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Theorem mgpval 20054
Description: Value of the multiplication group operation. (Contributed by Mario Carneiro, 21-Dec-2014.)
Hypotheses
Ref Expression
mgpval.1 𝑀 = (mulGrp‘𝑅)
mgpval.2 · = (.r𝑅)
Assertion
Ref Expression
mgpval 𝑀 = (𝑅 sSet ⟨(+g‘ndx), · ⟩)

Proof of Theorem mgpval
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 mgpval.1 . 2 𝑀 = (mulGrp‘𝑅)
2 id 22 . . . . 5 (𝑟 = 𝑅𝑟 = 𝑅)
3 fveq2 6817 . . . . . . 7 (𝑟 = 𝑅 → (.r𝑟) = (.r𝑅))
4 mgpval.2 . . . . . . 7 · = (.r𝑅)
53, 4eqtr4di 2783 . . . . . 6 (𝑟 = 𝑅 → (.r𝑟) = · )
65opeq2d 4830 . . . . 5 (𝑟 = 𝑅 → ⟨(+g‘ndx), (.r𝑟)⟩ = ⟨(+g‘ndx), · ⟩)
72, 6oveq12d 7359 . . . 4 (𝑟 = 𝑅 → (𝑟 sSet ⟨(+g‘ndx), (.r𝑟)⟩) = (𝑅 sSet ⟨(+g‘ndx), · ⟩))
8 df-mgp 20052 . . . 4 mulGrp = (𝑟 ∈ V ↦ (𝑟 sSet ⟨(+g‘ndx), (.r𝑟)⟩))
9 ovex 7374 . . . 4 (𝑅 sSet ⟨(+g‘ndx), · ⟩) ∈ V
107, 8, 9fvmpt 6924 . . 3 (𝑅 ∈ V → (mulGrp‘𝑅) = (𝑅 sSet ⟨(+g‘ndx), · ⟩))
11 fvprc 6809 . . . 4 𝑅 ∈ V → (mulGrp‘𝑅) = ∅)
12 reldmsets 17068 . . . . 5 Rel dom sSet
1312ovprc1 7380 . . . 4 𝑅 ∈ V → (𝑅 sSet ⟨(+g‘ndx), · ⟩) = ∅)
1411, 13eqtr4d 2768 . . 3 𝑅 ∈ V → (mulGrp‘𝑅) = (𝑅 sSet ⟨(+g‘ndx), · ⟩))
1510, 14pm2.61i 182 . 2 (mulGrp‘𝑅) = (𝑅 sSet ⟨(+g‘ndx), · ⟩)
161, 15eqtri 2753 1 𝑀 = (𝑅 sSet ⟨(+g‘ndx), · ⟩)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1541  wcel 2110  Vcvv 3434  c0 4281  cop 4580  cfv 6477  (class class class)co 7341   sSet csts 17066  ndxcnx 17096  +gcplusg 17153  .rcmulr 17154  mulGrpcmgp 20051
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-iota 6433  df-fun 6479  df-fv 6485  df-ov 7344  df-oprab 7345  df-mpo 7346  df-sets 17067  df-mgp 20052
This theorem is referenced by:  mgpplusg  20055  mgpbas  20056  mgpsca  20057  mgptset  20058  mgpds  20060  mgpress  20061
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