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

Theorem mgpval 19220
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 6643 . . . . . . 7 (𝑟 = 𝑅 → (.r𝑟) = (.r𝑅))
4 mgpval.2 . . . . . . 7 · = (.r𝑅)
53, 4syl6eqr 2874 . . . . . 6 (𝑟 = 𝑅 → (.r𝑟) = · )
65opeq2d 4783 . . . . 5 (𝑟 = 𝑅 → ⟨(+g‘ndx), (.r𝑟)⟩ = ⟨(+g‘ndx), · ⟩)
72, 6oveq12d 7148 . . . 4 (𝑟 = 𝑅 → (𝑟 sSet ⟨(+g‘ndx), (.r𝑟)⟩) = (𝑅 sSet ⟨(+g‘ndx), · ⟩))
8 df-mgp 19218 . . . 4 mulGrp = (𝑟 ∈ V ↦ (𝑟 sSet ⟨(+g‘ndx), (.r𝑟)⟩))
9 ovex 7163 . . . 4 (𝑅 sSet ⟨(+g‘ndx), · ⟩) ∈ V
107, 8, 9fvmpt 6741 . . 3 (𝑅 ∈ V → (mulGrp‘𝑅) = (𝑅 sSet ⟨(+g‘ndx), · ⟩))
11 fvprc 6636 . . . 4 𝑅 ∈ V → (mulGrp‘𝑅) = ∅)
12 reldmsets 16489 . . . . 5 Rel dom sSet
1312ovprc1 7169 . . . 4 𝑅 ∈ V → (𝑅 sSet ⟨(+g‘ndx), · ⟩) = ∅)
1411, 13eqtr4d 2859 . . 3 𝑅 ∈ V → (mulGrp‘𝑅) = (𝑅 sSet ⟨(+g‘ndx), · ⟩))
1510, 14pm2.61i 185 . 2 (mulGrp‘𝑅) = (𝑅 sSet ⟨(+g‘ndx), · ⟩)
161, 15eqtri 2844 1 𝑀 = (𝑅 sSet ⟨(+g‘ndx), · ⟩)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1538  wcel 2115  Vcvv 3471  c0 4266  cop 4546  cfv 6328  (class class class)co 7130  ndxcnx 16458   sSet csts 16459  +gcplusg 16543  .rcmulr 16544  mulGrpcmgp 19217
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303
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 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-sets 16468  df-mgp 19218
This theorem is referenced by:  mgpplusg  19221  mgplem  19222  mgpress  19228
  Copyright terms: Public domain W3C validator