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Theorem isring 18320
Description: The predicate "is a (unital) ring." Definition of ring with unit in [Schechter] p. 187. (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)
Hypotheses
Ref Expression
isring.b 𝐵 = (Base‘𝑅)
isring.g 𝐺 = (mulGrp‘𝑅)
isring.p + = (+g𝑅)
isring.t · = (.r𝑅)
Assertion
Ref Expression
isring (𝑅 ∈ Ring ↔ (𝑅 ∈ Grp ∧ 𝐺 ∈ Mnd ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐵   𝑥, + ,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧   𝑥, · ,𝑦,𝑧
Allowed substitution hints:   𝐺(𝑥,𝑦,𝑧)

Proof of Theorem isring
Dummy variables 𝑝 𝑏 𝑟 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6088 . . . . . 6 (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅))
2 isring.g . . . . . 6 𝐺 = (mulGrp‘𝑅)
31, 2syl6eqr 2661 . . . . 5 (𝑟 = 𝑅 → (mulGrp‘𝑟) = 𝐺)
43eleq1d 2671 . . . 4 (𝑟 = 𝑅 → ((mulGrp‘𝑟) ∈ Mnd ↔ 𝐺 ∈ Mnd))
5 fvex 6098 . . . . . 6 (Base‘𝑟) ∈ V
65a1i 11 . . . . 5 (𝑟 = 𝑅 → (Base‘𝑟) ∈ V)
7 fveq2 6088 . . . . . 6 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
8 isring.b . . . . . 6 𝐵 = (Base‘𝑅)
97, 8syl6eqr 2661 . . . . 5 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
10 fvex 6098 . . . . . . 7 (+g𝑟) ∈ V
1110a1i 11 . . . . . 6 ((𝑟 = 𝑅𝑏 = 𝐵) → (+g𝑟) ∈ V)
12 simpl 471 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = 𝐵) → 𝑟 = 𝑅)
1312fveq2d 6092 . . . . . . 7 ((𝑟 = 𝑅𝑏 = 𝐵) → (+g𝑟) = (+g𝑅))
14 isring.p . . . . . . 7 + = (+g𝑅)
1513, 14syl6eqr 2661 . . . . . 6 ((𝑟 = 𝑅𝑏 = 𝐵) → (+g𝑟) = + )
16 fvex 6098 . . . . . . . 8 (.r𝑟) ∈ V
1716a1i 11 . . . . . . 7 (((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑝 = + ) → (.r𝑟) ∈ V)
18 simpll 785 . . . . . . . . 9 (((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑝 = + ) → 𝑟 = 𝑅)
1918fveq2d 6092 . . . . . . . 8 (((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑝 = + ) → (.r𝑟) = (.r𝑅))
20 isring.t . . . . . . . 8 · = (.r𝑅)
2119, 20syl6eqr 2661 . . . . . . 7 (((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑝 = + ) → (.r𝑟) = · )
22 simpllr 794 . . . . . . . 8 ((((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → 𝑏 = 𝐵)
23 simpr 475 . . . . . . . . . . . . 13 ((((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → 𝑡 = · )
24 eqidd 2610 . . . . . . . . . . . . 13 ((((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → 𝑥 = 𝑥)
25 simplr 787 . . . . . . . . . . . . . 14 ((((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → 𝑝 = + )
2625oveqd 6544 . . . . . . . . . . . . 13 ((((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → (𝑦𝑝𝑧) = (𝑦 + 𝑧))
2723, 24, 26oveq123d 6548 . . . . . . . . . . . 12 ((((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → (𝑥𝑡(𝑦𝑝𝑧)) = (𝑥 · (𝑦 + 𝑧)))
2823oveqd 6544 . . . . . . . . . . . . 13 ((((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → (𝑥𝑡𝑦) = (𝑥 · 𝑦))
2923oveqd 6544 . . . . . . . . . . . . 13 ((((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → (𝑥𝑡𝑧) = (𝑥 · 𝑧))
3025, 28, 29oveq123d 6548 . . . . . . . . . . . 12 ((((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))
3127, 30eqeq12d 2624 . . . . . . . . . . 11 ((((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ↔ (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧))))
3225oveqd 6544 . . . . . . . . . . . . 13 ((((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → (𝑥𝑝𝑦) = (𝑥 + 𝑦))
33 eqidd 2610 . . . . . . . . . . . . 13 ((((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → 𝑧 = 𝑧)
3423, 32, 33oveq123d 6548 . . . . . . . . . . . 12 ((((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥 + 𝑦) · 𝑧))
3523oveqd 6544 . . . . . . . . . . . . 13 ((((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → (𝑦𝑡𝑧) = (𝑦 · 𝑧))
3625, 29, 35oveq123d 6548 . . . . . . . . . . . 12 ((((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧)) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
3734, 36eqeq12d 2624 . . . . . . . . . . 11 ((((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → (((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧)) ↔ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))))
3831, 37anbi12d 742 . . . . . . . . . 10 ((((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → (((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ↔ ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))))
3922, 38raleqbidv 3128 . . . . . . . . 9 ((((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → (∀𝑧𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ↔ ∀𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))))
4022, 39raleqbidv 3128 . . . . . . . 8 ((((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → (∀𝑦𝑏𝑧𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ↔ ∀𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))))
4122, 40raleqbidv 3128 . . . . . . 7 ((((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → (∀𝑥𝑏𝑦𝑏𝑧𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))))
4217, 21, 41sbcied2 3439 . . . . . 6 (((𝑟 = 𝑅𝑏 = 𝐵) ∧ 𝑝 = + ) → ([(.r𝑟) / 𝑡]𝑥𝑏𝑦𝑏𝑧𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))))
4311, 15, 42sbcied2 3439 . . . . 5 ((𝑟 = 𝑅𝑏 = 𝐵) → ([(+g𝑟) / 𝑝][(.r𝑟) / 𝑡]𝑥𝑏𝑦𝑏𝑧𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))))
446, 9, 43sbcied2 3439 . . . 4 (𝑟 = 𝑅 → ([(Base‘𝑟) / 𝑏][(+g𝑟) / 𝑝][(.r𝑟) / 𝑡]𝑥𝑏𝑦𝑏𝑧𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))))
454, 44anbi12d 742 . . 3 (𝑟 = 𝑅 → (((mulGrp‘𝑟) ∈ Mnd ∧ [(Base‘𝑟) / 𝑏][(+g𝑟) / 𝑝][(.r𝑟) / 𝑡]𝑥𝑏𝑦𝑏𝑧𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧)))) ↔ (𝐺 ∈ Mnd ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))))))
46 df-ring 18318 . . 3 Ring = {𝑟 ∈ Grp ∣ ((mulGrp‘𝑟) ∈ Mnd ∧ [(Base‘𝑟) / 𝑏][(+g𝑟) / 𝑝][(.r𝑟) / 𝑡]𝑥𝑏𝑦𝑏𝑧𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))))}
4745, 46elrab2 3332 . 2 (𝑅 ∈ Ring ↔ (𝑅 ∈ Grp ∧ (𝐺 ∈ Mnd ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))))))
48 3anass 1034 . 2 ((𝑅 ∈ Grp ∧ 𝐺 ∈ Mnd ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))) ↔ (𝑅 ∈ Grp ∧ (𝐺 ∈ Mnd ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))))))
4947, 48bitr4i 265 1 (𝑅 ∈ Ring ↔ (𝑅 ∈ Grp ∧ 𝐺 ∈ Mnd ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))))
Colors of variables: wff setvar class
Syntax hints:  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wral 2895  Vcvv 3172  [wsbc 3401  cfv 5790  (class class class)co 6527  Basecbs 15641  +gcplusg 15714  .rcmulr 15715  Mndcmnd 17063  Grpcgrp 17191  mulGrpcmgp 18258  Ringcrg 18316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-nul 4712
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-iota 5754  df-fv 5798  df-ov 6530  df-ring 18318
This theorem is referenced by:  ringgrp  18321  ringmgp  18322  ringi  18329  ringpropd  18351  isringd  18354  ringsrg  18358  ring1  18371  prdsringd  18381  ringrng  41664  isringrng  41666  2zrngnring  41737  cznnring  41743
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