MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  frgpval Structured version   Visualization version   GIF version

Theorem frgpval 19148
Description: Value of the free group construction. (Contributed by Mario Carneiro, 1-Oct-2015.)
Hypotheses
Ref Expression
frgpval.m 𝐺 = (freeGrp‘𝐼)
frgpval.b 𝑀 = (freeMnd‘(𝐼 × 2o))
frgpval.r = ( ~FG𝐼)
Assertion
Ref Expression
frgpval (𝐼𝑉𝐺 = (𝑀 /s ))

Proof of Theorem frgpval
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 frgpval.m . 2 𝐺 = (freeGrp‘𝐼)
2 elex 3426 . . 3 (𝐼𝑉𝐼 ∈ V)
3 xpeq1 5565 . . . . . . 7 (𝑖 = 𝐼 → (𝑖 × 2o) = (𝐼 × 2o))
43fveq2d 6721 . . . . . 6 (𝑖 = 𝐼 → (freeMnd‘(𝑖 × 2o)) = (freeMnd‘(𝐼 × 2o)))
5 frgpval.b . . . . . 6 𝑀 = (freeMnd‘(𝐼 × 2o))
64, 5eqtr4di 2796 . . . . 5 (𝑖 = 𝐼 → (freeMnd‘(𝑖 × 2o)) = 𝑀)
7 fveq2 6717 . . . . . 6 (𝑖 = 𝐼 → ( ~FG𝑖) = ( ~FG𝐼))
8 frgpval.r . . . . . 6 = ( ~FG𝐼)
97, 8eqtr4di 2796 . . . . 5 (𝑖 = 𝐼 → ( ~FG𝑖) = )
106, 9oveq12d 7231 . . . 4 (𝑖 = 𝐼 → ((freeMnd‘(𝑖 × 2o)) /s ( ~FG𝑖)) = (𝑀 /s ))
11 df-frgp 19100 . . . 4 freeGrp = (𝑖 ∈ V ↦ ((freeMnd‘(𝑖 × 2o)) /s ( ~FG𝑖)))
12 ovex 7246 . . . 4 (𝑀 /s ) ∈ V
1310, 11, 12fvmpt 6818 . . 3 (𝐼 ∈ V → (freeGrp‘𝐼) = (𝑀 /s ))
142, 13syl 17 . 2 (𝐼𝑉 → (freeGrp‘𝐼) = (𝑀 /s ))
151, 14syl5eq 2790 1 (𝐼𝑉𝐺 = (𝑀 /s ))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  wcel 2110  Vcvv 3408   × cxp 5549  cfv 6380  (class class class)co 7213  2oc2o 8196   /s cqus 17010  freeMndcfrmd 18274   ~FG cefg 19096  freeGrpcfrgp 19097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-iota 6338  df-fun 6382  df-fv 6388  df-ov 7216  df-frgp 19100
This theorem is referenced by:  frgp0  19150  frgpeccl  19151  frgpadd  19153  frgpupf  19163  frgpup1  19165  frgpup3lem  19167  frgpnabllem2  19259
  Copyright terms: Public domain W3C validator