15542 df-acs 15544 df-mgm 16537 df-sgrp 16576 df-mnd 16586 df-mhm 16631 df-submnd 16632 df-grp 16722 df-minusg 16723 df-mulg 16725 df-subg 16863 df-ghm 16930 df-gim 16972 df-oppg 17046 df-symg 17068 df-pmtr 17132 df-psgn 17181 df-cmn 17481 df-abl 17482 df-mgp 17773 df-ur 17785 df-ring 17831 df-cring 17832 df-oppr 17900 df-dvdsr 17918 df-unit 17919 df-invr 17949 df-dvr 17960 df-rnghom 17992 df-drng 18026 df-subrg 18055 df-cnfld 19020 df-zring 19089 df-zrh 19124 |
This theorem is referenced by: madetsumid 19535 mdetleib2 19662 mdetf 19669 mdetdiaglem 19672 mdetrlin 19676 mdetrsca 19677 mdetralt 19682 mdetunilem7 19692 mdetunilem8 19693 |
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