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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 16352 | . 2 ⊢ .r = Slot 3 | |
4 | 3nn 11454 | . 2 ⊢ 3 ∈ ℕ | |
5 | 1lt3 11555 | . 2 ⊢ 1 < 3 | |
6 | 1, 2, 3, 4, 5 | resslem 16329 | 1 ⊢ (𝐴 ∈ 𝑉 → · = (.r‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 ‘cfv 6135 (class class class)co 6922 3c3 11431 ↾s cress 16256 .rcmulr 16339 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-mulr 16352 |
This theorem is referenced by: mgpress 18887 subrg1 19182 subrgmcl 19184 subrgdvds 19186 subrguss 19187 subrginv 19188 subrgdv 19189 subrgunit 19190 subrgugrp 19191 issubrg2 19192 subrgpropd 19206 abvres 19231 sralmod 19584 issubassa 19721 resspsrmul 19814 resspsrvsca 19815 mplmul 19840 ressmplmul 19855 mplmulr 19987 ply1mulr 19993 ressply1mul 19997 nn0srg 20212 rge0srg 20213 zringmulr 20223 remulr 20354 dmatcrng 20713 scmatcrng 20732 scmatsrng1 20734 scmatmhm 20745 clmmul 23282 isclmp 23304 cphsubrglem 23384 ipcau2 23440 qabvexp 25767 ostthlem2 25769 padicabv 25771 ostth2lem2 25775 ostth3 25779 ress1r 30351 rdivmuldivd 30353 suborng 30377 xrge0slmod 30406 xrge0iifmhm 30583 qqhrhm 30631 cnfldsrngmul 42786 lidlmmgm 42940 lidlmsgrp 42941 lidlrng 42942 zlidlring 42943 uzlidlring 42944 aacllem 43653 |
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