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Theorem frlmval 27207
 Description: Value of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Hypothesis
Ref Expression
frlmval.f freeLMod
Assertion
Ref Expression
frlmval m ringLMod

Proof of Theorem frlmval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frlmval.f . 2 freeLMod
2 elex 2966 . . 3
3 elex 2966 . . 3
4 id 21 . . . . 5
5 fveq2 5731 . . . . . . 7 ringLMod ringLMod
65sneqd 3829 . . . . . 6 ringLMod ringLMod
76xpeq2d 4905 . . . . 5 ringLMod ringLMod
84, 7oveq12d 6102 . . . 4 m ringLMod m ringLMod
9 xpeq1 4895 . . . . 5 ringLMod ringLMod
109oveq2d 6100 . . . 4 m ringLMod m ringLMod
11 df-frlm 27205 . . . 4 freeLMod m ringLMod
12 ovex 6109 . . . 4 m ringLMod
138, 10, 11, 12ovmpt2 6212 . . 3 freeLMod m ringLMod
142, 3, 13syl2an 465 . 2 freeLMod m ringLMod
151, 14syl5eq 2482 1 m ringLMod
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   wceq 1653   wcel 1726  cvv 2958  csn 3816   cxp 4879  cfv 5457  (class class class)co 6084  ringLModcrglmod 16246   m cdsmm 27188   freeLMod cfrlm 27203 This theorem is referenced by:  frlmlmod  27208  frlmpws  27209  frlmlss  27210  frlmpwsfi  27211  frlmbas  27214 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406 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 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-frlm 27205
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