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Theorem scmatghm 21136
 Description: There is a group homomorphism from the additive group of a ring to the additive group of the ring of scalar matrices over this ring. (Contributed by AV, 22-Dec-2019.)
Hypotheses
Ref Expression
scmatrhmval.k 𝐾 = (Base‘𝑅)
scmatrhmval.a 𝐴 = (𝑁 Mat 𝑅)
scmatrhmval.o 1 = (1r𝐴)
scmatrhmval.t = ( ·𝑠𝐴)
scmatrhmval.f 𝐹 = (𝑥𝐾 ↦ (𝑥 1 ))
scmatrhmval.c 𝐶 = (𝑁 ScMat 𝑅)
scmatghm.s 𝑆 = (𝐴s 𝐶)
Assertion
Ref Expression
scmatghm ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
Distinct variable groups:   𝑥,𝐾   𝑥,𝑅   𝑥, 1   𝑥,   𝑥,𝐶   𝑥,𝑁
Allowed substitution hints:   𝐴(𝑥)   𝑆(𝑥)   𝐹(𝑥)

Proof of Theorem scmatghm
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 scmatrhmval.k . 2 𝐾 = (Base‘𝑅)
2 eqid 2821 . 2 (Base‘𝑆) = (Base‘𝑆)
3 eqid 2821 . 2 (+g𝑅) = (+g𝑅)
4 eqid 2821 . 2 (+g𝑆) = (+g𝑆)
5 ringgrp 19296 . . 3 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
65adantl 484 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 ∈ Grp)
7 scmatrhmval.a . . . 4 𝐴 = (𝑁 Mat 𝑅)
8 eqid 2821 . . . 4 (Base‘𝐴) = (Base‘𝐴)
9 eqid 2821 . . . 4 (0g𝑅) = (0g𝑅)
10 scmatrhmval.c . . . 4 𝐶 = (𝑁 ScMat 𝑅)
117, 8, 1, 9, 10scmatsgrp 21122 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ (SubGrp‘𝐴))
12 scmatghm.s . . . 4 𝑆 = (𝐴s 𝐶)
1312subggrp 18276 . . 3 (𝐶 ∈ (SubGrp‘𝐴) → 𝑆 ∈ Grp)
1411, 13syl 17 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ Grp)
15 scmatrhmval.o . . . 4 1 = (1r𝐴)
16 scmatrhmval.t . . . 4 = ( ·𝑠𝐴)
17 scmatrhmval.f . . . 4 𝐹 = (𝑥𝐾 ↦ (𝑥 1 ))
181, 7, 15, 16, 17, 10scmatf 21132 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐹:𝐾𝐶)
197, 10, 12scmatstrbas 21129 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘𝑆) = 𝐶)
2019feq3d 6496 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝐹:𝐾⟶(Base‘𝑆) ↔ 𝐹:𝐾𝐶))
2118, 20mpbird 259 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐹:𝐾⟶(Base‘𝑆))
227matsca2 21023 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 = (Scalar‘𝐴))
2310ovexi 7184 . . . . . . . . . 10 𝐶 ∈ V
24 eqid 2821 . . . . . . . . . . 11 (Scalar‘𝐴) = (Scalar‘𝐴)
2512, 24resssca 16644 . . . . . . . . . 10 (𝐶 ∈ V → (Scalar‘𝐴) = (Scalar‘𝑆))
2623, 25mp1i 13 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Scalar‘𝐴) = (Scalar‘𝑆))
2722, 26eqtrd 2856 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 = (Scalar‘𝑆))
2827fveq2d 6669 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (+g𝑅) = (+g‘(Scalar‘𝑆)))
2928oveqd 7167 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑦(+g𝑅)𝑧) = (𝑦(+g‘(Scalar‘𝑆))𝑧))
3029oveq1d 7165 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑦(+g𝑅)𝑧) 1 ) = ((𝑦(+g‘(Scalar‘𝑆))𝑧) 1 ))
3130adantr 483 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) → ((𝑦(+g𝑅)𝑧) 1 ) = ((𝑦(+g‘(Scalar‘𝑆))𝑧) 1 ))
327matlmod 21032 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ LMod)
337, 10scmatlss 21128 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ (LSubSp‘𝐴))
34 eqid 2821 . . . . . . . 8 (LSubSp‘𝐴) = (LSubSp‘𝐴)
3512, 34lsslmod 19726 . . . . . . 7 ((𝐴 ∈ LMod ∧ 𝐶 ∈ (LSubSp‘𝐴)) → 𝑆 ∈ LMod)
3632, 33, 35syl2anc 586 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ LMod)
3736adantr 483 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) → 𝑆 ∈ LMod)
3827fveq2d 6669 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘𝑅) = (Base‘(Scalar‘𝑆)))
391, 38syl5eq 2868 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐾 = (Base‘(Scalar‘𝑆)))
4039eleq2d 2898 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑦𝐾𝑦 ∈ (Base‘(Scalar‘𝑆))))
4140biimpd 231 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑦𝐾𝑦 ∈ (Base‘(Scalar‘𝑆))))
4241adantrd 494 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑦𝐾𝑧𝐾) → 𝑦 ∈ (Base‘(Scalar‘𝑆))))
4342imp 409 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) → 𝑦 ∈ (Base‘(Scalar‘𝑆)))
4439eleq2d 2898 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑧𝐾𝑧 ∈ (Base‘(Scalar‘𝑆))))
4544biimpd 231 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑧𝐾𝑧 ∈ (Base‘(Scalar‘𝑆))))
4645adantld 493 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑦𝐾𝑧𝐾) → 𝑧 ∈ (Base‘(Scalar‘𝑆))))
4746imp 409 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) → 𝑧 ∈ (Base‘(Scalar‘𝑆)))
487, 8, 1, 9, 10scmatid 21117 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r𝐴) ∈ 𝐶)
4915a1i 11 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 1 = (1r𝐴))
5048, 49, 193eltr4d 2928 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 1 ∈ (Base‘𝑆))
5150adantr 483 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) → 1 ∈ (Base‘𝑆))
52 eqid 2821 . . . . . 6 (Scalar‘𝑆) = (Scalar‘𝑆)
5312, 16ressvsca 16645 . . . . . . 7 (𝐶 ∈ V → = ( ·𝑠𝑆))
5423, 53ax-mp 5 . . . . . 6 = ( ·𝑠𝑆)
55 eqid 2821 . . . . . 6 (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
56 eqid 2821 . . . . . 6 (+g‘(Scalar‘𝑆)) = (+g‘(Scalar‘𝑆))
572, 4, 52, 54, 55, 56lmodvsdir 19652 . . . . 5 ((𝑆 ∈ LMod ∧ (𝑦 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑧 ∈ (Base‘(Scalar‘𝑆)) ∧ 1 ∈ (Base‘𝑆))) → ((𝑦(+g‘(Scalar‘𝑆))𝑧) 1 ) = ((𝑦 1 )(+g𝑆)(𝑧 1 )))
5837, 43, 47, 51, 57syl13anc 1368 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) → ((𝑦(+g‘(Scalar‘𝑆))𝑧) 1 ) = ((𝑦 1 )(+g𝑆)(𝑧 1 )))
5931, 58eqtrd 2856 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) → ((𝑦(+g𝑅)𝑧) 1 ) = ((𝑦 1 )(+g𝑆)(𝑧 1 )))
60 simpr 487 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 ∈ Ring)
6160adantr 483 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) → 𝑅 ∈ Ring)
6260anim1i 616 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) → (𝑅 ∈ Ring ∧ (𝑦𝐾𝑧𝐾)))
63 3anass 1091 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑦𝐾𝑧𝐾) ↔ (𝑅 ∈ Ring ∧ (𝑦𝐾𝑧𝐾)))
6462, 63sylibr 236 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) → (𝑅 ∈ Ring ∧ 𝑦𝐾𝑧𝐾))
651, 3ringacl 19322 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑦𝐾𝑧𝐾) → (𝑦(+g𝑅)𝑧) ∈ 𝐾)
6664, 65syl 17 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) → (𝑦(+g𝑅)𝑧) ∈ 𝐾)
671, 7, 15, 16, 17scmatrhmval 21130 . . . 4 ((𝑅 ∈ Ring ∧ (𝑦(+g𝑅)𝑧) ∈ 𝐾) → (𝐹‘(𝑦(+g𝑅)𝑧)) = ((𝑦(+g𝑅)𝑧) 1 ))
6861, 66, 67syl2anc 586 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) → (𝐹‘(𝑦(+g𝑅)𝑧)) = ((𝑦(+g𝑅)𝑧) 1 ))
691, 7, 15, 16, 17scmatrhmval 21130 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑦𝐾) → (𝐹𝑦) = (𝑦 1 ))
7069ad2ant2lr 746 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) → (𝐹𝑦) = (𝑦 1 ))
711, 7, 15, 16, 17scmatrhmval 21130 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑧𝐾) → (𝐹𝑧) = (𝑧 1 ))
7271ad2ant2l 744 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) → (𝐹𝑧) = (𝑧 1 ))
7370, 72oveq12d 7168 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) → ((𝐹𝑦)(+g𝑆)(𝐹𝑧)) = ((𝑦 1 )(+g𝑆)(𝑧 1 )))
7459, 68, 733eqtr4d 2866 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) → (𝐹‘(𝑦(+g𝑅)𝑧)) = ((𝐹𝑦)(+g𝑆)(𝐹𝑧)))
751, 2, 3, 4, 6, 14, 21, 74isghmd 18361 1 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  Vcvv 3495   ↦ cmpt 5139  ⟶wf 6346  ‘cfv 6350  (class class class)co 7150  Fincfn 8503  Basecbs 16477   ↾s cress 16478  +gcplusg 16559  Scalarcsca 16562   ·𝑠 cvsca 16563  0gc0g 16707  Grpcgrp 18097  SubGrpcsubg 18267   GrpHom cghm 18349  1rcur 19245  Ringcrg 19291  LModclmod 19628  LSubSpclss 19697   Mat cmat 21010   ScMat cscmat 21092 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-ot 4570  df-uni 4833  df-int 4870  df-iun 4914  df-iin 4915  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-se 5510  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-isom 6359  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-of 7403  df-om 7575  df-1st 7683  df-2nd 7684  df-supp 7825  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-er 8283  df-map 8402  df-ixp 8456  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-fsupp 8828  df-sup 8900  df-oi 8968  df-card 9362  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-2 11694  df-3 11695  df-4 11696  df-5 11697  df-6 11698  df-7 11699  df-8 11700  df-9 11701  df-n0 11892  df-z 11976  df-dec 12093  df-uz 12238  df-fz 12887  df-fzo 13028  df-seq 13364  df-hash 13685  df-struct 16479  df-ndx 16480  df-slot 16481  df-base 16483  df-sets 16484  df-ress 16485  df-plusg 16572  df-mulr 16573  df-sca 16575  df-vsca 16576  df-ip 16577  df-tset 16578  df-ple 16579  df-ds 16581  df-hom 16583  df-cco 16584  df-0g 16709  df-gsum 16710  df-prds 16715  df-pws 16717  df-mre 16851  df-mrc 16852  df-acs 16854  df-mgm 17846  df-sgrp 17895  df-mnd 17906  df-mhm 17950  df-submnd 17951  df-grp 18100  df-minusg 18101  df-sbg 18102  df-mulg 18219  df-subg 18270  df-ghm 18350  df-cntz 18441  df-cmn 18902  df-abl 18903  df-mgp 19234  df-ur 19246  df-ring 19293  df-subrg 19527  df-lmod 19630  df-lss 19698  df-sra 19938  df-rgmod 19939  df-dsmm 20870  df-frlm 20885  df-mamu 20989  df-mat 21011  df-dmat 21093  df-scmat 21094 This theorem is referenced by:  scmatrhm  21138
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