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Theorem frgpval 17994
 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 3185 . . 3 (𝐼𝑉𝐼 ∈ V)
3 xpeq1 5052 . . . . . . 7 (𝑖 = 𝐼 → (𝑖 × 2𝑜) = (𝐼 × 2𝑜))
43fveq2d 6107 . . . . . 6 (𝑖 = 𝐼 → (freeMnd‘(𝑖 × 2𝑜)) = (freeMnd‘(𝐼 × 2𝑜)))
5 frgpval.b . . . . . 6 𝑀 = (freeMnd‘(𝐼 × 2𝑜))
64, 5syl6eqr 2662 . . . . 5 (𝑖 = 𝐼 → (freeMnd‘(𝑖 × 2𝑜)) = 𝑀)
7 fveq2 6103 . . . . . 6 (𝑖 = 𝐼 → ( ~FG𝑖) = ( ~FG𝐼))
8 frgpval.r . . . . . 6 = ( ~FG𝐼)
97, 8syl6eqr 2662 . . . . 5 (𝑖 = 𝐼 → ( ~FG𝑖) = )
106, 9oveq12d 6567 . . . 4 (𝑖 = 𝐼 → ((freeMnd‘(𝑖 × 2𝑜)) /s ( ~FG𝑖)) = (𝑀 /s ))
11 df-frgp 17946 . . . 4 freeGrp = (𝑖 ∈ V ↦ ((freeMnd‘(𝑖 × 2𝑜)) /s ( ~FG𝑖)))
12 ovex 6577 . . . 4 (𝑀 /s ) ∈ V
1310, 11, 12fvmpt 6191 . . 3 (𝐼 ∈ V → (freeGrp‘𝐼) = (𝑀 /s ))
142, 13syl 17 . 2 (𝐼𝑉 → (freeGrp‘𝐼) = (𝑀 /s ))
151, 14syl5eq 2656 1 (𝐼𝑉𝐺 = (𝑀 /s ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173   × cxp 5036  ‘cfv 5804  (class class class)co 6549  2𝑜c2o 7441   /s cqus 15988  freeMndcfrmd 17207   ~FG cefg 17942  freeGrpcfrgp 17943 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-frgp 17946 This theorem is referenced by:  frgp0  17996  frgpeccl  17997  frgpadd  17999  frgpupf  18009  frgpup1  18011  frgpup3lem  18013  frgpnabllem2  18100
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