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Theorem isrngod 36407
Description: Conditions that determine a ring. (Changed label from isringd 20017 to isrngod 36407-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 5669 . . . . 5 (πœ‘ β†’ (𝑋 Γ— 𝑋) = (ran 𝐺 Γ— ran 𝐺))
54, 3feq23d 6667 . . . 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 1129 . . . . . 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 3197 . . . . 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 3312 . . . . . . 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 3318 . . . . . 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 3318 . . . . 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 513 . . . . . . 7 ((πœ‘ ∧ 𝑦 ∈ 𝑋) β†’ ((π‘ˆπ»π‘¦) = 𝑦 ∧ (π‘¦π»π‘ˆ) = 𝑦))
2019ralrimiva 3140 . . . . . 6 (πœ‘ β†’ βˆ€π‘¦ ∈ 𝑋 ((π‘ˆπ»π‘¦) = 𝑦 ∧ (π‘¦π»π‘ˆ) = 𝑦))
21 oveq1 7368 . . . . . . . . 9 (π‘₯ = π‘ˆ β†’ (π‘₯𝐻𝑦) = (π‘ˆπ»π‘¦))
2221eqeq1d 2735 . . . . . . . 8 (π‘₯ = π‘ˆ β†’ ((π‘₯𝐻𝑦) = 𝑦 ↔ (π‘ˆπ»π‘¦) = 𝑦))
2322ovanraleqv 7385 . . . . . . 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 3583 . . . . . 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 585 . . . . 5 (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦))
263raleqdv 3312 . . . . . 6 (πœ‘ β†’ (βˆ€π‘¦ ∈ 𝑋 ((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦) ↔ βˆ€π‘¦ ∈ ran 𝐺((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦)))
273, 26rexeqbidv 3319 . . . . 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 513 . . 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 516 . 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 7845 . . . . . 6 (𝐺 ∈ AbelOp β†’ ran 𝐺 ∈ V)
321, 31syl 17 . . . . 5 (πœ‘ β†’ ran 𝐺 ∈ V)
3332, 32xpexd 7689 . . . 4 (πœ‘ β†’ (ran 𝐺 Γ— ran 𝐺) ∈ V)
346, 33fexd 7181 . . 3 (πœ‘ β†’ 𝐻 ∈ V)
35 eqid 2733 . . . 4 ran 𝐺 = ran 𝐺
3635isrngo 36406 . . 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 257 1 (πœ‘ β†’ ⟨𝐺, 𝐻⟩ ∈ RingOps)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3447  βŸ¨cop 4596   Γ— cxp 5635  ran crn 5638  βŸΆwf 6496  (class class class)co 7361  AbelOpcablo 29535  RingOpscrngo 36403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-rngo 36404
This theorem is referenced by:  iscringd  36507
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