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Theorem mendrng 30737
 Description: The module endomorphism algebra is a ring. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Hypothesis
Ref Expression
mendassa.a MEndo
Assertion
Ref Expression
mendrng

Proof of Theorem mendrng
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mendassa.a . . . 4 MEndo
21mendbas 30729 . . 3 LMHom
32a1i 11 . 2 LMHom
4 eqidd 2463 . 2
5 eqidd 2463 . 2
6 eqid 2462 . . . . . 6
7 eqid 2462 . . . . . 6
81, 2, 6, 7mendplusg 30731 . . . . 5 LMHom LMHom
96lmhmplusg 17468 . . . . 5 LMHom LMHom LMHom
108, 9eqeltrd 2550 . . . 4 LMHom LMHom LMHom
11103adant1 1009 . . 3 LMHom LMHom LMHom
12 simpr1 997 . . . . . 6 LMHom LMHom LMHom LMHom
13 simpr2 998 . . . . . 6 LMHom LMHom LMHom LMHom
1412, 13, 9syl2anc 661 . . . . 5 LMHom LMHom LMHom LMHom
15 simpr3 999 . . . . 5 LMHom LMHom LMHom LMHom
161, 2, 6, 7mendplusg 30731 . . . . 5 LMHom LMHom
1714, 15, 16syl2anc 661 . . . 4 LMHom LMHom LMHom
1812, 13, 8syl2anc 661 . . . . 5 LMHom LMHom LMHom
1918oveq1d 6292 . . . 4 LMHom LMHom LMHom
206lmhmplusg 17468 . . . . . . 7 LMHom LMHom LMHom
2113, 15, 20syl2anc 661 . . . . . 6 LMHom LMHom LMHom LMHom
221, 2, 6, 7mendplusg 30731 . . . . . 6 LMHom LMHom
2312, 21, 22syl2anc 661 . . . . 5 LMHom LMHom LMHom
241, 2, 6, 7mendplusg 30731 . . . . . . 7 LMHom LMHom
2513, 15, 24syl2anc 661 . . . . . 6 LMHom LMHom LMHom
2625oveq2d 6293 . . . . 5 LMHom LMHom LMHom
27 lmodgrp 17297 . . . . . . . 8
28 grpmnd 15858 . . . . . . . 8
2927, 28syl 16 . . . . . . 7
3029adantr 465 . . . . . 6 LMHom LMHom LMHom
31 eqid 2462 . . . . . . . . 9
3231, 31lmhmf 17458 . . . . . . . 8 LMHom
3312, 32syl 16 . . . . . . 7 LMHom LMHom LMHom
34 fvex 5869 . . . . . . . 8
3534, 34elmap 7439 . . . . . . 7
3633, 35sylibr 212 . . . . . 6 LMHom LMHom LMHom
3731, 31lmhmf 17458 . . . . . . . 8 LMHom
3813, 37syl 16 . . . . . . 7 LMHom LMHom LMHom
3934, 34elmap 7439 . . . . . . 7
4038, 39sylibr 212 . . . . . 6 LMHom LMHom LMHom
4131, 31lmhmf 17458 . . . . . . . 8 LMHom
4215, 41syl 16 . . . . . . 7 LMHom LMHom LMHom
4334, 34elmap 7439 . . . . . . 7
4442, 43sylibr 212 . . . . . 6 LMHom LMHom LMHom
4531, 6mndvass 18656 . . . . . 6
4630, 36, 40, 44, 45syl13anc 1225 . . . . 5 LMHom LMHom LMHom
4723, 26, 463eqtr4d 2513 . . . 4 LMHom LMHom LMHom
4817, 19, 473eqtr4d 2513 . . 3 LMHom LMHom LMHom
49 id 22 . . . 4
50 eqidd 2463 . . . 4 Scalar Scalar
51 eqid 2462 . . . . 5
52 eqid 2462 . . . . 5 Scalar Scalar
5351, 31, 52, 520lmhm 17464 . . . 4 Scalar Scalar LMHom
5449, 49, 50, 53syl3anc 1223 . . 3 LMHom
551, 2, 6, 7mendplusg 30731 . . . . 5 LMHom LMHom
5654, 55sylan 471 . . . 4 LMHom
5732, 35sylibr 212 . . . . 5 LMHom
5831, 6, 51mndvlid 18657 . . . . 5
5929, 57, 58syl2an 477 . . . 4 LMHom
6056, 59eqtrd 2503 . . 3 LMHom
61 eqid 2462 . . . . 5
6261invlmhm 17466 . . . 4 LMHom
63 lmhmco 17467 . . . 4 LMHom LMHom LMHom
6462, 63sylan 471 . . 3 LMHom LMHom
651, 2, 6, 7mendplusg 30731 . . . . 5 LMHom LMHom
6664, 65sylancom 667 . . . 4 LMHom
6731, 6, 61, 51grpvlinv 18659 . . . . 5
6827, 57, 67syl2an 477 . . . 4 LMHom
6966, 68eqtrd 2503 . . 3 LMHom
703, 4, 11, 48, 54, 60, 64, 69isgrpd 15871 . 2
71 eqid 2462 . . . . 5
721, 2, 71mendmulr 30733 . . . 4 LMHom LMHom
73 lmhmco 17467 . . . 4 LMHom LMHom LMHom
7472, 73eqeltrd 2550 . . 3 LMHom LMHom LMHom
75743adant1 1009 . 2 LMHom LMHom LMHom
76 coass 5519 . . 3
7712, 13, 72syl2anc 661 . . . . 5 LMHom LMHom LMHom
7877oveq1d 6292 . . . 4 LMHom LMHom LMHom
7912, 13, 73syl2anc 661 . . . . 5 LMHom LMHom LMHom LMHom
801, 2, 71mendmulr 30733 . . . . 5 LMHom LMHom
8179, 15, 80syl2anc 661 . . . 4 LMHom LMHom LMHom
8278, 81eqtrd 2503 . . 3 LMHom LMHom LMHom
831, 2, 71mendmulr 30733 . . . . . 6 LMHom LMHom
8413, 15, 83syl2anc 661 . . . . 5 LMHom LMHom LMHom
8584oveq2d 6293 . . . 4 LMHom LMHom LMHom
86 lmhmco 17467 . . . . . 6 LMHom LMHom LMHom
8713, 15, 86syl2anc 661 . . . . 5 LMHom LMHom LMHom LMHom
881, 2, 71mendmulr 30733 . . . . 5 LMHom LMHom
8912, 87, 88syl2anc 661 . . . 4 LMHom LMHom LMHom
9085, 89eqtrd 2503 . . 3 LMHom LMHom LMHom
9176, 82, 903eqtr4a 2529 . 2 LMHom LMHom LMHom
921, 2, 71mendmulr 30733 . . . 4 LMHom LMHom
9312, 21, 92syl2anc 661 . . 3 LMHom LMHom LMHom
9425oveq2d 6293 . . 3 LMHom LMHom LMHom
95 lmhmco 17467 . . . . . 6 LMHom LMHom LMHom
9612, 15, 95syl2anc 661 . . . . 5 LMHom LMHom LMHom LMHom
971, 2, 6, 7mendplusg 30731 . . . . 5 LMHom LMHom
9879, 96, 97syl2anc 661 . . . 4 LMHom LMHom LMHom
991, 2, 71mendmulr 30733 . . . . . 6 LMHom LMHom
10012, 15, 99syl2anc 661 . . . . 5 LMHom LMHom LMHom
10177, 100oveq12d 6295 . . . 4 LMHom LMHom LMHom
102 lmghm 17455 . . . . . 6 LMHom
103 ghmmhm 16067 . . . . . 6 MndHom
10412, 102, 1033syl 20 . . . . 5 LMHom LMHom LMHom MndHom
10531, 6, 6mhmvlin 18661 . . . . 5 MndHom
106104, 40, 44, 105syl3anc 1223 . . . 4 LMHom LMHom LMHom
10798, 101, 1063eqtr4d 2513 . . 3 LMHom LMHom LMHom
10893, 94, 1073eqtr4d 2513 . 2 LMHom LMHom LMHom
1091, 2, 71mendmulr 30733 . . . 4 LMHom LMHom
11014, 15, 109syl2anc 661 . . 3 LMHom LMHom LMHom
11118oveq1d 6292 . . 3 LMHom LMHom LMHom
1121, 2, 6, 7mendplusg 30731 . . . . 5 LMHom LMHom
11396, 87, 112syl2anc 661 . . . 4 LMHom LMHom LMHom
114100, 84oveq12d 6295 . . . 4 LMHom LMHom LMHom
115 ffn 5724 . . . . . 6
11612, 32, 1153syl 20 . . . . 5 LMHom LMHom LMHom
117 ffn 5724 . . . . . 6
11813, 37, 1173syl 20 . . . . 5 LMHom LMHom LMHom
11934a1i 11 . . . . 5 LMHom LMHom LMHom
120 inidm 3702 . . . . 5
121116, 118, 42, 119, 119, 119, 120ofco 6537 . . . 4 LMHom LMHom LMHom
122113, 114, 1213eqtr4d 2513 . . 3 LMHom LMHom LMHom
123110, 111, 1223eqtr4d 2513 . 2 LMHom LMHom LMHom
12431idlmhm 17465 . 2 LMHom
1251, 2, 71mendmulr 30733 . . . 4 LMHom LMHom
126124, 125sylan 471 . . 3 LMHom
12732adantl 466 . . . 4 LMHom
128 fcoi2 5753 . . . 4
129127, 128syl 16 . . 3 LMHom
130126, 129eqtrd 2503 . 2 LMHom
131 id 22 . . . 4 LMHom LMHom
1321, 2, 71mendmulr 30733 . . . 4 LMHom LMHom
133131, 124, 132syl2anr 478 . . 3 LMHom
134 fcoi1 5752 . . . 4
135127, 134syl 16 . . 3 LMHom
136133, 135eqtrd 2503 . 2 LMHom
1373, 4, 5, 70, 75, 91, 108, 123, 124, 130, 136isrngd 17015 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   w3a 968   wceq 1374   wcel 1762  cvv 3108  csn 4022   cid 4785   cxp 4992   cres 4996   ccom 4998   wfn 5576  wf 5577  cfv 5581  (class class class)co 6277   cof 6515   cmap 7412  cbs 14481   cplusg 14546  cmulr 14547  Scalarcsca 14549  c0g 14686  cmnd 15717  cgrp 15718  cminusg 15719   MndHom cmhm 15770   cghm 16054  crg 16981  clmod 17290   LMHom clmhm 17443  MEndocmend 30720 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6517  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-2 10585  df-3 10586  df-4 10587  df-5 10588  df-6 10589  df-n0 10787  df-z 10856  df-uz 11074  df-fz 11664  df-struct 14483  df-ndx 14484  df-slot 14485  df-base 14486  df-sets 14487  df-plusg 14559  df-mulr 14560  df-sca 14562  df-vsca 14563  df-0g 14688  df-mnd 15723  df-mhm 15772  df-grp 15853  df-minusg 15854  df-ghm 16055  df-cmn 16591  df-abl 16592  df-mgp 16927  df-ur 16939  df-rng 16983  df-lmod 17292  df-lmhm 17446  df-mend 30721 This theorem is referenced by:  mendlmod  30738  mendassa  30739
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