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Mirrors > Home > ILE Home > Th. List > srgass | Unicode version |
Description: Associative law for the multiplication operation of a semiring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
Ref | Expression |
---|---|
srgcl.b |
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srgcl.t |
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Ref | Expression |
---|---|
srgass |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2177 |
. . . . 5
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2 | 1 | srgmgp 12964 |
. . . 4
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3 | 2 | adantr 276 |
. . 3
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4 | simpr1 1003 |
. . . 4
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5 | srgcl.b |
. . . . . 6
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6 | 1, 5 | mgpbasg 12950 |
. . . . 5
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7 | 6 | adantr 276 |
. . . 4
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8 | 4, 7 | eleqtrd 2256 |
. . 3
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9 | simpr2 1004 |
. . . 4
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10 | 9, 7 | eleqtrd 2256 |
. . 3
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11 | simpr3 1005 |
. . . 4
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12 | 11, 7 | eleqtrd 2256 |
. . 3
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13 | eqid 2177 |
. . . 4
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14 | eqid 2177 |
. . . 4
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15 | 13, 14 | mndass 12704 |
. . 3
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16 | 3, 8, 10, 12, 15 | syl13anc 1240 |
. 2
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17 | srgcl.t |
. . . . . 6
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18 | 1, 17 | mgpplusgg 12948 |
. . . . 5
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19 | 18 | adantr 276 |
. . . 4
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20 | 19 | oveqd 5885 |
. . 3
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21 | 19 | oveqd 5885 |
. . . 4
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22 | 21 | oveq1d 5883 |
. . 3
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23 | 20, 22 | eqtrd 2210 |
. 2
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24 | 19 | oveqd 5885 |
. . 3
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25 | 19 | oveqd 5885 |
. . . 4
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26 | 25 | oveq2d 5884 |
. . 3
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27 | 24, 26 | eqtrd 2210 |
. 2
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28 | 16, 23, 27 | 3eqtr4d 2220 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-cnex 7880 ax-resscn 7881 ax-1cn 7882 ax-1re 7883 ax-icn 7884 ax-addcl 7885 ax-addrcl 7886 ax-mulcl 7887 ax-addcom 7889 ax-addass 7891 ax-i2m1 7894 ax-0lt1 7895 ax-0id 7897 ax-rnegex 7898 ax-pre-ltirr 7901 ax-pre-ltadd 7905 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-iota 5173 df-fun 5213 df-fn 5214 df-fv 5219 df-riota 5824 df-ov 5871 df-oprab 5872 df-mpo 5873 df-pnf 7971 df-mnf 7972 df-ltxr 7974 df-inn 8896 df-2 8954 df-3 8955 df-ndx 12435 df-slot 12436 df-base 12438 df-sets 12439 df-plusg 12518 df-mulr 12519 df-0g 12642 df-sgrp 12687 df-mnd 12697 df-mgp 12945 df-srg 12960 |
This theorem is referenced by: srgpcomp 12986 srgpcompp 12987 srgpcomppsc 12988 |
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