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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 | mulridx 17235 | . 2 ⊢ .r = Slot (.r‘ndx) | |
4 | basendxnmulrndx 17236 | . . 3 ⊢ (Base‘ndx) ≠ (.r‘ndx) | |
5 | 4 | necomi 2987 | . 2 ⊢ (.r‘ndx) ≠ (Base‘ndx) |
6 | 1, 2, 3, 5 | resseqnbas 17182 | 1 ⊢ (𝐴 ∈ 𝑉 → · = (.r‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ‘cfv 6533 (class class class)co 7401 ndxcnx 17122 Basecbs 17140 ↾s cress 17169 .rcmulr 17194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-mulr 17207 |
This theorem is referenced by: mgpress 20039 mgpressOLD 20040 rdivmuldivd 20300 subrngmcl 20442 issubrng2 20443 subrngpropd 20453 subrg1 20469 subrgdvds 20473 subrguss 20474 subrginv 20475 subrgdv 20476 subrgunit 20477 subrgugrp 20478 issubrg2 20479 subrgpropd 20495 primefld 20641 abvres 20667 sralmod 21028 rnglidlmmgm 21088 rnglidlmsgrp 21089 rnglidlrng 21090 rngqiprngimfolem 21128 rngqiprnglinlem1 21129 rngqiprngimf1lem 21132 rngqiprngimf1 21138 rngqiprnglin 21140 rng2idl1cntr 21143 rngqiprngfulem5 21153 nn0srg 21294 rge0srg 21295 zringmulr 21307 pzriprnglem6 21336 remulr 21464 issubassa3 21720 resspsrmul 21838 resspsrvsca 21839 mplmulr 21868 ressmplmul 21886 ply1mulr 22058 ressply1mul 22063 dmatcrng 22314 scmatcrng 22333 scmatsrng1 22335 scmatmhm 22346 clmmul 24912 isclmp 24934 cphsubrglem 25015 ipcau2 25072 qabvexp 27463 ostthlem2 27465 padicabv 27467 ostth2lem2 27471 ostth3 27475 ress1r 32810 suborng 32860 xrge0slmod 32890 idlinsubrg 32980 evls1muld 33080 resssra 33119 drgextlsp 33125 fedgmullem1 33159 fedgmullem2 33160 extdg1id 33187 xrge0iifmhm 33374 qqhrhm 33424 imacrhmcl 41546 cnfldsrngmul 46992 zlidlring 47063 uzlidlring 47064 aacllem 48002 |
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