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Mirrors > Home > MPE Home > Th. List > evls1rhm | Structured version Visualization version Unicode version |
Description: Polynomial evaluation is a homomorphism (into the product ring). (Contributed by AV, 11-Sep-2019.) |
Ref | Expression |
---|---|
evls1rhm.q |
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evls1rhm.b |
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evls1rhm.t |
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evls1rhm.u |
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evls1rhm.w |
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Ref | Expression |
---|---|
evls1rhm |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evls1rhm.b |
. . . . . 6
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2 | 1 | subrgss 18021 |
. . . . 5
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3 | 2 | adantl 468 |
. . . 4
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4 | elpwg 3961 |
. . . . 5
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5 | 4 | adantl 468 |
. . . 4
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6 | 3, 5 | mpbird 236 |
. . 3
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7 | evls1rhm.q |
. . . 4
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8 | eqid 2453 |
. . . 4
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9 | 7, 8, 1 | evls1fval 18920 |
. . 3
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10 | 6, 9 | syldan 473 |
. 2
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11 | evls1rhm.t |
. . . . 5
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12 | eqid 2453 |
. . . . 5
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13 | 1, 11, 12 | evls1rhmlem 18922 |
. . . 4
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14 | 13 | adantr 467 |
. . 3
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15 | 1on 7194 |
. . . . 5
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16 | eqid 2453 |
. . . . . 6
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17 | eqid 2453 |
. . . . . 6
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18 | evls1rhm.u |
. . . . . 6
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19 | eqid 2453 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
20 | 16, 17, 18, 19, 1 | evlsrhm 18756 |
. . . . 5
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21 | 15, 20 | mp3an1 1353 |
. . . 4
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22 | eqidd 2454 |
. . . . 5
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23 | eqidd 2454 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
24 | evls1rhm.w |
. . . . . . 7
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25 | eqid 2453 |
. . . . . . 7
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26 | eqid 2453 |
. . . . . . 7
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27 | 24, 25, 26 | ply1bas 18800 |
. . . . . 6
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28 | 27 | a1i 11 |
. . . . 5
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29 | eqid 2453 |
. . . . . . . 8
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30 | 24, 17, 29 | ply1plusg 18830 |
. . . . . . 7
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31 | 30 | a1i 11 |
. . . . . 6
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32 | 31 | oveqdr 6319 |
. . . . 5
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33 | eqidd 2454 |
. . . . 5
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34 | eqid 2453 |
. . . . . . . 8
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35 | 24, 17, 34 | ply1mulr 18832 |
. . . . . . 7
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36 | 35 | a1i 11 |
. . . . . 6
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37 | 36 | oveqdr 6319 |
. . . . 5
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38 | eqidd 2454 |
. . . . 5
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39 | 22, 23, 28, 23, 32, 33, 37, 38 | rhmpropd 18055 |
. . . 4
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40 | 21, 39 | eleqtrrd 2534 |
. . 3
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41 | rhmco 17977 |
. . 3
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42 | 14, 40, 41 | syl2anc 667 |
. 2
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43 | 10, 42 | eqeltrd 2531 |
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 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-inf2 8151 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-rmo 2747 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-int 4238 df-iun 4283 df-iin 4284 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-se 4797 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-isom 5594 df-riota 6257 df-ov 6298 df-oprab 6299 df-mpt2 6300 df-of 6536 df-ofr 6537 df-om 6698 df-1st 6798 df-2nd 6799 df-supp 6920 df-wrecs 7033 df-recs 7095 df-rdg 7133 df-1o 7187 df-2o 7188 df-oadd 7191 df-er 7368 df-map 7479 df-pm 7480 df-ixp 7528 df-en 7575 df-dom 7576 df-sdom 7577 df-fin 7578 df-fsupp 7889 df-sup 7961 df-oi 8030 df-card 8378 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-3 10676 df-4 10677 df-5 10678 df-6 10679 df-7 10680 df-8 10681 df-9 10682 df-10 10683 df-n0 10877 df-z 10945 df-dec 11059 df-uz 11167 df-fz 11792 df-fzo 11923 df-seq 12221 df-hash 12523 df-struct 15135 df-ndx 15136 df-slot 15137 df-base 15138 df-sets 15139 df-ress 15140 df-plusg 15215 df-mulr 15216 df-sca 15218 df-vsca 15219 df-ip 15220 df-tset 15221 df-ple 15222 df-ds 15224 df-hom 15226 df-cco 15227 df-0g 15352 df-gsum 15353 df-prds 15358 df-pws 15360 df-mre 15504 df-mrc 15505 df-acs 15507 df-mgm 16500 df-sgrp 16539 df-mnd 16549 df-mhm 16594 df-submnd 16595 df-grp 16685 df-minusg 16686 df-sbg 16687 df-mulg 16688 df-subg 16826 df-ghm 16893 df-cntz 16983 df-cmn 17444 df-abl 17445 df-mgp 17736 df-ur 17748 df-srg 17752 df-ring 17794 df-cring 17795 df-rnghom 17955 df-subrg 18018 df-lmod 18105 df-lss 18168 df-lsp 18207 df-assa 18548 df-asp 18549 df-ascl 18550 df-psr 18592 df-mvr 18593 df-mpl 18594 df-opsr 18596 df-evls 18741 df-psr1 18785 df-ply1 18787 df-evls1 18916 |
This theorem is referenced by: evls1gsumadd 18925 evls1gsummul 18926 evls1varpw 18927 |
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