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Theorem isorng 30557
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 4057 . . 3 (𝑅 ∈ (Ring ∩ oGrp) ↔ (𝑅 ∈ Ring ∧ 𝑅 ∈ oGrp))
21anbi1i 614 . 2 ((𝑅 ∈ (Ring ∩ oGrp) ∧ ∀𝑎𝐵𝑏𝐵 (( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏))) ↔ ((𝑅 ∈ Ring ∧ 𝑅 ∈ oGrp) ∧ ∀𝑎𝐵𝑏𝐵 (( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏))))
3 fvexd 6514 . . . . 5 (𝑟 = 𝑅 → (.r𝑟) ∈ V)
4 simpr 477 . . . . . . . . . . 11 ((𝑟 = 𝑅𝑡 = (.r𝑟)) → 𝑡 = (.r𝑟))
5 simpl 475 . . . . . . . . . . . . 13 ((𝑟 = 𝑅𝑡 = (.r𝑟)) → 𝑟 = 𝑅)
65fveq2d 6503 . . . . . . . . . . . 12 ((𝑟 = 𝑅𝑡 = (.r𝑟)) → (.r𝑟) = (.r𝑅))
7 isorng.2 . . . . . . . . . . . 12 · = (.r𝑅)
86, 7syl6eqr 2832 . . . . . . . . . . 11 ((𝑟 = 𝑅𝑡 = (.r𝑟)) → (.r𝑟) = · )
94, 8eqtrd 2814 . . . . . . . . . 10 ((𝑟 = 𝑅𝑡 = (.r𝑟)) → 𝑡 = · )
109oveqd 6993 . . . . . . . . 9 ((𝑟 = 𝑅𝑡 = (.r𝑟)) → (𝑎𝑡𝑏) = (𝑎 · 𝑏))
1110breq2d 4941 . . . . . . . 8 ((𝑟 = 𝑅𝑡 = (.r𝑟)) → ( 0 𝑙(𝑎𝑡𝑏) ↔ 0 𝑙(𝑎 · 𝑏)))
1211imbi2d 333 . . . . . . 7 ((𝑟 = 𝑅𝑡 = (.r𝑟)) → ((( 0 𝑙𝑎0 𝑙𝑏) → 0 𝑙(𝑎𝑡𝑏)) ↔ (( 0 𝑙𝑎0 𝑙𝑏) → 0 𝑙(𝑎 · 𝑏))))
13122ralbidv 3149 . . . . . 6 ((𝑟 = 𝑅𝑡 = (.r𝑟)) → (∀𝑎𝐵𝑏𝐵 (( 0 𝑙𝑎0 𝑙𝑏) → 0 𝑙(𝑎𝑡𝑏)) ↔ ∀𝑎𝐵𝑏𝐵 (( 0 𝑙𝑎0 𝑙𝑏) → 0 𝑙(𝑎 · 𝑏))))
1413sbcbidv 3731 . . . . 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))))
153, 14sbcied 3718 . . . 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))))
16 fvexd 6514 . . . . . 6 (𝑟 = 𝑅 → (Base‘𝑟) ∈ V)
17 simpr 477 . . . . . . . . . . 11 ((𝑟 = 𝑅𝑣 = (Base‘𝑟)) → 𝑣 = (Base‘𝑟))
18 fveq2 6499 . . . . . . . . . . . . 13 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
19 isorng.0 . . . . . . . . . . . . 13 𝐵 = (Base‘𝑅)
2018, 19syl6eqr 2832 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
2120adantr 473 . . . . . . . . . . 11 ((𝑟 = 𝑅𝑣 = (Base‘𝑟)) → (Base‘𝑟) = 𝐵)
2217, 21eqtrd 2814 . . . . . . . . . 10 ((𝑟 = 𝑅𝑣 = (Base‘𝑟)) → 𝑣 = 𝐵)
23 raleq 3345 . . . . . . . . . . 11 (𝑣 = 𝐵 → (∀𝑏𝑣 ((𝑧𝑙𝑎𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏)) ↔ ∀𝑏𝐵 ((𝑧𝑙𝑎𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏))))
2423raleqbi1dv 3343 . . . . . . . . . 10 (𝑣 = 𝐵 → (∀𝑎𝑣𝑏𝑣 ((𝑧𝑙𝑎𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏)) ↔ ∀𝑎𝐵𝑏𝐵 ((𝑧𝑙𝑎𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏))))
2522, 24syl 17 . . . . . . . . 9 ((𝑟 = 𝑅𝑣 = (Base‘𝑟)) → (∀𝑎𝑣𝑏𝑣 ((𝑧𝑙𝑎𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏)) ↔ ∀𝑎𝐵𝑏𝐵 ((𝑧𝑙𝑎𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏))))
2625sbcbidv 3731 . . . . . . . 8 ((𝑟 = 𝑅𝑣 = (Base‘𝑟)) → ([(le‘𝑟) / 𝑙]𝑎𝑣𝑏𝑣 ((𝑧𝑙𝑎𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏)) ↔ [(le‘𝑟) / 𝑙]𝑎𝐵𝑏𝐵 ((𝑧𝑙𝑎𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏))))
2726sbcbidv 3731 . . . . . . 7 ((𝑟 = 𝑅𝑣 = (Base‘𝑟)) → ([(.r𝑟) / 𝑡][(le‘𝑟) / 𝑙]𝑎𝑣𝑏𝑣 ((𝑧𝑙𝑎𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏)) ↔ [(.r𝑟) / 𝑡][(le‘𝑟) / 𝑙]𝑎𝐵𝑏𝐵 ((𝑧𝑙𝑎𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏))))
2827sbcbidv 3731 . . . . . 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))))
2916, 28sbcied 3718 . . . . 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))))
30 fvexd 6514 . . . . . 6 (𝑟 = 𝑅 → (0g𝑟) ∈ V)
31 simpr 477 . . . . . . . . . . . . 13 ((𝑟 = 𝑅𝑧 = (0g𝑟)) → 𝑧 = (0g𝑟))
32 fveq2 6499 . . . . . . . . . . . . . . 15 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
33 isorng.1 . . . . . . . . . . . . . . 15 0 = (0g𝑅)
3432, 33syl6eqr 2832 . . . . . . . . . . . . . 14 (𝑟 = 𝑅 → (0g𝑟) = 0 )
3534adantr 473 . . . . . . . . . . . . 13 ((𝑟 = 𝑅𝑧 = (0g𝑟)) → (0g𝑟) = 0 )
3631, 35eqtrd 2814 . . . . . . . . . . . 12 ((𝑟 = 𝑅𝑧 = (0g𝑟)) → 𝑧 = 0 )
3736breq1d 4939 . . . . . . . . . . 11 ((𝑟 = 𝑅𝑧 = (0g𝑟)) → (𝑧𝑙𝑎0 𝑙𝑎))
3836breq1d 4939 . . . . . . . . . . 11 ((𝑟 = 𝑅𝑧 = (0g𝑟)) → (𝑧𝑙𝑏0 𝑙𝑏))
3937, 38anbi12d 621 . . . . . . . . . 10 ((𝑟 = 𝑅𝑧 = (0g𝑟)) → ((𝑧𝑙𝑎𝑧𝑙𝑏) ↔ ( 0 𝑙𝑎0 𝑙𝑏)))
4036breq1d 4939 . . . . . . . . . 10 ((𝑟 = 𝑅𝑧 = (0g𝑟)) → (𝑧𝑙(𝑎𝑡𝑏) ↔ 0 𝑙(𝑎𝑡𝑏)))
4139, 40imbi12d 337 . . . . . . . . 9 ((𝑟 = 𝑅𝑧 = (0g𝑟)) → (((𝑧𝑙𝑎𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏)) ↔ (( 0 𝑙𝑎0 𝑙𝑏) → 0 𝑙(𝑎𝑡𝑏))))
42412ralbidv 3149 . . . . . . . 8 ((𝑟 = 𝑅𝑧 = (0g𝑟)) → (∀𝑎𝐵𝑏𝐵 ((𝑧𝑙𝑎𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏)) ↔ ∀𝑎𝐵𝑏𝐵 (( 0 𝑙𝑎0 𝑙𝑏) → 0 𝑙(𝑎𝑡𝑏))))
4342sbcbidv 3731 . . . . . . 7 ((𝑟 = 𝑅𝑧 = (0g𝑟)) → ([(le‘𝑟) / 𝑙]𝑎𝐵𝑏𝐵 ((𝑧𝑙𝑎𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏)) ↔ [(le‘𝑟) / 𝑙]𝑎𝐵𝑏𝐵 (( 0 𝑙𝑎0 𝑙𝑏) → 0 𝑙(𝑎𝑡𝑏))))
4443sbcbidv 3731 . . . . . 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))))
4530, 44sbcied 3718 . . . . 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))))
4629, 45bitr2d 272 . . . 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))))
47 fvexd 6514 . . . . 5 (𝑟 = 𝑅 → (le‘𝑟) ∈ V)
48 simpr 477 . . . . . . . . . 10 ((𝑟 = 𝑅𝑙 = (le‘𝑟)) → 𝑙 = (le‘𝑟))
49 simpl 475 . . . . . . . . . . . 12 ((𝑟 = 𝑅𝑙 = (le‘𝑟)) → 𝑟 = 𝑅)
5049fveq2d 6503 . . . . . . . . . . 11 ((𝑟 = 𝑅𝑙 = (le‘𝑟)) → (le‘𝑟) = (le‘𝑅))
51 isorng.3 . . . . . . . . . . 11 = (le‘𝑅)
5250, 51syl6eqr 2832 . . . . . . . . . 10 ((𝑟 = 𝑅𝑙 = (le‘𝑟)) → (le‘𝑟) = )
5348, 52eqtrd 2814 . . . . . . . . 9 ((𝑟 = 𝑅𝑙 = (le‘𝑟)) → 𝑙 = )
5453breqd 4940 . . . . . . . 8 ((𝑟 = 𝑅𝑙 = (le‘𝑟)) → ( 0 𝑙𝑎0 𝑎))
5553breqd 4940 . . . . . . . 8 ((𝑟 = 𝑅𝑙 = (le‘𝑟)) → ( 0 𝑙𝑏0 𝑏))
5654, 55anbi12d 621 . . . . . . 7 ((𝑟 = 𝑅𝑙 = (le‘𝑟)) → (( 0 𝑙𝑎0 𝑙𝑏) ↔ ( 0 𝑎0 𝑏)))
5753breqd 4940 . . . . . . 7 ((𝑟 = 𝑅𝑙 = (le‘𝑟)) → ( 0 𝑙(𝑎 · 𝑏) ↔ 0 (𝑎 · 𝑏)))
5856, 57imbi12d 337 . . . . . 6 ((𝑟 = 𝑅𝑙 = (le‘𝑟)) → ((( 0 𝑙𝑎0 𝑙𝑏) → 0 𝑙(𝑎 · 𝑏)) ↔ (( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏))))
59582ralbidv 3149 . . . . 5 ((𝑟 = 𝑅𝑙 = (le‘𝑟)) → (∀𝑎𝐵𝑏𝐵 (( 0 𝑙𝑎0 𝑙𝑏) → 0 𝑙(𝑎 · 𝑏)) ↔ ∀𝑎𝐵𝑏𝐵 (( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏))))
6047, 59sbcied 3718 . . . 4 (𝑟 = 𝑅 → ([(le‘𝑟) / 𝑙]𝑎𝐵𝑏𝐵 (( 0 𝑙𝑎0 𝑙𝑏) → 0 𝑙(𝑎 · 𝑏)) ↔ ∀𝑎𝐵𝑏𝐵 (( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏))))
6115, 46, 603bitr3d 301 . . 3 (𝑟 = 𝑅 → ([(Base‘𝑟) / 𝑣][(0g𝑟) / 𝑧][(.r𝑟) / 𝑡][(le‘𝑟) / 𝑙]𝑎𝑣𝑏𝑣 ((𝑧𝑙𝑎𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏)) ↔ ∀𝑎𝐵𝑏𝐵 (( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏))))
62 df-orng 30555 . . 3 oRing = {𝑟 ∈ (Ring ∩ oGrp) ∣ [(Base‘𝑟) / 𝑣][(0g𝑟) / 𝑧][(.r𝑟) / 𝑡][(le‘𝑟) / 𝑙]𝑎𝑣𝑏𝑣 ((𝑧𝑙𝑎𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏))}
6361, 62elrab2 3599 . 2 (𝑅 ∈ oRing ↔ (𝑅 ∈ (Ring ∩ oGrp) ∧ ∀𝑎𝐵𝑏𝐵 (( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏))))
64 df-3an 1070 . 2 ((𝑅 ∈ Ring ∧ 𝑅 ∈ oGrp ∧ ∀𝑎𝐵𝑏𝐵 (( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏))) ↔ ((𝑅 ∈ Ring ∧ 𝑅 ∈ oGrp) ∧ ∀𝑎𝐵𝑏𝐵 (( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏))))
652, 63, 643bitr4i 295 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 198  wa 387  w3a 1068   = wceq 1507  wcel 2050  wral 3088  Vcvv 3415  [wsbc 3681  cin 3828   class class class wbr 4929  cfv 6188  (class class class)co 6976  Basecbs 16339  .rcmulr 16422  lecple 16428  0gc0g 16569  Ringcrg 19020  oGrpcogrp 30423  oRingcorng 30553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2750  ax-nul 5067
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3682  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-iota 6152  df-fv 6196  df-ov 6979  df-orng 30555
This theorem is referenced by:  orngring  30558  orngogrp  30559  orngmul  30561  suborng  30573  reofld  30598
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