Theorem opprmulfvalg 14313

Description: Value of the multiplication operation of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)

Hypotheses

Ref	Expression
opprval.1	⊢ 𝐵 = (Base‘𝑅)
opprval.2	⊢ · = (.r‘𝑅)
opprval.3	⊢ 𝑂 = (oppr‘𝑅)
opprmulfval.4	⊢ ∙ = (.r‘𝑂)

Assertion

Ref	Expression
opprmulfvalg	⊢ (𝑅 ∈ 𝑉 → ∙ = tpos · )

Proof of Theorem opprmulfvalg

Step	Hyp	Ref	Expression
1		opprmulfval.4	. 2 ⊢ ∙ = (.r‘𝑂)
2		opprval.1	. . . . 5 ⊢ 𝐵 = (Base‘𝑅)
3		opprval.2	. . . . 5 ⊢ · = (.r‘𝑅)
4		opprval.3	. . . . 5 ⊢ 𝑂 = (oppr‘𝑅)
5	2, 3, 4	opprvalg 14312	. . . 4 ⊢ (𝑅 ∈ 𝑉 → 𝑂 = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩))
6	5	fveq2d 5679	. . 3 ⊢ (𝑅 ∈ 𝑉 → (.r‘𝑂) = (.r‘(𝑅 sSet ⟨(.r‘ndx), tpos · ⟩)))
7		mulrslid 13429	. . . . . . 7 ⊢ (.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ)
8	7	slotex 13323	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (.r‘𝑅) ∈ V)
9	3, 8	eqeltrid 2321	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → · ∈ V)
10		tposexg 6502	. . . . 5 ⊢ ( · ∈ V → tpos · ∈ V)
11	9, 10	syl 14	. . . 4 ⊢ (𝑅 ∈ 𝑉 → tpos · ∈ V)
12	7	setsslid 13347	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ tpos · ∈ V) → tpos · = (.r‘(𝑅 sSet ⟨(.r‘ndx), tpos · ⟩)))
13	11, 12	mpdan 421	. . 3 ⊢ (𝑅 ∈ 𝑉 → tpos · = (.r‘(𝑅 sSet ⟨(.r‘ndx), tpos · ⟩)))
14	6, 13	eqtr4d 2270	. 2 ⊢ (𝑅 ∈ 𝑉 → (.r‘𝑂) = tpos · )
15	1, 14	eqtrid 2279	1 ⊢ (𝑅 ∈ 𝑉 → ∙ = tpos · )

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2205  Vcvv 2815  ⟨cop 3697  ‘cfv 5357  (class class class)co 6058  tpos ctpos 6488  ndxcnx 13293   sSet csts 13294  Basecbs 13296  ._rcmulr 13375  opp_rcoppr 14310

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-tpos 6489  df-inn 9255  df-2 9313  df-3 9314  df-ndx 13299  df-slot 13300  df-sets 13303  df-mulr 13388  df-oppr 14311

This theorem is referenced by:  opprmulg  14314