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Theorem isrngod 36761
Description: Conditions that determine a ring. (Changed label from isringd 20104 to isrngod 36761-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 (πœ‘ β†’ 𝐺 ∈ AbelOp)
isringod.2 (πœ‘ β†’ 𝑋 = ran 𝐺)
isringod.3 (πœ‘ β†’ 𝐻:(𝑋 Γ— 𝑋)βŸΆπ‘‹)
isringod.4 ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) β†’ ((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)))
isringod.5 ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) β†’ (π‘₯𝐻(𝑦𝐺𝑧)) = ((π‘₯𝐻𝑦)𝐺(π‘₯𝐻𝑧)))
isringod.6 ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) β†’ ((π‘₯𝐺𝑦)𝐻𝑧) = ((π‘₯𝐻𝑧)𝐺(𝑦𝐻𝑧)))
isringod.7 (πœ‘ β†’ π‘ˆ ∈ 𝑋)
isringod.8 ((πœ‘ ∧ 𝑦 ∈ 𝑋) β†’ (π‘ˆπ»π‘¦) = 𝑦)
isringod.9 ((πœ‘ ∧ 𝑦 ∈ 𝑋) β†’ (π‘¦π»π‘ˆ) = 𝑦)
Assertion
Ref Expression
isrngod (πœ‘ β†’ ⟨𝐺, 𝐻⟩ ∈ RingOps)
Distinct variable groups:   πœ‘,π‘₯,𝑦,𝑧   π‘₯,𝐺,𝑦,𝑧   π‘₯,𝐻,𝑦,𝑧   π‘₯,𝑋,𝑦,𝑧   π‘₯,π‘ˆ,𝑦
Allowed substitution hint:   π‘ˆ(𝑧)

Proof of Theorem isrngod
StepHypRef Expression
1 isringod.1 . . 3 (πœ‘ β†’ 𝐺 ∈ AbelOp)
2 isringod.3 . . . 4 (πœ‘ β†’ 𝐻:(𝑋 Γ— 𝑋)βŸΆπ‘‹)
3 isringod.2 . . . . . 6 (πœ‘ β†’ 𝑋 = ran 𝐺)
43sqxpeqd 5708 . . . . 5 (πœ‘ β†’ (𝑋 Γ— 𝑋) = (ran 𝐺 Γ— ran 𝐺))
54, 3feq23d 6712 . . . 4 (πœ‘ β†’ (𝐻:(𝑋 Γ— 𝑋)βŸΆπ‘‹ ↔ 𝐻:(ran 𝐺 Γ— ran 𝐺)⟢ran 𝐺))
62, 5mpbid 231 . . 3 (πœ‘ β†’ 𝐻:(ran 𝐺 Γ— ran 𝐺)⟢ran 𝐺)
7 isringod.4 . . . . . . 7 ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) β†’ ((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)))
8 isringod.5 . . . . . . 7 ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) β†’ (π‘₯𝐻(𝑦𝐺𝑧)) = ((π‘₯𝐻𝑦)𝐺(π‘₯𝐻𝑧)))
9 isringod.6 . . . . . . 7 ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) β†’ ((π‘₯𝐺𝑦)𝐻𝑧) = ((π‘₯𝐻𝑧)𝐺(𝑦𝐻𝑧)))
107, 8, 93jca 1128 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) β†’ (((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)) ∧ (π‘₯𝐻(𝑦𝐺𝑧)) = ((π‘₯𝐻𝑦)𝐺(π‘₯𝐻𝑧)) ∧ ((π‘₯𝐺𝑦)𝐻𝑧) = ((π‘₯𝐻𝑧)𝐺(𝑦𝐻𝑧))))
1110ralrimivvva 3203 . . . . 5 (πœ‘ β†’ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 (((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)) ∧ (π‘₯𝐻(𝑦𝐺𝑧)) = ((π‘₯𝐻𝑦)𝐺(π‘₯𝐻𝑧)) ∧ ((π‘₯𝐺𝑦)𝐻𝑧) = ((π‘₯𝐻𝑧)𝐺(𝑦𝐻𝑧))))
123raleqdv 3325 . . . . . . 7 (πœ‘ β†’ (βˆ€π‘§ ∈ 𝑋 (((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)) ∧ (π‘₯𝐻(𝑦𝐺𝑧)) = ((π‘₯𝐻𝑦)𝐺(π‘₯𝐻𝑧)) ∧ ((π‘₯𝐺𝑦)𝐻𝑧) = ((π‘₯𝐻𝑧)𝐺(𝑦𝐻𝑧))) ↔ βˆ€π‘§ ∈ ran 𝐺(((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)) ∧ (π‘₯𝐻(𝑦𝐺𝑧)) = ((π‘₯𝐻𝑦)𝐺(π‘₯𝐻𝑧)) ∧ ((π‘₯𝐺𝑦)𝐻𝑧) = ((π‘₯𝐻𝑧)𝐺(𝑦𝐻𝑧)))))
133, 12raleqbidv 3342 . . . . . 6 (πœ‘ β†’ (βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 (((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)) ∧ (π‘₯𝐻(𝑦𝐺𝑧)) = ((π‘₯𝐻𝑦)𝐺(π‘₯𝐻𝑧)) ∧ ((π‘₯𝐺𝑦)𝐻𝑧) = ((π‘₯𝐻𝑧)𝐺(𝑦𝐻𝑧))) ↔ βˆ€π‘¦ ∈ ran πΊβˆ€π‘§ ∈ ran 𝐺(((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)) ∧ (π‘₯𝐻(𝑦𝐺𝑧)) = ((π‘₯𝐻𝑦)𝐺(π‘₯𝐻𝑧)) ∧ ((π‘₯𝐺𝑦)𝐻𝑧) = ((π‘₯𝐻𝑧)𝐺(𝑦𝐻𝑧)))))
143, 13raleqbidv 3342 . . . . 5 (πœ‘ β†’ (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 (((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)) ∧ (π‘₯𝐻(𝑦𝐺𝑧)) = ((π‘₯𝐻𝑦)𝐺(π‘₯𝐻𝑧)) ∧ ((π‘₯𝐺𝑦)𝐻𝑧) = ((π‘₯𝐻𝑧)𝐺(𝑦𝐻𝑧))) ↔ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran πΊβˆ€π‘§ ∈ ran 𝐺(((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)) ∧ (π‘₯𝐻(𝑦𝐺𝑧)) = ((π‘₯𝐻𝑦)𝐺(π‘₯𝐻𝑧)) ∧ ((π‘₯𝐺𝑦)𝐻𝑧) = ((π‘₯𝐻𝑧)𝐺(𝑦𝐻𝑧)))))
1511, 14mpbid 231 . . . 4 (πœ‘ β†’ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran πΊβˆ€π‘§ ∈ ran 𝐺(((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)) ∧ (π‘₯𝐻(𝑦𝐺𝑧)) = ((π‘₯𝐻𝑦)𝐺(π‘₯𝐻𝑧)) ∧ ((π‘₯𝐺𝑦)𝐻𝑧) = ((π‘₯𝐻𝑧)𝐺(𝑦𝐻𝑧))))
16 isringod.7 . . . . . 6 (πœ‘ β†’ π‘ˆ ∈ 𝑋)
17 isringod.8 . . . . . . . 8 ((πœ‘ ∧ 𝑦 ∈ 𝑋) β†’ (π‘ˆπ»π‘¦) = 𝑦)
18 isringod.9 . . . . . . . 8 ((πœ‘ ∧ 𝑦 ∈ 𝑋) β†’ (π‘¦π»π‘ˆ) = 𝑦)
1917, 18jca 512 . . . . . . 7 ((πœ‘ ∧ 𝑦 ∈ 𝑋) β†’ ((π‘ˆπ»π‘¦) = 𝑦 ∧ (π‘¦π»π‘ˆ) = 𝑦))
2019ralrimiva 3146 . . . . . 6 (πœ‘ β†’ βˆ€π‘¦ ∈ 𝑋 ((π‘ˆπ»π‘¦) = 𝑦 ∧ (π‘¦π»π‘ˆ) = 𝑦))
21 oveq1 7415 . . . . . . . . 9 (π‘₯ = π‘ˆ β†’ (π‘₯𝐻𝑦) = (π‘ˆπ»π‘¦))
2221eqeq1d 2734 . . . . . . . 8 (π‘₯ = π‘ˆ β†’ ((π‘₯𝐻𝑦) = 𝑦 ↔ (π‘ˆπ»π‘¦) = 𝑦))
2322ovanraleqv 7432 . . . . . . 7 (π‘₯ = π‘ˆ β†’ (βˆ€π‘¦ ∈ 𝑋 ((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦) ↔ βˆ€π‘¦ ∈ 𝑋 ((π‘ˆπ»π‘¦) = 𝑦 ∧ (π‘¦π»π‘ˆ) = 𝑦)))
2423rspcev 3612 . . . . . 6 ((π‘ˆ ∈ 𝑋 ∧ βˆ€π‘¦ ∈ 𝑋 ((π‘ˆπ»π‘¦) = 𝑦 ∧ (π‘¦π»π‘ˆ) = 𝑦)) β†’ βˆƒπ‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦))
2516, 20, 24syl2anc 584 . . . . 5 (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦))
263raleqdv 3325 . . . . . 6 (πœ‘ β†’ (βˆ€π‘¦ ∈ 𝑋 ((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦) ↔ βˆ€π‘¦ ∈ ran 𝐺((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦)))
273, 26rexeqbidv 3343 . . . . 5 (πœ‘ β†’ (βˆƒπ‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦) ↔ βˆƒπ‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦)))
2825, 27mpbid 231 . . . 4 (πœ‘ β†’ βˆƒπ‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦))
2915, 28jca 512 . . 3 (πœ‘ β†’ (βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran πΊβˆ€π‘§ ∈ ran 𝐺(((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)) ∧ (π‘₯𝐻(𝑦𝐺𝑧)) = ((π‘₯𝐻𝑦)𝐺(π‘₯𝐻𝑧)) ∧ ((π‘₯𝐺𝑦)𝐻𝑧) = ((π‘₯𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ βˆƒπ‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦)))
301, 6, 29jca31 515 . 2 (πœ‘ β†’ ((𝐺 ∈ AbelOp ∧ 𝐻:(ran 𝐺 Γ— ran 𝐺)⟢ran 𝐺) ∧ (βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran πΊβˆ€π‘§ ∈ ran 𝐺(((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)) ∧ (π‘₯𝐻(𝑦𝐺𝑧)) = ((π‘₯𝐻𝑦)𝐺(π‘₯𝐻𝑧)) ∧ ((π‘₯𝐺𝑦)𝐻𝑧) = ((π‘₯𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ βˆƒπ‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦))))
31 rnexg 7894 . . . . . 6 (𝐺 ∈ AbelOp β†’ ran 𝐺 ∈ V)
321, 31syl 17 . . . . 5 (πœ‘ β†’ ran 𝐺 ∈ V)
3332, 32xpexd 7737 . . . 4 (πœ‘ β†’ (ran 𝐺 Γ— ran 𝐺) ∈ V)
346, 33fexd 7228 . . 3 (πœ‘ β†’ 𝐻 ∈ V)
35 eqid 2732 . . . 4 ran 𝐺 = ran 𝐺
3635isrngo 36760 . . 3 (𝐻 ∈ V β†’ (⟨𝐺, 𝐻⟩ ∈ RingOps ↔ ((𝐺 ∈ AbelOp ∧ 𝐻:(ran 𝐺 Γ— ran 𝐺)⟢ran 𝐺) ∧ (βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran πΊβˆ€π‘§ ∈ ran 𝐺(((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)) ∧ (π‘₯𝐻(𝑦𝐺𝑧)) = ((π‘₯𝐻𝑦)𝐺(π‘₯𝐻𝑧)) ∧ ((π‘₯𝐺𝑦)𝐻𝑧) = ((π‘₯𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ βˆƒπ‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦)))))
3734, 36syl 17 . 2 (πœ‘ β†’ (⟨𝐺, 𝐻⟩ ∈ RingOps ↔ ((𝐺 ∈ AbelOp ∧ 𝐻:(ran 𝐺 Γ— ran 𝐺)⟢ran 𝐺) ∧ (βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran πΊβˆ€π‘§ ∈ ran 𝐺(((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)) ∧ (π‘₯𝐻(𝑦𝐺𝑧)) = ((π‘₯𝐻𝑦)𝐺(π‘₯𝐻𝑧)) ∧ ((π‘₯𝐺𝑦)𝐻𝑧) = ((π‘₯𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ βˆƒπ‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦)))))
3830, 37mpbird 256 1 (πœ‘ β†’ ⟨𝐺, 𝐻⟩ ∈ RingOps)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3474  βŸ¨cop 4634   Γ— cxp 5674  ran crn 5677  βŸΆwf 6539  (class class class)co 7408  AbelOpcablo 29792  RingOpscrngo 36757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-rngo 36758
This theorem is referenced by:  iscringd  36861
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