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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 17324 | . 2 ⊢ +g = Slot (+g‘ndx) | |
| 4 | basendxnplusgndx 17327 | . . 3 ⊢ (Base‘ndx) ≠ (+g‘ndx) | |
| 5 | 4 | necomi 2995 | . 2 ⊢ (+g‘ndx) ≠ (Base‘ndx) |
| 6 | 1, 2, 3, 5 | resseqnbas 17287 | 1 ⊢ (𝐴 ∈ 𝑉 → + = (+g‘𝐻)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 ndxcnx 17230 Basecbs 17247 ↾s cress 17274 +gcplusg 17297 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 |
| This theorem is referenced by: issstrmgm 18666 gsumress 18695 issubmgm2 18716 resmgmhm 18724 resmgmhm2 18725 resmgmhm2b 18726 issubmnd 18774 ress0g 18775 submnd0 18776 resmhm 18833 resmhm2 18834 resmhm2b 18835 smndex1mgm 18920 smndex1sgrp 18921 smndex1mnd 18923 smndex1id 18924 ressmulgnn 19094 ressmulgnnd 19096 submmulg 19136 subg0 19150 subginv 19151 subgcl 19154 subgsub 19156 subgmulg 19158 issubg2 19159 nmznsg 19186 resghm 19250 subgga 19318 gasubg 19320 resscntz 19351 symgplusg 19400 sylow2blem2 19639 sylow3lem6 19650 subglsm 19691 pj1ghm 19721 subgabl 19854 subcmn 19855 submcmn2 19857 cntrcmnd 19860 cycsubmcmn 19907 ringidss 20274 opprsubg 20352 unitgrp 20383 unitlinv 20393 unitrinv 20394 invrpropd 20418 rhmunitinv 20511 issubrng2 20558 subrngpropd 20568 subrgugrp 20591 issubrg2 20592 subrgpropd 20608 isdrng2 20743 drngmclOLD 20751 drngid2 20752 isdrngd 20765 isdrngdOLD 20767 cntzsdrg 20803 abvres 20832 islss3 20957 sralmod 21194 rnglidlrng 21257 rngqiprngghmlem3 21299 xrs1mnd 21422 xrs10 21423 xrs1cmn 21424 xrge0subm 21425 cnmsubglem 21448 expmhm 21454 nn0srg 21455 rge0srg 21456 zringplusg 21465 expghm 21486 psgnghm 21598 psgnco 21601 evpmodpmf1o 21614 replusg 21628 phlssphl 21677 frlmplusgval 21784 resspsradd 21995 mplplusg 22027 ressmpladd 22047 mhpmulcl 22153 ply1plusg 22225 ressply1add 22231 evls1addd 22375 mdetralt 22614 invrvald 22682 submtmd 24112 imasdsf1olem 24383 xrge0gsumle 24855 clmadd 25107 isclmp 25130 ipcau2 25268 reefgim 26494 efabl 26592 efsubm 26593 dchrptlem2 27309 dchrsum2 27312 qabvle 27669 padicabv 27674 ostth2lem2 27678 ostth3 27682 ressplusf 32948 xrge0plusg 33018 submomnd 33087 ringinvval 33239 dvrcan5 33240 xrge0slmod 33376 idlinsubrg 33459 zringfrac 33582 drgextlsp 33644 fedgmullem2 33681 algextdeglem8 33765 2sqr3minply 33791 qqhghm 33989 qqhrhm 33990 esumpfinvallem 34075 lcdvadd 41599 primrootsunit1 42098 aks6d1c6isolem2 42176 mhphflem 42606 deg1mhm 43212 sge0tsms 46395 cnfldsrngadd 48078 amgmlemALT 49322 |
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