Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 | df-plusg 16566 | . 2 ⊢ +g = Slot 2 | |
4 | 2nn 11698 | . 2 ⊢ 2 ∈ ℕ | |
5 | 1lt2 11796 | . 2 ⊢ 1 < 2 | |
6 | 1, 2, 3, 4, 5 | resslem 16545 | 1 ⊢ (𝐴 ∈ 𝑉 → + = (+g‘𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ‘cfv 6348 (class class class)co 7145 2c2 11680 ↾s cress 16472 +gcplusg 16553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 |
This theorem is referenced by: issstrmgm 17851 gsumress 17880 issubmnd 17926 ress0g 17927 submnd0 17928 resmhm 17973 resmhm2 17974 resmhm2b 17975 submmulg 18209 subg0 18223 subginv 18224 subgcl 18227 subgsub 18229 subgmulg 18231 issubg2 18232 nmznsg 18258 resghm 18312 subgga 18368 gasubg 18370 resscntz 18400 sylow2blem2 18675 sylow3lem6 18686 subglsm 18728 pj1ghm 18758 subgabl 18885 subcmn 18886 submcmn2 18888 cntrcmnd 18891 cycsubmcmn 18937 ringidss 19256 opprsubg 19315 unitgrp 19346 unitlinv 19356 unitrinv 19357 invrpropd 19377 isdrng2 19441 drngmcl 19444 drngid2 19447 isdrngd 19456 subrgugrp 19483 issubrg2 19484 subrgpropd 19499 cntzsdrg 19510 abvres 19539 islss3 19660 sralmod 19888 resspsradd 20124 mpladd 20150 ressmpladd 20166 mplplusg 20316 ply1plusg 20321 ressply1add 20326 xrs1mnd 20511 xrs10 20512 xrs1cmn 20513 xrge0subm 20514 cnmsubglem 20536 expmhm 20542 nn0srg 20543 rge0srg 20544 zringplusg 20552 expghm 20571 psgnghm 20652 psgnco 20655 evpmodpmf1o 20668 replusg 20682 phlssphl 20731 frlmplusgval 20836 mdetralt 21145 invrvald 21213 submtmd 22640 imasdsf1olem 22910 xrge0gsumle 23368 clmadd 23605 isclmp 23628 ipcau2 23764 reefgim 24965 efabl 25061 efsubm 25062 dchrptlem2 25768 dchrsum2 25771 qabvle 26128 padicabv 26133 ostth2lem2 26137 ostth3 26141 ressplusf 30564 ressmulgnn 30597 xrge0plusg 30601 submomnd 30638 ringinvval 30790 dvrcan5 30791 rhmunitinv 30822 xrge0slmod 30844 drgextlsp 30895 fedgmullem2 30925 qqhghm 31128 qqhrhm 31129 esumpfinvallem 31232 lcdvadd 38613 deg1mhm 39685 sge0tsms 42539 cnfldsrngadd 43914 issubmgm2 43934 resmgmhm 43942 resmgmhm2 43943 resmgmhm2b 43944 smndex1mgm 44007 smndex1sgrp 44008 smndex1mnd 44010 smndex1id 44011 lidlrng 44126 amgmlemALT 44832 |
Copyright terms: Public domain | W3C validator |