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Mirrors > Home > MPE Home > Th. List > ressmulr | Structured version Visualization version GIF version |
Description: .r is unaffected by restriction. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
Ref | Expression |
---|---|
ressmulr.1 | ⊢ 𝑆 = (𝑅 ↾s 𝐴) |
ressmulr.2 | ⊢ · = (.r‘𝑅) |
Ref | Expression |
---|---|
ressmulr | ⊢ (𝐴 ∈ 𝑉 → · = (.r‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ressmulr.1 | . 2 ⊢ 𝑆 = (𝑅 ↾s 𝐴) | |
2 | ressmulr.2 | . 2 ⊢ · = (.r‘𝑅) | |
3 | df-mulr 15782 | . 2 ⊢ .r = Slot 3 | |
4 | 3nn 11063 | . 2 ⊢ 3 ∈ ℕ | |
5 | 1lt3 11073 | . 2 ⊢ 1 < 3 | |
6 | 1, 2, 3, 4, 5 | resslem 15760 | 1 ⊢ (𝐴 ∈ 𝑉 → · = (.r‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 3c3 10948 ↾s cress 15696 .rcmulr 15769 |
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-3 10957 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-mulr 15782 |
This theorem is referenced by: mgpress 18323 subrg1 18613 subrgmcl 18615 subrgdvds 18617 subrguss 18618 subrginv 18619 subrgdv 18620 subrgunit 18621 subrgugrp 18622 issubrg2 18623 subrgpropd 18637 abvres 18662 sralmod 19008 issubassa 19145 resspsrmul 19238 resspsrvsca 19239 mplmul 19264 ressmplmul 19279 mplmulr 19412 ply1mulr 19418 ressply1mul 19422 nn0srg 19635 rge0srg 19636 zringmulr 19646 remulr 19776 dmatcrng 20127 scmatcrng 20146 scmatsrng1 20148 scmatmhm 20159 clmmul 22683 isclmp 22705 cphsubrglem 22785 ipcau2 22841 qabvexp 25115 ostthlem2 25117 padicabv 25119 ostth2lem2 25123 ostth3 25127 ress1r 29120 rdivmuldivd 29122 suborng 29146 xrge0slmod 29175 xrge0iifmhm 29313 qqhrhm 29361 cnfldsrngmul 41561 lidlmmgm 41715 lidlmsgrp 41716 lidlrng 41717 zlidlring 41718 uzlidlring 41719 aacllem 42356 |
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