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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 | df-plusg 15781 | . 2 ⊢ +g = Slot 2 | |
4 | 2nn 11062 | . 2 ⊢ 2 ∈ ℕ | |
5 | 1lt2 11071 | . 2 ⊢ 1 < 2 | |
6 | 1, 2, 3, 4, 5 | resslem 15760 | 1 ⊢ (𝐴 ∈ 𝑉 → + = (+g‘𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 2c2 10947 ↾s cress 15696 +gcplusg 15768 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 |
This theorem is referenced by: issstrmgm 17075 gsumress 17099 issubmnd 17141 ress0g 17142 submnd0 17143 resmhm 17182 resmhm2 17183 resmhm2b 17184 submmulg 17409 subg0 17423 subginv 17424 subgcl 17427 subgsub 17429 subgmulg 17431 issubg2 17432 nmznsg 17461 resghm 17499 subgga 17556 gasubg 17558 resscntz 17587 sylow2blem2 17859 sylow3lem6 17870 subglsm 17909 pj1ghm 17939 subgabl 18064 subcmn 18065 submcmn2 18067 ringidss 18400 opprsubg 18459 unitgrp 18490 unitlinv 18500 unitrinv 18501 invrpropd 18521 isdrng2 18580 drngmcl 18583 drngid2 18586 isdrngd 18595 subrgugrp 18622 issubrg2 18623 subrgpropd 18637 abvres 18662 islss3 18780 sralmod 19008 resspsradd 19237 mpladd 19263 ressmpladd 19278 mplplusg 19411 ply1plusg 19416 ressply1add 19421 xrs1mnd 19603 xrs10 19604 xrs1cmn 19605 xrge0subm 19606 cnmsubglem 19628 expmhm 19634 nn0srg 19635 rge0srg 19636 zringplusg 19644 expghm 19663 psgnghm 19745 psgnco 19748 evpmodpmf1o 19761 replusg 19775 frlmplusgval 19926 mdetralt 20233 invrvald 20301 submtmd 21718 imasdsf1olem 21988 xrge0gsumle 22444 clmadd 22682 isclmp 22705 ipcau2 22841 reefgim 24008 efabl 24100 efsubm 24101 dchrptlem2 24790 dchrsum2 24793 qabvle 25114 padicabv 25119 ostth2lem2 25123 ostth3 25127 ressplusf 28981 ressmulgnn 29014 xrge0plusg 29018 submomnd 29041 ringinvval 29123 dvrcan5 29124 rhmunitinv 29153 xrge0slmod 29175 qqhghm 29360 qqhrhm 29361 esumpfinvallem 29463 lcdvadd 35904 cntzsdrg 36791 deg1mhm 36804 sge0tsms 39273 cnfldsrngadd 41560 issubmgm2 41580 resmgmhm 41588 resmgmhm2 41589 resmgmhm2b 41590 lidlrng 41717 amgmlemALT 42358 |
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