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Theorem frgpval 18880
 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 3459 . . 3 (𝐼𝑉𝐼 ∈ V)
3 xpeq1 5534 . . . . . . 7 (𝑖 = 𝐼 → (𝑖 × 2o) = (𝐼 × 2o))
43fveq2d 6650 . . . . . 6 (𝑖 = 𝐼 → (freeMnd‘(𝑖 × 2o)) = (freeMnd‘(𝐼 × 2o)))
5 frgpval.b . . . . . 6 𝑀 = (freeMnd‘(𝐼 × 2o))
64, 5eqtr4di 2851 . . . . 5 (𝑖 = 𝐼 → (freeMnd‘(𝑖 × 2o)) = 𝑀)
7 fveq2 6646 . . . . . 6 (𝑖 = 𝐼 → ( ~FG𝑖) = ( ~FG𝐼))
8 frgpval.r . . . . . 6 = ( ~FG𝐼)
97, 8eqtr4di 2851 . . . . 5 (𝑖 = 𝐼 → ( ~FG𝑖) = )
106, 9oveq12d 7154 . . . 4 (𝑖 = 𝐼 → ((freeMnd‘(𝑖 × 2o)) /s ( ~FG𝑖)) = (𝑀 /s ))
11 df-frgp 18832 . . . 4 freeGrp = (𝑖 ∈ V ↦ ((freeMnd‘(𝑖 × 2o)) /s ( ~FG𝑖)))
12 ovex 7169 . . . 4 (𝑀 /s ) ∈ V
1310, 11, 12fvmpt 6746 . . 3 (𝐼 ∈ V → (freeGrp‘𝐼) = (𝑀 /s ))
142, 13syl 17 . 2 (𝐼𝑉 → (freeGrp‘𝐼) = (𝑀 /s ))
151, 14syl5eq 2845 1 (𝐼𝑉𝐺 = (𝑀 /s ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  Vcvv 3441   × cxp 5518  ‘cfv 6325  (class class class)co 7136  2oc2o 8082   /s cqus 16773  freeMndcfrmd 18007   ~FG cefg 18828  freeGrpcfrgp 18829 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-iota 6284  df-fun 6327  df-fv 6333  df-ov 7139  df-frgp 18832 This theorem is referenced by:  frgp0  18882  frgpeccl  18883  frgpadd  18885  frgpupf  18895  frgpup1  18897  frgpup3lem  18899  frgpnabllem2  18991
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