Theorem opprmulfvalg 14082

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 14081	. . . 4 ⊢ (𝑅 ∈ 𝑉 → 𝑂 = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩))
6	5	fveq2d 5643	. . 3 ⊢ (𝑅 ∈ 𝑉 → (.r‘𝑂) = (.r‘(𝑅 sSet ⟨(.r‘ndx), tpos · ⟩)))
7		mulrslid 13214	. . . . . . 7 ⊢ (.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ)
8	7	slotex 13108	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (.r‘𝑅) ∈ V)
9	3, 8	eqeltrid 2318	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → · ∈ V)
10		tposexg 6423	. . . . 5 ⊢ ( · ∈ V → tpos · ∈ V)
11	9, 10	syl 14	. . . 4 ⊢ (𝑅 ∈ 𝑉 → tpos · ∈ V)
12	7	setsslid 13132	. . . 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 2267	. 2 ⊢ (𝑅 ∈ 𝑉 → (.r‘𝑂) = tpos · )
15	1, 14	eqtrid 2276	1 ⊢ (𝑅 ∈ 𝑉 → ∙ = tpos · )

Syntax hints:   → wi 4   = wceq 1397   ∈ wcel 2202  Vcvv 2802  ⟨cop 3672  ‘cfv 5326  (class class class)co 6017  tpos ctpos 6409  ndxcnx 13078   sSet csts 13079  Basecbs 13081  ._rcmulr 13160  opp_rcoppr 14079

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-tpos 6410  df-inn 9143  df-2 9201  df-3 9202  df-ndx 13084  df-slot 13085  df-sets 13088  df-mulr 13173  df-oppr 14080

This theorem is referenced by:  opprmulg  14083