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Theorem matval 26877
 Description: Value of the matrix algebra. (Contributed by Stefan O'Rear, 4-Sep-2015.)
Hypotheses
Ref Expression
matval.a Mat
matval.g freeLMod
matval.t maMul
Assertion
Ref Expression
matval sSet

Proof of Theorem matval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 matval.a . 2 Mat
2 elex 2796 . . 3
3 id 19 . . . . . . 7
4 id 19 . . . . . . . 8
54, 4xpeq12d 4714 . . . . . . 7
63, 5oveqan12rd 5878 . . . . . 6 freeLMod freeLMod
7 matval.g . . . . . 6 freeLMod
86, 7syl6eqr 2333 . . . . 5 freeLMod
94, 4opeq12d 3804 . . . . . . . . . 10
109, 4opeq12d 3804 . . . . . . . . 9
11 df-ot 3650 . . . . . . . . 9
12 df-ot 3650 . . . . . . . . 9
1310, 11, 123eqtr4g 2340 . . . . . . . 8
143, 13oveqan12rd 5878 . . . . . . 7 maMul maMul
15 matval.t . . . . . . 7 maMul
1614, 15syl6eqr 2333 . . . . . 6 maMul
1716opeq2d 3803 . . . . 5 maMul
188, 17oveq12d 5876 . . . 4 freeLMod sSet maMul sSet
19 df-mat 26854 . . . 4 Mat freeLMod sSet maMul
20 ovex 5883 . . . 4 sSet
2118, 19, 20ovmpt2a 5978 . . 3 Mat sSet
222, 21sylan2 460 . 2 Mat sSet
231, 22syl5eq 2327 1 sSet
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358   wceq 1623   wcel 1684  cvv 2788  cop 3643  cotp 3644   cxp 4687  cfv 5255  (class class class)co 5858  cfn 6863  cnx 13145   sSet csts 13146  cmulr 13209   freeLMod cfrlm 26624   maMul cmmul 26851   Mat cmat 26852 This theorem is referenced by:  matmulr  26879  matbas  26880  matplusg  26881  matsca  26882  matvsca  26883 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-ot 3650  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-mat 26854
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