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Theorem mgpval 20172
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 6863 . . . . . . 7 (𝑟 = 𝑅 → (.r𝑟) = (.r𝑅))
4 mgpval.2 . . . . . . 7 · = (.r𝑅)
53, 4eqtr4di 2814 . . . . . 6 (𝑟 = 𝑅 → (.r𝑟) = · )
65opeq2d 4837 . . . . 5 (𝑟 = 𝑅 → ⟨(+g‘ndx), (.r𝑟)⟩ = ⟨(+g‘ndx), · ⟩)
72, 6oveq12d 7410 . . . 4 (𝑟 = 𝑅 → (𝑟 sSet ⟨(+g‘ndx), (.r𝑟)⟩) = (𝑅 sSet ⟨(+g‘ndx), · ⟩))
8 df-mgp 20170 . . . 4 mulGrp = (𝑟 ∈ V ↦ (𝑟 sSet ⟨(+g‘ndx), (.r𝑟)⟩))
9 ovex 7425 . . . 4 (𝑅 sSet ⟨(+g‘ndx), · ⟩) ∈ V
107, 8, 9fvmpt 6971 . . 3 (𝑅 ∈ V → (mulGrp‘𝑅) = (𝑅 sSet ⟨(+g‘ndx), · ⟩))
11 fvprc 6855 . . . 4 𝑅 ∈ V → (mulGrp‘𝑅) = ∅)
12 reldmsets 17184 . . . . 5 Rel dom sSet
1312ovprc1 7431 . . . 4 𝑅 ∈ V → (𝑅 sSet ⟨(+g‘ndx), · ⟩) = ∅)
1411, 13eqtr4d 2799 . . 3 𝑅 ∈ V → (mulGrp‘𝑅) = (𝑅 sSet ⟨(+g‘ndx), · ⟩))
1510, 14pm2.61i 183 . 2 (mulGrp‘𝑅) = (𝑅 sSet ⟨(+g‘ndx), · ⟩)
161, 15eqtri 2784 1 𝑀 = (𝑅 sSet ⟨(+g‘ndx), · ⟩)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1559  wcel 2141  Vcvv 3453  c0 4285  cop 4587  cfv 6517  (class class class)co 7392   sSet csts 17182  ndxcnx 17212  +gcplusg 17269  .rcmulr 17270  mulGrpcmgp 20169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389
This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-iota 6473  df-fun 6519  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-sets 17183  df-mgp 20170
This theorem is referenced by:  mgpplusg  20173  mgpbas  20174  mgpsca  20175  mgptset  20176  mgpds  20178  mgpress  20179
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