Theorem mgpval 20179

Description: Value of the multiplication group operation. (Contributed by Mario Carneiro, 21-Dec-2014.)

Hypotheses

Ref	Expression
mgpval.1	⊢ 𝑀 = (mulGrp‘𝑅)
mgpval.2	⊢ · = (.r‘𝑅)

Assertion

Ref	Expression
mgpval	⊢ 𝑀 = (𝑅 sSet ⟨(+g‘ndx), · ⟩)

Proof of Theorem mgpval

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mgpval.1	. 2 ⊢ 𝑀 = (mulGrp‘𝑅)
2		id 22	. . . . 5 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅)
3		fveq2 6861	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅))
4		mgpval.2	. . . . . . 7 ⊢ · = (.r‘𝑅)
5	3, 4	eqtr4di 2814	. . . . . 6 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = · )
6	5	opeq2d 4835	. . . . 5 ⊢ (𝑟 = 𝑅 → ⟨(+g‘ndx), (.r‘𝑟)⟩ = ⟨(+g‘ndx), · ⟩)
7	2, 6	oveq12d 7408	. . . 4 ⊢ (𝑟 = 𝑅 → (𝑟 sSet ⟨(+g‘ndx), (.r‘𝑟)⟩) = (𝑅 sSet ⟨(+g‘ndx), · ⟩))
8		df-mgp 20177	. . . 4 ⊢ mulGrp = (𝑟 ∈ V ↦ (𝑟 sSet ⟨(+g‘ndx), (.r‘𝑟)⟩))
9		ovex 7423	. . . 4 ⊢ (𝑅 sSet ⟨(+g‘ndx), · ⟩) ∈ V
10	7, 8, 9	fvmpt 6969	. . 3 ⊢ (𝑅 ∈ V → (mulGrp‘𝑅) = (𝑅 sSet ⟨(+g‘ndx), · ⟩))
11		fvprc 6853	. . . 4 ⊢ (¬ 𝑅 ∈ V → (mulGrp‘𝑅) = ∅)
12		reldmsets 17191	. . . . 5 ⊢ Rel dom sSet
13	12	ovprc1 7429	. . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑅 sSet ⟨(+g‘ndx), · ⟩) = ∅)
14	11, 13	eqtr4d 2799	. . 3 ⊢ (¬ 𝑅 ∈ V → (mulGrp‘𝑅) = (𝑅 sSet ⟨(+g‘ndx), · ⟩))
15	10, 14	pm2.61i 183	. 2 ⊢ (mulGrp‘𝑅) = (𝑅 sSet ⟨(+g‘ndx), · ⟩)
16	1, 15	eqtri 2784	1 ⊢ 𝑀 = (𝑅 sSet ⟨(+g‘ndx), · ⟩)

Syntax hints:  ¬ wn 3   = wceq 1559   ∈ wcel 2141  Vcvv 3453  ∅c0 4283  ⟨cop 4585  ‘cfv 6515  (class class class)co 7390   sSet csts 17189  ndxcnx 17219  +_gcplusg 17276  ._rcmulr 17277  mulGrpcmgp 20176

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 5243  ax-nul 5253  ax-pr 5387

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 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-iota 6471  df-fun 6517  df-fv 6523  df-ov 7393  df-oprab 7394  df-mpo 7395  df-sets 17190  df-mgp 20177

This theorem is referenced by:  mgpplusg  20180  mgpbas  20181  mgpsca  20182  mgptset  20183  mgpds  20185  mgpress  20186