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Theorem lmhmlem 16107
 Description: Non-quantified consequences of a left module homomorphism. (Contributed by Stefan O'Rear, 1-Jan-2015.)
Hypotheses
Ref Expression
lmhmlem.k Scalar
lmhmlem.l Scalar
Assertion
Ref Expression
lmhmlem LMHom

Proof of Theorem lmhmlem
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmhmlem.k . . 3 Scalar
2 lmhmlem.l . . 3 Scalar
3 eqid 2438 . . 3
4 eqid 2438 . . 3
5 eqid 2438 . . 3
6 eqid 2438 . . 3
71, 2, 3, 4, 5, 6islmhm 16105 . 2 LMHom
8 3simpa 955 . . 3
98anim2i 554 . 2
107, 9sylbi 189 1 LMHom
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   w3a 937   wceq 1653   wcel 1726  wral 2707  cfv 5456  (class class class)co 6083  cbs 13471  Scalarcsca 13534  cvsca 13535   cghm 15005  clmod 15952   LMHom clmhm 16097 This theorem is referenced by:  lmhmsca  16108  lmghm  16109  lmhmlmod2  16110  lmhmlmod1  16111 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-lmhm 16100
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