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Theorem isorng 29130
 Description: An ordered ring is a ring with a total ordering compatible with its operations. (Contributed by Thierry Arnoux, 18-Jan-2018.)
Hypotheses
Ref Expression
isorng.0 𝐵 = (Base‘𝑅)
isorng.1 0 = (0g𝑅)
isorng.2 · = (.r𝑅)
isorng.3 = (le‘𝑅)
Assertion
Ref Expression
isorng (𝑅 ∈ oRing ↔ (𝑅 ∈ Ring ∧ 𝑅 ∈ oGrp ∧ ∀𝑎𝐵𝑏𝐵 (( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏))))
Distinct variable groups:   𝑎,𝑏,𝐵   𝑅,𝑎,𝑏
Allowed substitution hints:   · (𝑎,𝑏)   (𝑎,𝑏)   0 (𝑎,𝑏)

Proof of Theorem isorng
Dummy variables 𝑙 𝑟 𝑡 𝑣 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elin 3758 . . 3 (𝑅 ∈ (Ring ∩ oGrp) ↔ (𝑅 ∈ Ring ∧ 𝑅 ∈ oGrp))
21anbi1i 727 . 2 ((𝑅 ∈ (Ring ∩ oGrp) ∧ ∀𝑎𝐵𝑏𝐵 (( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏))) ↔ ((𝑅 ∈ Ring ∧ 𝑅 ∈ oGrp) ∧ ∀𝑎𝐵𝑏𝐵 (( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏))))
3 fvex 6113 . . . . . 6 (.r𝑟) ∈ V
43a1i 11 . . . . 5 (𝑟 = 𝑅 → (.r𝑟) ∈ V)
5 simpr 476 . . . . . . . . . . 11 ((𝑟 = 𝑅𝑡 = (.r𝑟)) → 𝑡 = (.r𝑟))
6 simpl 472 . . . . . . . . . . . . 13 ((𝑟 = 𝑅𝑡 = (.r𝑟)) → 𝑟 = 𝑅)
76fveq2d 6107 . . . . . . . . . . . 12 ((𝑟 = 𝑅𝑡 = (.r𝑟)) → (.r𝑟) = (.r𝑅))
8 isorng.2 . . . . . . . . . . . 12 · = (.r𝑅)
97, 8syl6eqr 2662 . . . . . . . . . . 11 ((𝑟 = 𝑅𝑡 = (.r𝑟)) → (.r𝑟) = · )
105, 9eqtrd 2644 . . . . . . . . . 10 ((𝑟 = 𝑅𝑡 = (.r𝑟)) → 𝑡 = · )
1110oveqd 6566 . . . . . . . . 9 ((𝑟 = 𝑅𝑡 = (.r𝑟)) → (𝑎𝑡𝑏) = (𝑎 · 𝑏))
1211breq2d 4595 . . . . . . . 8 ((𝑟 = 𝑅𝑡 = (.r𝑟)) → ( 0 𝑙(𝑎𝑡𝑏) ↔ 0 𝑙(𝑎 · 𝑏)))
1312imbi2d 329 . . . . . . 7 ((𝑟 = 𝑅𝑡 = (.r𝑟)) → ((( 0 𝑙𝑎0 𝑙𝑏) → 0 𝑙(𝑎𝑡𝑏)) ↔ (( 0 𝑙𝑎0 𝑙𝑏) → 0 𝑙(𝑎 · 𝑏))))
14132ralbidv 2972 . . . . . 6 ((𝑟 = 𝑅𝑡 = (.r𝑟)) → (∀𝑎𝐵𝑏𝐵 (( 0 𝑙𝑎0 𝑙𝑏) → 0 𝑙(𝑎𝑡𝑏)) ↔ ∀𝑎𝐵𝑏𝐵 (( 0 𝑙𝑎0 𝑙𝑏) → 0 𝑙(𝑎 · 𝑏))))
1514sbcbidv 3457 . . . . 5 ((𝑟 = 𝑅𝑡 = (.r𝑟)) → ([(le‘𝑟) / 𝑙]𝑎𝐵𝑏𝐵 (( 0 𝑙𝑎0 𝑙𝑏) → 0 𝑙(𝑎𝑡𝑏)) ↔ [(le‘𝑟) / 𝑙]𝑎𝐵𝑏𝐵 (( 0 𝑙𝑎0 𝑙𝑏) → 0 𝑙(𝑎 · 𝑏))))
164, 15sbcied 3439 . . . 4 (𝑟 = 𝑅 → ([(.r𝑟) / 𝑡][(le‘𝑟) / 𝑙]𝑎𝐵𝑏𝐵 (( 0 𝑙𝑎0 𝑙𝑏) → 0 𝑙(𝑎𝑡𝑏)) ↔ [(le‘𝑟) / 𝑙]𝑎𝐵𝑏𝐵 (( 0 𝑙𝑎0 𝑙𝑏) → 0 𝑙(𝑎 · 𝑏))))
17 fvex 6113 . . . . . . 7 (Base‘𝑟) ∈ V
1817a1i 11 . . . . . 6 (𝑟 = 𝑅 → (Base‘𝑟) ∈ V)
19 simpr 476 . . . . . . . . . . 11 ((𝑟 = 𝑅𝑣 = (Base‘𝑟)) → 𝑣 = (Base‘𝑟))
20 fveq2 6103 . . . . . . . . . . . . 13 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
21 isorng.0 . . . . . . . . . . . . 13 𝐵 = (Base‘𝑅)
2220, 21syl6eqr 2662 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
2322adantr 480 . . . . . . . . . . 11 ((𝑟 = 𝑅𝑣 = (Base‘𝑟)) → (Base‘𝑟) = 𝐵)
2419, 23eqtrd 2644 . . . . . . . . . 10 ((𝑟 = 𝑅𝑣 = (Base‘𝑟)) → 𝑣 = 𝐵)
25 raleq 3115 . . . . . . . . . . 11 (𝑣 = 𝐵 → (∀𝑏𝑣 ((𝑧𝑙𝑎𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏)) ↔ ∀𝑏𝐵 ((𝑧𝑙𝑎𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏))))
2625raleqbi1dv 3123 . . . . . . . . . 10 (𝑣 = 𝐵 → (∀𝑎𝑣𝑏𝑣 ((𝑧𝑙𝑎𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏)) ↔ ∀𝑎𝐵𝑏𝐵 ((𝑧𝑙𝑎𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏))))
2724, 26syl 17 . . . . . . . . 9 ((𝑟 = 𝑅𝑣 = (Base‘𝑟)) → (∀𝑎𝑣𝑏𝑣 ((𝑧𝑙𝑎𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏)) ↔ ∀𝑎𝐵𝑏𝐵 ((𝑧𝑙𝑎𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏))))
2827sbcbidv 3457 . . . . . . . 8 ((𝑟 = 𝑅𝑣 = (Base‘𝑟)) → ([(le‘𝑟) / 𝑙]𝑎𝑣𝑏𝑣 ((𝑧𝑙𝑎𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏)) ↔ [(le‘𝑟) / 𝑙]𝑎𝐵𝑏𝐵 ((𝑧𝑙𝑎𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏))))
2928sbcbidv 3457 . . . . . . 7 ((𝑟 = 𝑅𝑣 = (Base‘𝑟)) → ([(.r𝑟) / 𝑡][(le‘𝑟) / 𝑙]𝑎𝑣𝑏𝑣 ((𝑧𝑙𝑎𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏)) ↔ [(.r𝑟) / 𝑡][(le‘𝑟) / 𝑙]𝑎𝐵𝑏𝐵 ((𝑧𝑙𝑎𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏))))
3029sbcbidv 3457 . . . . . 6 ((𝑟 = 𝑅𝑣 = (Base‘𝑟)) → ([(0g𝑟) / 𝑧][(.r𝑟) / 𝑡][(le‘𝑟) / 𝑙]𝑎𝑣𝑏𝑣 ((𝑧𝑙𝑎𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏)) ↔ [(0g𝑟) / 𝑧][(.r𝑟) / 𝑡][(le‘𝑟) / 𝑙]𝑎𝐵𝑏𝐵 ((𝑧𝑙𝑎𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏))))
3118, 30sbcied 3439 . . . . 5 (𝑟 = 𝑅 → ([(Base‘𝑟) / 𝑣][(0g𝑟) / 𝑧][(.r𝑟) / 𝑡][(le‘𝑟) / 𝑙]𝑎𝑣𝑏𝑣 ((𝑧𝑙𝑎𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏)) ↔ [(0g𝑟) / 𝑧][(.r𝑟) / 𝑡][(le‘𝑟) / 𝑙]𝑎𝐵𝑏𝐵 ((𝑧𝑙𝑎𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏))))
32 fvex 6113 . . . . . . 7 (0g𝑟) ∈ V
3332a1i 11 . . . . . 6 (𝑟 = 𝑅 → (0g𝑟) ∈ V)
34 simpr 476 . . . . . . . . . . . . 13 ((𝑟 = 𝑅𝑧 = (0g𝑟)) → 𝑧 = (0g𝑟))
35 fveq2 6103 . . . . . . . . . . . . . . 15 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
36 isorng.1 . . . . . . . . . . . . . . 15 0 = (0g𝑅)
3735, 36syl6eqr 2662 . . . . . . . . . . . . . 14 (𝑟 = 𝑅 → (0g𝑟) = 0 )
3837adantr 480 . . . . . . . . . . . . 13 ((𝑟 = 𝑅𝑧 = (0g𝑟)) → (0g𝑟) = 0 )
3934, 38eqtrd 2644 . . . . . . . . . . . 12 ((𝑟 = 𝑅𝑧 = (0g𝑟)) → 𝑧 = 0 )
4039breq1d 4593 . . . . . . . . . . 11 ((𝑟 = 𝑅𝑧 = (0g𝑟)) → (𝑧𝑙𝑎0 𝑙𝑎))
4139breq1d 4593 . . . . . . . . . . 11 ((𝑟 = 𝑅𝑧 = (0g𝑟)) → (𝑧𝑙𝑏0 𝑙𝑏))
4240, 41anbi12d 743 . . . . . . . . . 10 ((𝑟 = 𝑅𝑧 = (0g𝑟)) → ((𝑧𝑙𝑎𝑧𝑙𝑏) ↔ ( 0 𝑙𝑎0 𝑙𝑏)))
4339breq1d 4593 . . . . . . . . . 10 ((𝑟 = 𝑅𝑧 = (0g𝑟)) → (𝑧𝑙(𝑎𝑡𝑏) ↔ 0 𝑙(𝑎𝑡𝑏)))
4442, 43imbi12d 333 . . . . . . . . 9 ((𝑟 = 𝑅𝑧 = (0g𝑟)) → (((𝑧𝑙𝑎𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏)) ↔ (( 0 𝑙𝑎0 𝑙𝑏) → 0 𝑙(𝑎𝑡𝑏))))
45442ralbidv 2972 . . . . . . . 8 ((𝑟 = 𝑅𝑧 = (0g𝑟)) → (∀𝑎𝐵𝑏𝐵 ((𝑧𝑙𝑎𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏)) ↔ ∀𝑎𝐵𝑏𝐵 (( 0 𝑙𝑎0 𝑙𝑏) → 0 𝑙(𝑎𝑡𝑏))))
4645sbcbidv 3457 . . . . . . 7 ((𝑟 = 𝑅𝑧 = (0g𝑟)) → ([(le‘𝑟) / 𝑙]𝑎𝐵𝑏𝐵 ((𝑧𝑙𝑎𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏)) ↔ [(le‘𝑟) / 𝑙]𝑎𝐵𝑏𝐵 (( 0 𝑙𝑎0 𝑙𝑏) → 0 𝑙(𝑎𝑡𝑏))))
4746sbcbidv 3457 . . . . . 6 ((𝑟 = 𝑅𝑧 = (0g𝑟)) → ([(.r𝑟) / 𝑡][(le‘𝑟) / 𝑙]𝑎𝐵𝑏𝐵 ((𝑧𝑙𝑎𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏)) ↔ [(.r𝑟) / 𝑡][(le‘𝑟) / 𝑙]𝑎𝐵𝑏𝐵 (( 0 𝑙𝑎0 𝑙𝑏) → 0 𝑙(𝑎𝑡𝑏))))
4833, 47sbcied 3439 . . . . 5 (𝑟 = 𝑅 → ([(0g𝑟) / 𝑧][(.r𝑟) / 𝑡][(le‘𝑟) / 𝑙]𝑎𝐵𝑏𝐵 ((𝑧𝑙𝑎𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏)) ↔ [(.r𝑟) / 𝑡][(le‘𝑟) / 𝑙]𝑎𝐵𝑏𝐵 (( 0 𝑙𝑎0 𝑙𝑏) → 0 𝑙(𝑎𝑡𝑏))))
4931, 48bitr2d 268 . . . 4 (𝑟 = 𝑅 → ([(.r𝑟) / 𝑡][(le‘𝑟) / 𝑙]𝑎𝐵𝑏𝐵 (( 0 𝑙𝑎0 𝑙𝑏) → 0 𝑙(𝑎𝑡𝑏)) ↔ [(Base‘𝑟) / 𝑣][(0g𝑟) / 𝑧][(.r𝑟) / 𝑡][(le‘𝑟) / 𝑙]𝑎𝑣𝑏𝑣 ((𝑧𝑙𝑎𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏))))
50 fvex 6113 . . . . . 6 (le‘𝑟) ∈ V
5150a1i 11 . . . . 5 (𝑟 = 𝑅 → (le‘𝑟) ∈ V)
52 simpr 476 . . . . . . . . . 10 ((𝑟 = 𝑅𝑙 = (le‘𝑟)) → 𝑙 = (le‘𝑟))
53 simpl 472 . . . . . . . . . . . 12 ((𝑟 = 𝑅𝑙 = (le‘𝑟)) → 𝑟 = 𝑅)
5453fveq2d 6107 . . . . . . . . . . 11 ((𝑟 = 𝑅𝑙 = (le‘𝑟)) → (le‘𝑟) = (le‘𝑅))
55 isorng.3 . . . . . . . . . . 11 = (le‘𝑅)
5654, 55syl6eqr 2662 . . . . . . . . . 10 ((𝑟 = 𝑅𝑙 = (le‘𝑟)) → (le‘𝑟) = )
5752, 56eqtrd 2644 . . . . . . . . 9 ((𝑟 = 𝑅𝑙 = (le‘𝑟)) → 𝑙 = )
5857breqd 4594 . . . . . . . 8 ((𝑟 = 𝑅𝑙 = (le‘𝑟)) → ( 0 𝑙𝑎0 𝑎))
5957breqd 4594 . . . . . . . 8 ((𝑟 = 𝑅𝑙 = (le‘𝑟)) → ( 0 𝑙𝑏0 𝑏))
6058, 59anbi12d 743 . . . . . . 7 ((𝑟 = 𝑅𝑙 = (le‘𝑟)) → (( 0 𝑙𝑎0 𝑙𝑏) ↔ ( 0 𝑎0 𝑏)))
6157breqd 4594 . . . . . . 7 ((𝑟 = 𝑅𝑙 = (le‘𝑟)) → ( 0 𝑙(𝑎 · 𝑏) ↔ 0 (𝑎 · 𝑏)))
6260, 61imbi12d 333 . . . . . 6 ((𝑟 = 𝑅𝑙 = (le‘𝑟)) → ((( 0 𝑙𝑎0 𝑙𝑏) → 0 𝑙(𝑎 · 𝑏)) ↔ (( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏))))
63622ralbidv 2972 . . . . 5 ((𝑟 = 𝑅𝑙 = (le‘𝑟)) → (∀𝑎𝐵𝑏𝐵 (( 0 𝑙𝑎0 𝑙𝑏) → 0 𝑙(𝑎 · 𝑏)) ↔ ∀𝑎𝐵𝑏𝐵 (( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏))))
6451, 63sbcied 3439 . . . 4 (𝑟 = 𝑅 → ([(le‘𝑟) / 𝑙]𝑎𝐵𝑏𝐵 (( 0 𝑙𝑎0 𝑙𝑏) → 0 𝑙(𝑎 · 𝑏)) ↔ ∀𝑎𝐵𝑏𝐵 (( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏))))
6516, 49, 643bitr3d 297 . . 3 (𝑟 = 𝑅 → ([(Base‘𝑟) / 𝑣][(0g𝑟) / 𝑧][(.r𝑟) / 𝑡][(le‘𝑟) / 𝑙]𝑎𝑣𝑏𝑣 ((𝑧𝑙𝑎𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏)) ↔ ∀𝑎𝐵𝑏𝐵 (( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏))))
66 df-orng 29128 . . 3 oRing = {𝑟 ∈ (Ring ∩ oGrp) ∣ [(Base‘𝑟) / 𝑣][(0g𝑟) / 𝑧][(.r𝑟) / 𝑡][(le‘𝑟) / 𝑙]𝑎𝑣𝑏𝑣 ((𝑧𝑙𝑎𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏))}
6765, 66elrab2 3333 . 2 (𝑅 ∈ oRing ↔ (𝑅 ∈ (Ring ∩ oGrp) ∧ ∀𝑎𝐵𝑏𝐵 (( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏))))
68 df-3an 1033 . 2 ((𝑅 ∈ Ring ∧ 𝑅 ∈ oGrp ∧ ∀𝑎𝐵𝑏𝐵 (( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏))) ↔ ((𝑅 ∈ Ring ∧ 𝑅 ∈ oGrp) ∧ ∀𝑎𝐵𝑏𝐵 (( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏))))
692, 67, 683bitr4i 291 1 (𝑅 ∈ oRing ↔ (𝑅 ∈ Ring ∧ 𝑅 ∈ oGrp ∧ ∀𝑎𝐵𝑏𝐵 (( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  [wsbc 3402   ∩ cin 3539   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  .rcmulr 15769  lecple 15775  0gc0g 15923  Ringcrg 18370  oGrpcogrp 29029  oRingcorng 29126 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-orng 29128 This theorem is referenced by:  orngring  29131  orngogrp  29132  orngmul  29134  suborng  29146  reofld  29171
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