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Mirrors > Home > MPE Home > Th. List > ressplusg | Structured version Visualization version GIF version |
Description: +g is unaffected by restriction. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
Ref | Expression |
---|---|
ressplusg.1 | ⊢ 𝐻 = (𝐺 ↾s 𝐴) |
ressplusg.2 | ⊢ + = (+g‘𝐺) |
Ref | Expression |
---|---|
ressplusg | ⊢ (𝐴 ∈ 𝑉 → + = (+g‘𝐻)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ressplusg.1 | . 2 ⊢ 𝐻 = (𝐺 ↾s 𝐴) | |
2 | ressplusg.2 | . 2 ⊢ + = (+g‘𝐺) | |
3 | plusgid 16829 | . 2 ⊢ +g = Slot (+g‘ndx) | |
4 | basendxnplusgndx 16830 | . . 3 ⊢ (Base‘ndx) ≠ (+g‘ndx) | |
5 | 4 | necomi 2995 | . 2 ⊢ (+g‘ndx) ≠ (Base‘ndx) |
6 | 1, 2, 3, 5 | resseqnbas 16793 | 1 ⊢ (𝐴 ∈ 𝑉 → + = (+g‘𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 ‘cfv 6380 (class class class)co 7213 ndxcnx 16744 Basecbs 16760 ↾s cress 16784 +gcplusg 16802 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 |
This theorem is referenced by: issstrmgm 18125 gsumress 18154 issubmnd 18200 ress0g 18201 submnd0 18202 resmhm 18247 resmhm2 18248 resmhm2b 18249 smndex1mgm 18334 smndex1sgrp 18335 smndex1mnd 18337 smndex1id 18338 submmulg 18535 subg0 18549 subginv 18550 subgcl 18553 subgsub 18555 subgmulg 18557 issubg2 18558 nmznsg 18584 resghm 18638 subgga 18694 gasubg 18696 resscntz 18726 symgplusg 18775 sylow2blem2 19010 sylow3lem6 19021 subglsm 19063 pj1ghm 19093 subgabl 19221 subcmn 19222 submcmn2 19224 cntrcmnd 19227 cycsubmcmn 19273 ringidss 19595 opprsubg 19654 unitgrp 19685 unitlinv 19695 unitrinv 19696 invrpropd 19716 isdrng2 19777 drngmcl 19780 drngid2 19783 isdrngd 19792 subrgugrp 19819 issubrg2 19820 subrgpropd 19835 cntzsdrg 19846 abvres 19875 islss3 19996 sralmod 20224 xrs1mnd 20401 xrs10 20402 xrs1cmn 20403 xrge0subm 20404 cnmsubglem 20426 expmhm 20432 nn0srg 20433 rge0srg 20434 zringplusg 20442 expghm 20462 psgnghm 20542 psgnco 20545 evpmodpmf1o 20558 replusg 20572 phlssphl 20621 frlmplusgval 20726 resspsradd 20941 mpladd 20969 ressmpladd 20986 mhpmulcl 21089 mplplusg 21141 ply1plusg 21146 ressply1add 21151 mdetralt 21505 invrvald 21573 submtmd 23001 imasdsf1olem 23271 xrge0gsumle 23730 clmadd 23971 isclmp 23994 ipcau2 24131 reefgim 25342 efabl 25439 efsubm 25440 dchrptlem2 26146 dchrsum2 26149 qabvle 26506 padicabv 26511 ostth2lem2 26515 ostth3 26519 ressplusf 30955 ressmulgnn 31011 xrge0plusg 31015 submomnd 31055 ringinvval 31208 dvrcan5 31209 rhmunitinv 31240 xrge0slmod 31262 idlinsubrg 31322 drgextlsp 31395 fedgmullem2 31425 qqhghm 31650 qqhrhm 31651 esumpfinvallem 31754 lcdvadd 39348 mhphflem 39994 deg1mhm 40735 sge0tsms 43593 cnfldsrngadd 44997 issubmgm2 45017 resmgmhm 45025 resmgmhm2 45026 resmgmhm2b 45027 lidlrng 45158 amgmlemALT 46178 |
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