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Mirrors > Home > MPE Home > Th. List > lsmub2 | Unicode version |
Description: Subgroup sum is an upper bound of its arguments. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.) |
Ref | Expression |
---|---|
lsmub1.p |
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Ref | Expression |
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lsmub2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subgsubm 14925 |
. 2
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2 | eqid 2412 |
. . 3
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3 | 2 | subgss 14908 |
. 2
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4 | lsmub1.p |
. . 3
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5 | 2, 4 | lsmub2x 15244 |
. 2
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6 | 1, 3, 5 | syl2an 464 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem is referenced by: lsmunss 15255 lsmlub 15260 lsmss1 15261 lsmss2b 15264 lsmdisj 15276 pj1rid 15297 dprd2da 15563 dprdsplit 15569 pgpfac1lem1 15595 pgpfac1lem3 15598 lspabs2 16155 lspindpi 16167 lshpnelb 29479 lsmsat 29503 lcvexchlem4 29532 lsatexch 29538 lsatcvat3 29547 dia2dimlem9 31567 dihjustlem 31711 dihord1 31713 dihord2b 31715 dihord5b 31754 dochexmidlem7 31961 mapdlsm 32159 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1552 ax-5 1563 ax-17 1623 ax-9 1662 ax-8 1683 ax-13 1723 ax-14 1725 ax-6 1740 ax-7 1745 ax-11 1757 ax-12 1946 ax-ext 2393 ax-rep 4288 ax-sep 4298 ax-nul 4306 ax-pow 4345 ax-pr 4371 ax-un 4668 ax-cnex 9010 ax-resscn 9011 ax-1cn 9012 ax-icn 9013 ax-addcl 9014 ax-addrcl 9015 ax-mulcl 9016 ax-mulrcl 9017 ax-mulcom 9018 ax-addass 9019 ax-mulass 9020 ax-distr 9021 ax-i2m1 9022 ax-1ne0 9023 ax-1rid 9024 ax-rnegex 9025 ax-rrecex 9026 ax-cnre 9027 ax-pre-lttri 9028 ax-pre-lttrn 9029 ax-pre-ltadd 9030 ax-pre-mulgt0 9031 |
This theorem depends on definitions: df-bi 178 df-or 360 df-an 361 df-3or 937 df-3an 938 df-tru 1325 df-ex 1548 df-nf 1551 df-sb 1656 df-eu 2266 df-mo 2267 df-clab 2399 df-cleq 2405 df-clel 2408 df-nfc 2537 df-ne 2577 df-nel 2578 df-ral 2679 df-rex 2680 df-reu 2681 df-rmo 2682 df-rab 2683 df-v 2926 df-sbc 3130 df-csb 3220 df-dif 3291 df-un 3293 df-in 3295 df-ss 3302 df-pss 3304 df-nul 3597 df-if 3708 df-pw 3769 df-sn 3788 df-pr 3789 df-tp 3790 df-op 3791 df-uni 3984 df-iun 4063 df-br 4181 df-opab 4235 df-mpt 4236 df-tr 4271 df-eprel 4462 df-id 4466 df-po 4471 df-so 4472 df-fr 4509 df-we 4511 df-ord 4552 df-on 4553 df-lim 4554 df-suc 4555 df-om 4813 df-xp 4851 df-rel 4852 df-cnv 4853 df-co 4854 df-dm 4855 df-rn 4856 df-res 4857 df-ima 4858 df-iota 5385 df-fun 5423 df-fn 5424 df-f 5425 df-f1 5426 df-fo 5427 df-f1o 5428 df-fv 5429 df-ov 6051 df-oprab 6052 df-mpt2 6053 df-1st 6316 df-2nd 6317 df-riota 6516 df-recs 6600 df-rdg 6635 df-er 6872 df-en 7077 df-dom 7078 df-sdom 7079 df-pnf 9086 df-mnf 9087 df-xr 9088 df-ltxr 9089 df-le 9090 df-sub 9257 df-neg 9258 df-nn 9965 df-2 10022 df-ndx 13435 df-slot 13436 df-base 13437 df-sets 13438 df-ress 13439 df-plusg 13505 df-0g 13690 df-mnd 14653 df-submnd 14702 df-grp 14775 df-minusg 14776 df-subg 14904 df-lsm 15233 |
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