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Theorem mendrng 27468
 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 27460 . . 3 LMHom
32a1i 11 . 2 LMHom
4 eqidd 2436 . 2
5 eqidd 2436 . 2
6 eqid 2435 . . . . . 6
7 eqid 2435 . . . . . 6
81, 2, 6, 7mendplusg 27462 . . . . 5 LMHom LMHom
96lmhmplusg 16112 . . . . 5 LMHom LMHom LMHom
108, 9eqeltrd 2509 . . . 4 LMHom LMHom LMHom
11103adant1 975 . . 3 LMHom LMHom LMHom
12 simpr1 963 . . . . . 6 LMHom LMHom LMHom LMHom
13 simpr2 964 . . . . . 6 LMHom LMHom LMHom LMHom
1412, 13, 9syl2anc 643 . . . . 5 LMHom LMHom LMHom LMHom
15 simpr3 965 . . . . 5 LMHom LMHom LMHom LMHom
161, 2, 6, 7mendplusg 27462 . . . . 5 LMHom LMHom
1714, 15, 16syl2anc 643 . . . 4 LMHom LMHom LMHom
1812, 13, 8syl2anc 643 . . . . 5 LMHom LMHom LMHom
1918oveq1d 6088 . . . 4 LMHom LMHom LMHom
206lmhmplusg 16112 . . . . . . 7 LMHom LMHom LMHom
2113, 15, 20syl2anc 643 . . . . . 6 LMHom LMHom LMHom LMHom
221, 2, 6, 7mendplusg 27462 . . . . . 6 LMHom LMHom
2312, 21, 22syl2anc 643 . . . . 5 LMHom LMHom LMHom
241, 2, 6, 7mendplusg 27462 . . . . . . 7 LMHom LMHom
2513, 15, 24syl2anc 643 . . . . . 6 LMHom LMHom LMHom
2625oveq2d 6089 . . . . 5 LMHom LMHom LMHom
27 lmodgrp 15949 . . . . . . . 8
28 grpmnd 14809 . . . . . . . 8
2927, 28syl 16 . . . . . . 7
3029adantr 452 . . . . . 6 LMHom LMHom LMHom
31 eqid 2435 . . . . . . . . 9
3231, 31lmhmf 16102 . . . . . . . 8 LMHom
3312, 32syl 16 . . . . . . 7 LMHom LMHom LMHom
34 fvex 5734 . . . . . . . 8
3534, 34elmap 7034 . . . . . . 7
3633, 35sylibr 204 . . . . . 6 LMHom LMHom LMHom
3731, 31lmhmf 16102 . . . . . . . 8 LMHom
3813, 37syl 16 . . . . . . 7 LMHom LMHom LMHom
3934, 34elmap 7034 . . . . . . 7
4038, 39sylibr 204 . . . . . 6 LMHom LMHom LMHom
4131, 31lmhmf 16102 . . . . . . . 8 LMHom
4215, 41syl 16 . . . . . . 7 LMHom LMHom LMHom
4334, 34elmap 7034 . . . . . . 7
4442, 43sylibr 204 . . . . . 6 LMHom LMHom LMHom
4531, 6mndvass 27415 . . . . . 6
4630, 36, 40, 44, 45syl13anc 1186 . . . . 5 LMHom LMHom LMHom
4723, 26, 463eqtr4d 2477 . . . 4 LMHom LMHom LMHom
4817, 19, 473eqtr4d 2477 . . 3 LMHom LMHom LMHom
49 id 20 . . . 4
50 eqidd 2436 . . . 4 Scalar Scalar
51 eqid 2435 . . . . 5
52 eqid 2435 . . . . 5 Scalar Scalar
5351, 31, 52, 520lmhm 16108 . . . 4 Scalar Scalar LMHom
5449, 49, 50, 53syl3anc 1184 . . 3 LMHom
551, 2, 6, 7mendplusg 27462 . . . . 5 LMHom LMHom
5654, 55sylan 458 . . . 4 LMHom
5732, 35sylibr 204 . . . . 5 LMHom
5831, 6, 51mndvlid 27416 . . . . 5
5929, 57, 58syl2an 464 . . . 4 LMHom
6056, 59eqtrd 2467 . . 3 LMHom
61 eqid 2435 . . . . 5
6261invlmhm 16110 . . . 4 LMHom
63 lmhmco 16111 . . . 4 LMHom LMHom LMHom
6462, 63sylan 458 . . 3 LMHom LMHom
651, 2, 6, 7mendplusg 27462 . . . . 5 LMHom LMHom
6664, 65sylancom 649 . . . 4 LMHom
6731, 6, 61, 51grpvlinv 27418 . . . . 5
6827, 57, 67syl2an 464 . . . 4 LMHom
6966, 68eqtrd 2467 . . 3 LMHom
703, 4, 11, 48, 54, 60, 64, 69isgrpd 14822 . 2
71 eqid 2435 . . . . 5
721, 2, 71mendmulr 27464 . . . 4 LMHom LMHom
73 lmhmco 16111 . . . 4 LMHom LMHom LMHom
7472, 73eqeltrd 2509 . . 3 LMHom LMHom LMHom
75743adant1 975 . 2 LMHom LMHom LMHom
76 coass 5380 . . 3
7712, 13, 72syl2anc 643 . . . . 5 LMHom LMHom LMHom
7877oveq1d 6088 . . . 4 LMHom LMHom LMHom
7912, 13, 73syl2anc 643 . . . . 5 LMHom LMHom LMHom LMHom
801, 2, 71mendmulr 27464 . . . . 5 LMHom LMHom
8179, 15, 80syl2anc 643 . . . 4 LMHom LMHom LMHom
8278, 81eqtrd 2467 . . 3 LMHom LMHom LMHom
831, 2, 71mendmulr 27464 . . . . . 6 LMHom LMHom
8413, 15, 83syl2anc 643 . . . . 5 LMHom LMHom LMHom
8584oveq2d 6089 . . . 4 LMHom LMHom LMHom
86 lmhmco 16111 . . . . . 6 LMHom LMHom LMHom
8713, 15, 86syl2anc 643 . . . . 5 LMHom LMHom LMHom LMHom
881, 2, 71mendmulr 27464 . . . . 5 LMHom LMHom
8912, 87, 88syl2anc 643 . . . 4 LMHom LMHom LMHom
9085, 89eqtrd 2467 . . 3 LMHom LMHom LMHom
9176, 82, 903eqtr4a 2493 . 2 LMHom LMHom LMHom
921, 2, 71mendmulr 27464 . . . 4 LMHom LMHom
9312, 21, 92syl2anc 643 . . 3 LMHom LMHom LMHom
9425oveq2d 6089 . . 3 LMHom LMHom LMHom
95 lmhmco 16111 . . . . . 6 LMHom LMHom LMHom
9612, 15, 95syl2anc 643 . . . . 5 LMHom LMHom LMHom LMHom
971, 2, 6, 7mendplusg 27462 . . . . 5 LMHom LMHom
9879, 96, 97syl2anc 643 . . . 4 LMHom LMHom LMHom
991, 2, 71mendmulr 27464 . . . . . 6 LMHom LMHom
10012, 15, 99syl2anc 643 . . . . 5 LMHom LMHom LMHom
10177, 100oveq12d 6091 . . . 4 LMHom LMHom LMHom
102 lmghm 16099 . . . . . 6 LMHom
103 ghmmhm 15008 . . . . . 6 MndHom
10412, 102, 1033syl 19 . . . . 5 LMHom LMHom LMHom MndHom
10531, 6, 6mhmvlin 27420 . . . . 5 MndHom
106104, 40, 44, 105syl3anc 1184 . . . 4 LMHom LMHom LMHom
10798, 101, 1063eqtr4d 2477 . . 3 LMHom LMHom LMHom
10893, 94, 1073eqtr4d 2477 . 2 LMHom LMHom LMHom
1091, 2, 71mendmulr 27464 . . . 4 LMHom LMHom
11014, 15, 109syl2anc 643 . . 3 LMHom LMHom LMHom
11118oveq1d 6088 . . 3 LMHom LMHom LMHom
1121, 2, 6, 7mendplusg 27462 . . . . 5 LMHom LMHom
11396, 87, 112syl2anc 643 . . . 4 LMHom LMHom LMHom
114100, 84oveq12d 6091 . . . 4 LMHom LMHom LMHom
115 ffn 5583 . . . . . 6
11612, 32, 1153syl 19 . . . . 5 LMHom LMHom LMHom
117 ffn 5583 . . . . . 6
11813, 37, 1173syl 19 . . . . 5 LMHom LMHom LMHom
11934a1i 11 . . . . 5 LMHom LMHom LMHom
120 inidm 3542 . . . . 5
121116, 118, 42, 119, 119, 119, 120ofco 6316 . . . 4 LMHom LMHom LMHom
122113, 114, 1213eqtr4d 2477 . . 3 LMHom LMHom LMHom
123110, 111, 1223eqtr4d 2477 . 2 LMHom LMHom LMHom
12431idlmhm 16109 . 2 LMHom
1251, 2, 71mendmulr 27464 . . . 4 LMHom LMHom
126124, 125sylan 458 . . 3 LMHom
12732adantl 453 . . . 4 LMHom
128 fcoi2 5610 . . . 4
129127, 128syl 16 . . 3 LMHom
130126, 129eqtrd 2467 . 2 LMHom
131 id 20 . . . 4 LMHom LMHom
1321, 2, 71mendmulr 27464 . . . 4 LMHom LMHom
133131, 124, 132syl2anr 465 . . 3 LMHom
134 fcoi1 5609 . . . 4
135127, 134syl 16 . . 3 LMHom
136133, 135eqtrd 2467 . 2 LMHom
1373, 4, 5, 70, 75, 91, 108, 123, 124, 130, 136isrngd 15690 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   w3a 936   wceq 1652   wcel 1725  cvv 2948  csn 3806   cid 4485   cxp 4868   cres 4872   ccom 4874   wfn 5441  wf 5442  cfv 5446  (class class class)co 6073   cof 6295   cmap 7010  cbs 13461   cplusg 13521  cmulr 13522  Scalarcsca 13524  c0g 13715  cmnd 14676  cgrp 14677  cminusg 14678   MndHom cmhm 14728   cghm 14995  crg 15652  clmod 15942   LMHom clmhm 16087  MEndocmend 27457 This theorem is referenced by:  mendlmod  27469  mendassa  27470 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-plusg 13534  df-mulr 13535  df-sca 13537  df-vsca 13538  df-0g 13719  df-mnd 14682  df-mhm 14730  df-grp 14804  df-minusg 14805  df-ghm 14996  df-cmn 15406  df-abl 15407  df-mgp 15641  df-rng 15655  df-ur 15657  df-lmod 15944  df-lmhm 16090  df-mend 27458
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