Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isrngod Structured version   Visualization version   GIF version

Theorem isrngod 34010
Description: Conditions that determine a ring. (Changed label from isringd 18785 to isrngod 34010-NM 2-Aug-2013.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
isringod.1 (𝜑𝐺 ∈ AbelOp)
isringod.2 (𝜑𝑋 = ran 𝐺)
isringod.3 (𝜑𝐻:(𝑋 × 𝑋)⟶𝑋)
isringod.4 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → ((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)))
isringod.5 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)))
isringod.6 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧)))
isringod.7 (𝜑𝑈𝑋)
isringod.8 ((𝜑𝑦𝑋) → (𝑈𝐻𝑦) = 𝑦)
isringod.9 ((𝜑𝑦𝑋) → (𝑦𝐻𝑈) = 𝑦)
Assertion
Ref Expression
isrngod (𝜑 → ⟨𝐺, 𝐻⟩ ∈ RingOps)
Distinct variable groups:   𝜑,𝑥,𝑦,𝑧   𝑥,𝐺,𝑦,𝑧   𝑥,𝐻,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧   𝑥,𝑈,𝑦
Allowed substitution hint:   𝑈(𝑧)

Proof of Theorem isrngod
StepHypRef Expression
1 isringod.1 . . 3 (𝜑𝐺 ∈ AbelOp)
2 isringod.3 . . . 4 (𝜑𝐻:(𝑋 × 𝑋)⟶𝑋)
3 isringod.2 . . . . . 6 (𝜑𝑋 = ran 𝐺)
43sqxpeqd 5298 . . . . 5 (𝜑 → (𝑋 × 𝑋) = (ran 𝐺 × ran 𝐺))
54, 3feq23d 6201 . . . 4 (𝜑 → (𝐻:(𝑋 × 𝑋)⟶𝑋𝐻:(ran 𝐺 × ran 𝐺)⟶ran 𝐺))
62, 5mpbid 222 . . 3 (𝜑𝐻:(ran 𝐺 × ran 𝐺)⟶ran 𝐺)
7 isringod.4 . . . . . . 7 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → ((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)))
8 isringod.5 . . . . . . 7 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)))
9 isringod.6 . . . . . . 7 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧)))
107, 8, 93jca 1123 . . . . . 6 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))))
1110ralrimivvva 3110 . . . . 5 (𝜑 → ∀𝑥𝑋𝑦𝑋𝑧𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))))
123raleqdv 3283 . . . . . . 7 (𝜑 → (∀𝑧𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ↔ ∀𝑧 ∈ ran 𝐺(((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧)))))
133, 12raleqbidv 3291 . . . . . 6 (𝜑 → (∀𝑦𝑋𝑧𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ↔ ∀𝑦 ∈ ran 𝐺𝑧 ∈ ran 𝐺(((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧)))))
143, 13raleqbidv 3291 . . . . 5 (𝜑 → (∀𝑥𝑋𝑦𝑋𝑧𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ↔ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺𝑧 ∈ ran 𝐺(((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧)))))
1511, 14mpbid 222 . . . 4 (𝜑 → ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺𝑧 ∈ ran 𝐺(((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))))
16 isringod.7 . . . . . 6 (𝜑𝑈𝑋)
17 isringod.8 . . . . . . . 8 ((𝜑𝑦𝑋) → (𝑈𝐻𝑦) = 𝑦)
18 isringod.9 . . . . . . . 8 ((𝜑𝑦𝑋) → (𝑦𝐻𝑈) = 𝑦)
1917, 18jca 555 . . . . . . 7 ((𝜑𝑦𝑋) → ((𝑈𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑈) = 𝑦))
2019ralrimiva 3104 . . . . . 6 (𝜑 → ∀𝑦𝑋 ((𝑈𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑈) = 𝑦))
21 oveq1 6820 . . . . . . . . . 10 (𝑥 = 𝑈 → (𝑥𝐻𝑦) = (𝑈𝐻𝑦))
2221eqeq1d 2762 . . . . . . . . 9 (𝑥 = 𝑈 → ((𝑥𝐻𝑦) = 𝑦 ↔ (𝑈𝐻𝑦) = 𝑦))
23 oveq2 6821 . . . . . . . . . 10 (𝑥 = 𝑈 → (𝑦𝐻𝑥) = (𝑦𝐻𝑈))
2423eqeq1d 2762 . . . . . . . . 9 (𝑥 = 𝑈 → ((𝑦𝐻𝑥) = 𝑦 ↔ (𝑦𝐻𝑈) = 𝑦))
2522, 24anbi12d 749 . . . . . . . 8 (𝑥 = 𝑈 → (((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦) ↔ ((𝑈𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑈) = 𝑦)))
2625ralbidv 3124 . . . . . . 7 (𝑥 = 𝑈 → (∀𝑦𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦) ↔ ∀𝑦𝑋 ((𝑈𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑈) = 𝑦)))
2726rspcev 3449 . . . . . 6 ((𝑈𝑋 ∧ ∀𝑦𝑋 ((𝑈𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑈) = 𝑦)) → ∃𝑥𝑋𝑦𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))
2816, 20, 27syl2anc 696 . . . . 5 (𝜑 → ∃𝑥𝑋𝑦𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))
293raleqdv 3283 . . . . . 6 (𝜑 → (∀𝑦𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦) ↔ ∀𝑦 ∈ ran 𝐺((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))
303, 29rexeqbidv 3292 . . . . 5 (𝜑 → (∃𝑥𝑋𝑦𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦) ↔ ∃𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))
3128, 30mpbid 222 . . . 4 (𝜑 → ∃𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))
3215, 31jca 555 . . 3 (𝜑 → (∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺𝑧 ∈ ran 𝐺(((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))
331, 6, 32jca31 558 . 2 (𝜑 → ((𝐺 ∈ AbelOp ∧ 𝐻:(ran 𝐺 × ran 𝐺)⟶ran 𝐺) ∧ (∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺𝑧 ∈ ran 𝐺(((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))))
34 rnexg 7263 . . . . . 6 (𝐺 ∈ AbelOp → ran 𝐺 ∈ V)
351, 34syl 17 . . . . 5 (𝜑 → ran 𝐺 ∈ V)
36 xpexg 7125 . . . . 5 ((ran 𝐺 ∈ V ∧ ran 𝐺 ∈ V) → (ran 𝐺 × ran 𝐺) ∈ V)
3735, 35, 36syl2anc 696 . . . 4 (𝜑 → (ran 𝐺 × ran 𝐺) ∈ V)
38 fex 6653 . . . 4 ((𝐻:(ran 𝐺 × ran 𝐺)⟶ran 𝐺 ∧ (ran 𝐺 × ran 𝐺) ∈ V) → 𝐻 ∈ V)
396, 37, 38syl2anc 696 . . 3 (𝜑𝐻 ∈ V)
40 eqid 2760 . . . 4 ran 𝐺 = ran 𝐺
4140isrngo 34009 . . 3 (𝐻 ∈ V → (⟨𝐺, 𝐻⟩ ∈ RingOps ↔ ((𝐺 ∈ AbelOp ∧ 𝐻:(ran 𝐺 × ran 𝐺)⟶ran 𝐺) ∧ (∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺𝑧 ∈ ran 𝐺(((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))))
4239, 41syl 17 . 2 (𝜑 → (⟨𝐺, 𝐻⟩ ∈ RingOps ↔ ((𝐺 ∈ AbelOp ∧ 𝐻:(ran 𝐺 × ran 𝐺)⟶ran 𝐺) ∧ (∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺𝑧 ∈ ran 𝐺(((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))))
4333, 42mpbird 247 1 (𝜑 → ⟨𝐺, 𝐻⟩ ∈ RingOps)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  wrex 3051  Vcvv 3340  cop 4327   × cxp 5264  ran crn 5267  wf 6045  (class class class)co 6813  AbelOpcablo 27707  RingOpscrngo 34006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-rngo 34007
This theorem is referenced by:  iscringd  34110
  Copyright terms: Public domain W3C validator