Theorem opprmulfvalg 14033

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 14032	. . . 4 ⊢ (𝑅 ∈ 𝑉 → 𝑂 = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩))
6	5	fveq2d 5631	. . 3 ⊢ (𝑅 ∈ 𝑉 → (.r‘𝑂) = (.r‘(𝑅 sSet ⟨(.r‘ndx), tpos · ⟩)))
7		mulrslid 13165	. . . . . . 7 ⊢ (.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ)
8	7	slotex 13059	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (.r‘𝑅) ∈ V)
9	3, 8	eqeltrid 2316	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → · ∈ V)
10		tposexg 6404	. . . . 5 ⊢ ( · ∈ V → tpos · ∈ V)
11	9, 10	syl 14	. . . 4 ⊢ (𝑅 ∈ 𝑉 → tpos · ∈ V)
12	7	setsslid 13083	. . . 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 2265	. 2 ⊢ (𝑅 ∈ 𝑉 → (.r‘𝑂) = tpos · )
15	1, 14	eqtrid 2274	1 ⊢ (𝑅 ∈ 𝑉 → ∙ = tpos · )

Syntax hints:   → wi 4   = wceq 1395   ∈ wcel 2200  Vcvv 2799  ⟨cop 3669  ‘cfv 5318  (class class class)co 6001  tpos ctpos 6390  ndxcnx 13029   sSet csts 13030  Basecbs 13032  ._rcmulr 13111  opp_rcoppr 14030

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1re 8093  ax-addrcl 8096

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-tpos 6391  df-inn 9111  df-2 9169  df-3 9170  df-ndx 13035  df-slot 13036  df-sets 13039  df-mulr 13124  df-oppr 14031

This theorem is referenced by:  opprmulg  14034