Nearby theorems
|
|
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > lsmub2x | Structured version Visualization version GIF 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 |
|---|---|
| lsmless2.v | β’ π΅ = (BaseβπΊ) |
| lsmless2.s | β’ β = (LSSumβπΊ) |
| Ref | Expression |
|---|---|
| lsmub2x | β’ ((π β (SubMndβπΊ) β§ π β π΅) β π β (π β π)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | submrcl 18719 | . . . . . 6 β’ (π β (SubMndβπΊ) β πΊ β Mnd) | |
| 2 | 1 | ad2antrr 722 | . . . . 5 β’ (((π β (SubMndβπΊ) β§ π β π΅) β§ π₯ β π) β πΊ β Mnd) |
| 3 | simpr 483 | . . . . . 6 β’ ((π β (SubMndβπΊ) β§ π β π΅) β π β π΅) | |
| 4 | 3 | sselda 3981 | . . . . 5 β’ (((π β (SubMndβπΊ) β§ π β π΅) β§ π₯ β π) β π₯ β π΅) |
| 5 | lsmless2.v | . . . . . 6 β’ π΅ = (BaseβπΊ) | |
| 6 | eqid 2730 | . . . . . 6 β’ (+gβπΊ) = (+gβπΊ) | |
| 7 | eqid 2730 | . . . . . 6 β’ (0gβπΊ) = (0gβπΊ) | |
| 8 | 5, 6, 7 | mndlid 18679 | . . . . 5 β’ ((πΊ β Mnd β§ π₯ β π΅) β ((0gβπΊ)(+gβπΊ)π₯) = π₯) |
| 9 | 2, 4, 8 | syl2anc 582 | . . . 4 β’ (((π β (SubMndβπΊ) β§ π β π΅) β§ π₯ β π) β ((0gβπΊ)(+gβπΊ)π₯) = π₯) |
| 10 | 5 | submss 18726 | . . . . . 6 β’ (π β (SubMndβπΊ) β π β π΅) |
| 11 | 10 | ad2antrr 722 | . . . . 5 β’ (((π β (SubMndβπΊ) β§ π β π΅) β§ π₯ β π) β π β π΅) |
| 12 | simplr 765 | . . . . 5 β’ (((π β (SubMndβπΊ) β§ π β π΅) β§ π₯ β π) β π β π΅) | |
| 13 | 7 | subm0cl 18728 | . . . . . 6 β’ (π β (SubMndβπΊ) β (0gβπΊ) β π) |
| 14 | 13 | ad2antrr 722 | . . . . 5 β’ (((π β (SubMndβπΊ) β§ π β π΅) β§ π₯ β π) β (0gβπΊ) β π) |
| 15 | simpr 483 | . . . . 5 β’ (((π β (SubMndβπΊ) β§ π β π΅) β§ π₯ β π) β π₯ β π) | |
| 16 | lsmless2.s | . . . . . 6 β’ β = (LSSumβπΊ) | |
| 17 | 5, 6, 16 | lsmelvalix 19550 | . . . . 5 β’ (((πΊ β Mnd β§ π β π΅ β§ π β π΅) β§ ((0gβπΊ) β π β§ π₯ β π)) β ((0gβπΊ)(+gβπΊ)π₯) β (π β π)) |
| 18 | 2, 11, 12, 14, 15, 17 | syl32anc 1376 | . . . 4 β’ (((π β (SubMndβπΊ) β§ π β π΅) β§ π₯ β π) β ((0gβπΊ)(+gβπΊ)π₯) β (π β π)) |
| 19 | 9, 18 | eqeltrrd 2832 | . . 3 β’ (((π β (SubMndβπΊ) β§ π β π΅) β§ π₯ β π) β π₯ β (π β π)) |
| 20 | 19 | ex 411 | . 2 β’ ((π β (SubMndβπΊ) β§ π β π΅) β (π₯ β π β π₯ β (π β π))) |
| 21 | 20 | ssrdv 3987 | 1 β’ ((π β (SubMndβπΊ) β§ π β π΅) β π β (π β π)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 394 = wceq 1539 β wcel 2104 β wss 3947 βcfv 6542 (class class class)co 7411 Basecbs 17148 +gcplusg 17201 0gc0g 17389 Mndcmnd 18659 SubMndcsubmnd 18704 LSSumclsm 19543 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-0g 17391 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18706 df-lsm 19545 |
| This theorem is referenced by: lsmub2 19567 |
| Copyright terms: Public domain | W3C validator |