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Mirrors > Home > MPE Home > Th. List > srgidmlem | Structured version Visualization version GIF version |
Description: Lemma for srglidm 20019 and srgridm 20020. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 27-Dec-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
Ref | Expression |
---|---|
srgidm.b | ⊢ 𝐵 = (Base‘𝑅) |
srgidm.t | ⊢ · = (.r‘𝑅) |
srgidm.u | ⊢ 1 = (1r‘𝑅) |
Ref | Expression |
---|---|
srgidmlem | ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵) → (( 1 · 𝑋) = 𝑋 ∧ (𝑋 · 1 ) = 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
2 | 1 | srgmgp 20008 | . 2 ⊢ (𝑅 ∈ SRing → (mulGrp‘𝑅) ∈ Mnd) |
3 | srgidm.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
4 | 1, 3 | mgpbas 19988 | . . 3 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅)) |
5 | srgidm.t | . . . 4 ⊢ · = (.r‘𝑅) | |
6 | 1, 5 | mgpplusg 19986 | . . 3 ⊢ · = (+g‘(mulGrp‘𝑅)) |
7 | srgidm.u | . . . 4 ⊢ 1 = (1r‘𝑅) | |
8 | 1, 7 | ringidval 20001 | . . 3 ⊢ 1 = (0g‘(mulGrp‘𝑅)) |
9 | 4, 6, 8 | mndlrid 18641 | . 2 ⊢ (((mulGrp‘𝑅) ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (( 1 · 𝑋) = 𝑋 ∧ (𝑋 · 1 ) = 𝑋)) |
10 | 2, 9 | sylan 581 | 1 ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵) → (( 1 · 𝑋) = 𝑋 ∧ (𝑋 · 1 ) = 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ‘cfv 6541 (class class class)co 7406 Basecbs 17141 .rcmulr 17195 Mndcmnd 18622 mulGrpcmgp 19982 1rcur 19999 SRingcsrg 20003 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17142 df-plusg 17207 df-0g 17384 df-mgm 18558 df-sgrp 18607 df-mnd 18623 df-mgp 19983 df-ur 20000 df-srg 20004 |
This theorem is referenced by: srglidm 20019 srgridm 20020 |
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