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

Theorem mgpval 18432
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 6158 . . . . . . 7 (𝑟 = 𝑅 → (.r𝑟) = (.r𝑅))
4 mgpval.2 . . . . . . 7 · = (.r𝑅)
53, 4syl6eqr 2673 . . . . . 6 (𝑟 = 𝑅 → (.r𝑟) = · )
65opeq2d 4384 . . . . 5 (𝑟 = 𝑅 → ⟨(+g‘ndx), (.r𝑟)⟩ = ⟨(+g‘ndx), · ⟩)
72, 6oveq12d 6633 . . . 4 (𝑟 = 𝑅 → (𝑟 sSet ⟨(+g‘ndx), (.r𝑟)⟩) = (𝑅 sSet ⟨(+g‘ndx), · ⟩))
8 df-mgp 18430 . . . 4 mulGrp = (𝑟 ∈ V ↦ (𝑟 sSet ⟨(+g‘ndx), (.r𝑟)⟩))
9 ovex 6643 . . . 4 (𝑅 sSet ⟨(+g‘ndx), · ⟩) ∈ V
107, 8, 9fvmpt 6249 . . 3 (𝑅 ∈ V → (mulGrp‘𝑅) = (𝑅 sSet ⟨(+g‘ndx), · ⟩))
11 fvprc 6152 . . . 4 𝑅 ∈ V → (mulGrp‘𝑅) = ∅)
12 reldmsets 15826 . . . . 5 Rel dom sSet
1312ovprc1 6649 . . . 4 𝑅 ∈ V → (𝑅 sSet ⟨(+g‘ndx), · ⟩) = ∅)
1411, 13eqtr4d 2658 . . 3 𝑅 ∈ V → (mulGrp‘𝑅) = (𝑅 sSet ⟨(+g‘ndx), · ⟩))
1510, 14pm2.61i 176 . 2 (mulGrp‘𝑅) = (𝑅 sSet ⟨(+g‘ndx), · ⟩)
161, 15eqtri 2643 1 𝑀 = (𝑅 sSet ⟨(+g‘ndx), · ⟩)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1480  wcel 1987  Vcvv 3190  c0 3897  cop 4161  cfv 5857  (class class class)co 6615  ndxcnx 15797   sSet csts 15798  +gcplusg 15881  .rcmulr 15882  mulGrpcmgp 18429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-iota 5820  df-fun 5859  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-sets 15806  df-mgp 18430
This theorem is referenced by:  mgpplusg  18433  mgplem  18434  mgpress  18440
  Copyright terms: Public domain W3C validator