Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isdrngo2 Structured version   Visualization version   GIF version

Theorem isdrngo2 34088
Description: A division ring is a ring in which 1 ≠ 0 and every nonzero element is invertible. (Contributed by Jeff Madsen, 8-Jun-2010.)
Hypotheses
Ref Expression
isdivrng1.1 𝐺 = (1st𝑅)
isdivrng1.2 𝐻 = (2nd𝑅)
isdivrng1.3 𝑍 = (GId‘𝐺)
isdivrng1.4 𝑋 = ran 𝐺
isdivrng2.5 𝑈 = (GId‘𝐻)
Assertion
Ref Expression
isdrngo2 (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)))
Distinct variable groups:   𝑥,𝐻,𝑦   𝑥,𝑋,𝑦   𝑥,𝑍,𝑦   𝑥,𝑅,𝑦   𝑥,𝑈,𝑦
Allowed substitution hints:   𝐺(𝑥,𝑦)

Proof of Theorem isdrngo2
Dummy variables 𝑢 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isdivrng1.1 . . 3 𝐺 = (1st𝑅)
2 isdivrng1.2 . . 3 𝐻 = (2nd𝑅)
3 isdivrng1.3 . . 3 𝑍 = (GId‘𝐺)
4 isdivrng1.4 . . 3 𝑋 = ran 𝐺
51, 2, 3, 4isdrngo1 34086 . 2 (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))
6 isdivrng2.5 . . . . . . 7 𝑈 = (GId‘𝐻)
71, 2, 4, 3, 6dvrunz 34084 . . . . . 6 (𝑅 ∈ DivRingOps → 𝑈𝑍)
85, 7sylbir 225 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → 𝑈𝑍)
9 grporndm 27694 . . . . . . . . . . . 12 ((𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp → ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = dom dom (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))))
109adantl 473 . . . . . . . . . . 11 ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = dom dom (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))))
11 difss 3880 . . . . . . . . . . . . . . . . 17 (𝑋 ∖ {𝑍}) ⊆ 𝑋
12 xpss12 5281 . . . . . . . . . . . . . . . . 17 (((𝑋 ∖ {𝑍}) ⊆ 𝑋 ∧ (𝑋 ∖ {𝑍}) ⊆ 𝑋) → ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})) ⊆ (𝑋 × 𝑋))
1311, 11, 12mp2an 710 . . . . . . . . . . . . . . . 16 ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})) ⊆ (𝑋 × 𝑋)
141, 2, 4rngosm 34030 . . . . . . . . . . . . . . . . 17 (𝑅 ∈ RingOps → 𝐻:(𝑋 × 𝑋)⟶𝑋)
15 fdm 6212 . . . . . . . . . . . . . . . . 17 (𝐻:(𝑋 × 𝑋)⟶𝑋 → dom 𝐻 = (𝑋 × 𝑋))
1614, 15syl 17 . . . . . . . . . . . . . . . 16 (𝑅 ∈ RingOps → dom 𝐻 = (𝑋 × 𝑋))
1713, 16syl5sseqr 3795 . . . . . . . . . . . . . . 15 (𝑅 ∈ RingOps → ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})) ⊆ dom 𝐻)
1817adantr 472 . . . . . . . . . . . . . 14 ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})) ⊆ dom 𝐻)
19 ssdmres 5578 . . . . . . . . . . . . . 14 (((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})) ⊆ dom 𝐻 ↔ dom (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))
2018, 19sylib 208 . . . . . . . . . . . . 13 ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → dom (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))
2120dmeqd 5481 . . . . . . . . . . . 12 ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → dom dom (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = dom ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))
22 dmxpid 5500 . . . . . . . . . . . 12 dom ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})) = (𝑋 ∖ {𝑍})
2321, 22syl6eq 2810 . . . . . . . . . . 11 ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → dom dom (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = (𝑋 ∖ {𝑍}))
2410, 23eqtrd 2794 . . . . . . . . . 10 ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = (𝑋 ∖ {𝑍}))
2524eleq2d 2825 . . . . . . . . 9 ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → (𝑥 ∈ ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ↔ 𝑥 ∈ (𝑋 ∖ {𝑍})))
2625biimpar 503 . . . . . . . 8 (((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → 𝑥 ∈ ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))))
27 eqid 2760 . . . . . . . . . . 11 ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))
28 eqid 2760 . . . . . . . . . . 11 (inv‘(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))) = (inv‘(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))))
2927, 28grpoinvcl 27708 . . . . . . . . . 10 (((𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp ∧ 𝑥 ∈ ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))) → ((inv‘(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))))‘𝑥) ∈ ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))))
3029adantll 752 . . . . . . . . 9 (((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ∧ 𝑥 ∈ ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))) → ((inv‘(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))))‘𝑥) ∈ ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))))
31 eqid 2760 . . . . . . . . . . . 12 (GId‘(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))) = (GId‘(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))))
3227, 31, 28grpolinv 27710 . . . . . . . . . . 11 (((𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp ∧ 𝑥 ∈ ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))) → (((inv‘(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))))‘𝑥)(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑥) = (GId‘(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))))
3332adantll 752 . . . . . . . . . 10 (((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ∧ 𝑥 ∈ ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))) → (((inv‘(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))))‘𝑥)(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑥) = (GId‘(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))))
342rngomndo 34065 . . . . . . . . . . . . . 14 (𝑅 ∈ RingOps → 𝐻 ∈ MndOp)
35 mndomgmid 34001 . . . . . . . . . . . . . 14 (𝐻 ∈ MndOp → 𝐻 ∈ (Magma ∩ ExId ))
3634, 35syl 17 . . . . . . . . . . . . 13 (𝑅 ∈ RingOps → 𝐻 ∈ (Magma ∩ ExId ))
3736adantr 472 . . . . . . . . . . . 12 ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → 𝐻 ∈ (Magma ∩ ExId ))
3811, 4sseqtri 3778 . . . . . . . . . . . . . 14 (𝑋 ∖ {𝑍}) ⊆ ran 𝐺
392, 1rngorn1eq 34064 . . . . . . . . . . . . . 14 (𝑅 ∈ RingOps → ran 𝐺 = ran 𝐻)
4038, 39syl5sseq 3794 . . . . . . . . . . . . 13 (𝑅 ∈ RingOps → (𝑋 ∖ {𝑍}) ⊆ ran 𝐻)
4140adantr 472 . . . . . . . . . . . 12 ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → (𝑋 ∖ {𝑍}) ⊆ ran 𝐻)
421rneqi 5507 . . . . . . . . . . . . . . . 16 ran 𝐺 = ran (1st𝑅)
434, 42eqtri 2782 . . . . . . . . . . . . . . 15 𝑋 = ran (1st𝑅)
4443, 2, 6rngo1cl 34069 . . . . . . . . . . . . . 14 (𝑅 ∈ RingOps → 𝑈𝑋)
4544adantr 472 . . . . . . . . . . . . 13 ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → 𝑈𝑋)
46 eldifsn 4462 . . . . . . . . . . . . 13 (𝑈 ∈ (𝑋 ∖ {𝑍}) ↔ (𝑈𝑋𝑈𝑍))
4745, 8, 46sylanbrc 701 . . . . . . . . . . . 12 ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → 𝑈 ∈ (𝑋 ∖ {𝑍}))
48 grpomndo 34005 . . . . . . . . . . . . . 14 ((𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp → (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ MndOp)
49 mndoismgmOLD 34000 . . . . . . . . . . . . . 14 ((𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ MndOp → (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ Magma)
5048, 49syl 17 . . . . . . . . . . . . 13 ((𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp → (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ Magma)
5150adantl 473 . . . . . . . . . . . 12 ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ Magma)
52 eqid 2760 . . . . . . . . . . . . 13 ran 𝐻 = ran 𝐻
53 eqid 2760 . . . . . . . . . . . . 13 (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))
5452, 6, 53exidresid 34009 . . . . . . . . . . . 12 (((𝐻 ∈ (Magma ∩ ExId ) ∧ (𝑋 ∖ {𝑍}) ⊆ ran 𝐻𝑈 ∈ (𝑋 ∖ {𝑍})) ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ Magma) → (GId‘(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))) = 𝑈)
5537, 41, 47, 51, 54syl31anc 1480 . . . . . . . . . . 11 ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → (GId‘(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))) = 𝑈)
5655adantr 472 . . . . . . . . . 10 (((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ∧ 𝑥 ∈ ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))) → (GId‘(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))) = 𝑈)
5733, 56eqtrd 2794 . . . . . . . . 9 (((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ∧ 𝑥 ∈ ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))) → (((inv‘(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))))‘𝑥)(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑥) = 𝑈)
58 oveq1 6821 . . . . . . . . . . 11 (𝑦 = ((inv‘(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))))‘𝑥) → (𝑦(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑥) = (((inv‘(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))))‘𝑥)(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑥))
5958eqeq1d 2762 . . . . . . . . . 10 (𝑦 = ((inv‘(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))))‘𝑥) → ((𝑦(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑥) = 𝑈 ↔ (((inv‘(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))))‘𝑥)(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑥) = 𝑈))
6059rspcev 3449 . . . . . . . . 9 ((((inv‘(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))))‘𝑥) ∈ ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∧ (((inv‘(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))))‘𝑥)(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑥) = 𝑈) → ∃𝑦 ∈ ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))(𝑦(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑥) = 𝑈)
6130, 57, 60syl2anc 696 . . . . . . . 8 (((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ∧ 𝑥 ∈ ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))) → ∃𝑦 ∈ ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))(𝑦(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑥) = 𝑈)
6226, 61syldan 488 . . . . . . 7 (((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → ∃𝑦 ∈ ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))(𝑦(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑥) = 𝑈)
6324adantr 472 . . . . . . . . 9 (((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = (𝑋 ∖ {𝑍}))
6463rexeqdv 3284 . . . . . . . 8 (((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (∃𝑦 ∈ ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))(𝑦(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑥) = 𝑈 ↔ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑥) = 𝑈))
65 ovres 6966 . . . . . . . . . . . 12 ((𝑦 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (𝑦(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑥) = (𝑦𝐻𝑥))
6665ancoms 468 . . . . . . . . . . 11 ((𝑥 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑦 ∈ (𝑋 ∖ {𝑍})) → (𝑦(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑥) = (𝑦𝐻𝑥))
6766eqeq1d 2762 . . . . . . . . . 10 ((𝑥 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑦 ∈ (𝑋 ∖ {𝑍})) → ((𝑦(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑥) = 𝑈 ↔ (𝑦𝐻𝑥) = 𝑈))
6867rexbidva 3187 . . . . . . . . 9 (𝑥 ∈ (𝑋 ∖ {𝑍}) → (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑥) = 𝑈 ↔ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈))
6968adantl 473 . . . . . . . 8 (((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑥) = 𝑈 ↔ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈))
7064, 69bitrd 268 . . . . . . 7 (((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (∃𝑦 ∈ ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))(𝑦(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑥) = 𝑈 ↔ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈))
7162, 70mpbid 222 . . . . . 6 (((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)
7271ralrimiva 3104 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)
738, 72jca 555 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈))
74 fvex 6363 . . . . . . . . 9 (1st𝑅) ∈ V
751, 74eqeltri 2835 . . . . . . . 8 𝐺 ∈ V
7675rnex 7266 . . . . . . 7 ran 𝐺 ∈ V
774, 76eqeltri 2835 . . . . . 6 𝑋 ∈ V
78 difexg 4960 . . . . . 6 (𝑋 ∈ V → (𝑋 ∖ {𝑍}) ∈ V)
7977, 78mp1i 13 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) → (𝑋 ∖ {𝑍}) ∈ V)
80 ffn 6206 . . . . . . . . 9 (𝐻:(𝑋 × 𝑋)⟶𝑋𝐻 Fn (𝑋 × 𝑋))
8114, 80syl 17 . . . . . . . 8 (𝑅 ∈ RingOps → 𝐻 Fn (𝑋 × 𝑋))
8281adantr 472 . . . . . . 7 ((𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) → 𝐻 Fn (𝑋 × 𝑋))
83 fnssres 6165 . . . . . . 7 ((𝐻 Fn (𝑋 × 𝑋) ∧ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})) ⊆ (𝑋 × 𝑋)) → (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) Fn ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))
8482, 13, 83sylancl 697 . . . . . 6 ((𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) → (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) Fn ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))
85 ovres 6966 . . . . . . . . 9 ((𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍})) → (𝑢(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑣) = (𝑢𝐻𝑣))
8685adantl 473 . . . . . . . 8 (((𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}))) → (𝑢(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑣) = (𝑢𝐻𝑣))
87 eldifi 3875 . . . . . . . . . . . 12 (𝑢 ∈ (𝑋 ∖ {𝑍}) → 𝑢𝑋)
88 eldifi 3875 . . . . . . . . . . . 12 (𝑣 ∈ (𝑋 ∖ {𝑍}) → 𝑣𝑋)
8987, 88anim12i 591 . . . . . . . . . . 11 ((𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍})) → (𝑢𝑋𝑣𝑋))
901, 2, 4rngocl 34031 . . . . . . . . . . . 12 ((𝑅 ∈ RingOps ∧ 𝑢𝑋𝑣𝑋) → (𝑢𝐻𝑣) ∈ 𝑋)
91903expb 1114 . . . . . . . . . . 11 ((𝑅 ∈ RingOps ∧ (𝑢𝑋𝑣𝑋)) → (𝑢𝐻𝑣) ∈ 𝑋)
9289, 91sylan2 492 . . . . . . . . . 10 ((𝑅 ∈ RingOps ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}))) → (𝑢𝐻𝑣) ∈ 𝑋)
9392adantlr 753 . . . . . . . . 9 (((𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}))) → (𝑢𝐻𝑣) ∈ 𝑋)
94 oveq2 6822 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑢 → (𝑦𝐻𝑥) = (𝑦𝐻𝑢))
9594eqeq1d 2762 . . . . . . . . . . . . . . 15 (𝑥 = 𝑢 → ((𝑦𝐻𝑥) = 𝑈 ↔ (𝑦𝐻𝑢) = 𝑈))
9695rexbidv 3190 . . . . . . . . . . . . . 14 (𝑥 = 𝑢 → (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈 ↔ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈))
9796rspcv 3445 . . . . . . . . . . . . 13 (𝑢 ∈ (𝑋 ∖ {𝑍}) → (∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈 → ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈))
9897imdistanri 729 . . . . . . . . . . . 12 ((∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈𝑢 ∈ (𝑋 ∖ {𝑍})) → (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈𝑢 ∈ (𝑋 ∖ {𝑍})))
99 eldifsn 4462 . . . . . . . . . . . . . . 15 (𝑣 ∈ (𝑋 ∖ {𝑍}) ↔ (𝑣𝑋𝑣𝑍))
100 ssrexv 3808 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑋 ∖ {𝑍}) ⊆ 𝑋 → (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈 → ∃𝑦𝑋 (𝑦𝐻𝑢) = 𝑈))
10111, 100ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈 → ∃𝑦𝑋 (𝑦𝐻𝑢) = 𝑈)
1021, 2, 3, 4, 6zerdivemp1x 34077 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ RingOps ∧ 𝑢𝑋 ∧ ∃𝑦𝑋 (𝑦𝐻𝑢) = 𝑈) → (𝑣𝑋 → ((𝑢𝐻𝑣) = 𝑍𝑣 = 𝑍)))
103101, 102syl3an3 1170 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ RingOps ∧ 𝑢𝑋 ∧ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈) → (𝑣𝑋 → ((𝑢𝐻𝑣) = 𝑍𝑣 = 𝑍)))
10487, 103syl3an2 1168 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∈ RingOps ∧ 𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈) → (𝑣𝑋 → ((𝑢𝐻𝑣) = 𝑍𝑣 = 𝑍)))
1051043expb 1114 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ RingOps ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈)) → (𝑣𝑋 → ((𝑢𝐻𝑣) = 𝑍𝑣 = 𝑍)))
106105imp 444 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ RingOps ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈)) ∧ 𝑣𝑋) → ((𝑢𝐻𝑣) = 𝑍𝑣 = 𝑍))
107106necon3d 2953 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ RingOps ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈)) ∧ 𝑣𝑋) → (𝑣𝑍 → (𝑢𝐻𝑣) ≠ 𝑍))
108107impr 650 . . . . . . . . . . . . . . 15 (((𝑅 ∈ RingOps ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈)) ∧ (𝑣𝑋𝑣𝑍)) → (𝑢𝐻𝑣) ≠ 𝑍)
10999, 108sylan2b 493 . . . . . . . . . . . . . 14 (((𝑅 ∈ RingOps ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈)) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍})) → (𝑢𝐻𝑣) ≠ 𝑍)
110109an32s 881 . . . . . . . . . . . . 13 (((𝑅 ∈ RingOps ∧ 𝑣 ∈ (𝑋 ∖ {𝑍})) ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈)) → (𝑢𝐻𝑣) ≠ 𝑍)
111110ancom2s 879 . . . . . . . . . . . 12 (((𝑅 ∈ RingOps ∧ 𝑣 ∈ (𝑋 ∖ {𝑍})) ∧ (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈𝑢 ∈ (𝑋 ∖ {𝑍}))) → (𝑢𝐻𝑣) ≠ 𝑍)
11298, 111sylan2 492 . . . . . . . . . . 11 (((𝑅 ∈ RingOps ∧ 𝑣 ∈ (𝑋 ∖ {𝑍})) ∧ (∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈𝑢 ∈ (𝑋 ∖ {𝑍}))) → (𝑢𝐻𝑣) ≠ 𝑍)
113112an42s 905 . . . . . . . . . 10 (((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈) ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}))) → (𝑢𝐻𝑣) ≠ 𝑍)
114113adantlrl 758 . . . . . . . . 9 (((𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}))) → (𝑢𝐻𝑣) ≠ 𝑍)
115 eldifsn 4462 . . . . . . . . 9 ((𝑢𝐻𝑣) ∈ (𝑋 ∖ {𝑍}) ↔ ((𝑢𝐻𝑣) ∈ 𝑋 ∧ (𝑢𝐻𝑣) ≠ 𝑍))
11693, 114, 115sylanbrc 701 . . . . . . . 8 (((𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}))) → (𝑢𝐻𝑣) ∈ (𝑋 ∖ {𝑍}))
11786, 116eqeltrd 2839 . . . . . . 7 (((𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}))) → (𝑢(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑣) ∈ (𝑋 ∖ {𝑍}))
118117ralrimivva 3109 . . . . . 6 ((𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) → ∀𝑢 ∈ (𝑋 ∖ {𝑍})∀𝑣 ∈ (𝑋 ∖ {𝑍})(𝑢(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑣) ∈ (𝑋 ∖ {𝑍}))
119 ffnov 6930 . . . . . 6 ((𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))):((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))⟶(𝑋 ∖ {𝑍}) ↔ ((𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) Fn ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})) ∧ ∀𝑢 ∈ (𝑋 ∖ {𝑍})∀𝑣 ∈ (𝑋 ∖ {𝑍})(𝑢(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑣) ∈ (𝑋 ∖ {𝑍})))
12084, 118, 119sylanbrc 701 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) → (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))):((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))⟶(𝑋 ∖ {𝑍}))
1211163adantr3 1177 . . . . . . 7 (((𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑤 ∈ (𝑋 ∖ {𝑍}))) → (𝑢𝐻𝑣) ∈ (𝑋 ∖ {𝑍}))
122 simpr3 1238 . . . . . . 7 (((𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑤 ∈ (𝑋 ∖ {𝑍}))) → 𝑤 ∈ (𝑋 ∖ {𝑍}))
123121, 122ovresd 6967 . . . . . 6 (((𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑤 ∈ (𝑋 ∖ {𝑍}))) → ((𝑢𝐻𝑣)(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑤) = ((𝑢𝐻𝑣)𝐻𝑤))
124853adant3 1127 . . . . . . . 8 ((𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑤 ∈ (𝑋 ∖ {𝑍})) → (𝑢(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑣) = (𝑢𝐻𝑣))
125124adantl 473 . . . . . . 7 (((𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑤 ∈ (𝑋 ∖ {𝑍}))) → (𝑢(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑣) = (𝑢𝐻𝑣))
126125oveq1d 6829 . . . . . 6 (((𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑤 ∈ (𝑋 ∖ {𝑍}))) → ((𝑢(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑣)(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑤) = ((𝑢𝐻𝑣)(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑤))
127 ovres 6966 . . . . . . . . . 10 ((𝑣 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑤 ∈ (𝑋 ∖ {𝑍})) → (𝑣(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑤) = (𝑣𝐻𝑤))
1281273adant1 1125 . . . . . . . . 9 ((𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑤 ∈ (𝑋 ∖ {𝑍})) → (𝑣(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑤) = (𝑣𝐻𝑤))
129128adantl 473 . . . . . . . 8 (((𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑤 ∈ (𝑋 ∖ {𝑍}))) → (𝑣(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑤) = (𝑣𝐻𝑤))
130129oveq2d 6830 . . . . . . 7 (((𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑤 ∈ (𝑋 ∖ {𝑍}))) → (𝑢𝐻(𝑣(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑤)) = (𝑢𝐻(𝑣𝐻𝑤)))
131 simpr1 1234 . . . . . . . 8 (((𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑤 ∈ (𝑋 ∖ {𝑍}))) → 𝑢 ∈ (𝑋 ∖ {𝑍}))
132 fovrn 6970 . . . . . . . . . 10 (((𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))):((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))⟶(𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑤 ∈ (𝑋 ∖ {𝑍})) → (𝑣(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑤) ∈ (𝑋 ∖ {𝑍}))
1331323adant3r1 1198 . . . . . . . . 9 (((𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))):((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))⟶(𝑋 ∖ {𝑍}) ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑤 ∈ (𝑋 ∖ {𝑍}))) → (𝑣(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑤) ∈ (𝑋 ∖ {𝑍}))
134120, 133sylan 489 . . . . . . . 8 (((𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑤 ∈ (𝑋 ∖ {𝑍}))) → (𝑣(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑤) ∈ (𝑋 ∖ {𝑍}))
135131, 134ovresd 6967 . . . . . . 7 (((𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑤 ∈ (𝑋 ∖ {𝑍}))) → (𝑢(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))(𝑣(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑤)) = (𝑢𝐻(𝑣(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑤)))
136 eldifi 3875 . . . . . . . . . 10 (𝑤 ∈ (𝑋 ∖ {𝑍}) → 𝑤𝑋)
13787, 88, 1363anim123i 1155 . . . . . . . . 9 ((𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑤 ∈ (𝑋 ∖ {𝑍})) → (𝑢𝑋𝑣𝑋𝑤𝑋))
1381, 2, 4rngoass 34036 . . . . . . . . 9 ((𝑅 ∈ RingOps ∧ (𝑢𝑋𝑣𝑋𝑤𝑋)) → ((𝑢𝐻𝑣)𝐻𝑤) = (𝑢𝐻(𝑣𝐻𝑤)))
139137, 138sylan2 492 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑤 ∈ (𝑋 ∖ {𝑍}))) → ((𝑢𝐻𝑣)𝐻𝑤) = (𝑢𝐻(𝑣𝐻𝑤)))
140139adantlr 753 . . . . . . 7 (((𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑤 ∈ (𝑋 ∖ {𝑍}))) → ((𝑢𝐻𝑣)𝐻𝑤) = (𝑢𝐻(𝑣𝐻𝑤)))
141130, 135, 1403eqtr4d 2804 . . . . . 6 (((𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑤 ∈ (𝑋 ∖ {𝑍}))) → (𝑢(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))(𝑣(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑤)) = ((𝑢𝐻𝑣)𝐻𝑤))
142123, 126, 1413eqtr4d 2804 . . . . 5 (((𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑤 ∈ (𝑋 ∖ {𝑍}))) → ((𝑢(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑣)(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑤) = (𝑢(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))(𝑣(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑤)))
14344anim1i 593 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑈𝑍) → (𝑈𝑋𝑈𝑍))
144143, 46sylibr 224 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑈𝑍) → 𝑈 ∈ (𝑋 ∖ {𝑍}))
145144adantrr 755 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) → 𝑈 ∈ (𝑋 ∖ {𝑍}))
146 ovres 6966 . . . . . . . 8 ((𝑈 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑢 ∈ (𝑋 ∖ {𝑍})) → (𝑈(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑢) = (𝑈𝐻𝑢))
147144, 146sylan 489 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝑈𝑍) ∧ 𝑢 ∈ (𝑋 ∖ {𝑍})) → (𝑈(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑢) = (𝑈𝐻𝑢))
1482, 43, 6rngolidm 34067 . . . . . . . . 9 ((𝑅 ∈ RingOps ∧ 𝑢𝑋) → (𝑈𝐻𝑢) = 𝑢)
14987, 148sylan2 492 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑢 ∈ (𝑋 ∖ {𝑍})) → (𝑈𝐻𝑢) = 𝑢)
150149adantlr 753 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝑈𝑍) ∧ 𝑢 ∈ (𝑋 ∖ {𝑍})) → (𝑈𝐻𝑢) = 𝑢)
151147, 150eqtrd 2794 . . . . . 6 (((𝑅 ∈ RingOps ∧ 𝑈𝑍) ∧ 𝑢 ∈ (𝑋 ∖ {𝑍})) → (𝑈(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑢) = 𝑢)
152151adantlrr 759 . . . . 5 (((𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) ∧ 𝑢 ∈ (𝑋 ∖ {𝑍})) → (𝑈(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑢) = 𝑢)
15396rspcva 3447 . . . . . . . . 9 ((𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈) → ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈)
154 oveq1 6821 . . . . . . . . . . . 12 (𝑦 = 𝑧 → (𝑦𝐻𝑢) = (𝑧𝐻𝑢))
155154eqeq1d 2762 . . . . . . . . . . 11 (𝑦 = 𝑧 → ((𝑦𝐻𝑢) = 𝑈 ↔ (𝑧𝐻𝑢) = 𝑈))
156155cbvrexv 3311 . . . . . . . . . 10 (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈 ↔ ∃𝑧 ∈ (𝑋 ∖ {𝑍})(𝑧𝐻𝑢) = 𝑈)
157 ovres 6966 . . . . . . . . . . . . . 14 ((𝑧 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑢 ∈ (𝑋 ∖ {𝑍})) → (𝑧(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑢) = (𝑧𝐻𝑢))
158157eqeq1d 2762 . . . . . . . . . . . . 13 ((𝑧 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑢 ∈ (𝑋 ∖ {𝑍})) → ((𝑧(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑢) = 𝑈 ↔ (𝑧𝐻𝑢) = 𝑈))
159158ancoms 468 . . . . . . . . . . . 12 ((𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑧 ∈ (𝑋 ∖ {𝑍})) → ((𝑧(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑢) = 𝑈 ↔ (𝑧𝐻𝑢) = 𝑈))
160159rexbidva 3187 . . . . . . . . . . 11 (𝑢 ∈ (𝑋 ∖ {𝑍}) → (∃𝑧 ∈ (𝑋 ∖ {𝑍})(𝑧(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑢) = 𝑈 ↔ ∃𝑧 ∈ (𝑋 ∖ {𝑍})(𝑧𝐻𝑢) = 𝑈))
161160biimpar 503 . . . . . . . . . 10 ((𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ ∃𝑧 ∈ (𝑋 ∖ {𝑍})(𝑧𝐻𝑢) = 𝑈) → ∃𝑧 ∈ (𝑋 ∖ {𝑍})(𝑧(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑢) = 𝑈)
162156, 161sylan2b 493 . . . . . . . . 9 ((𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈) → ∃𝑧 ∈ (𝑋 ∖ {𝑍})(𝑧(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑢) = 𝑈)
163153, 162syldan 488 . . . . . . . 8 ((𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈) → ∃𝑧 ∈ (𝑋 ∖ {𝑍})(𝑧(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑢) = 𝑈)
164163ancoms 468 . . . . . . 7 ((∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈𝑢 ∈ (𝑋 ∖ {𝑍})) → ∃𝑧 ∈ (𝑋 ∖ {𝑍})(𝑧(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑢) = 𝑈)
165164adantll 752 . . . . . 6 (((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈) ∧ 𝑢 ∈ (𝑋 ∖ {𝑍})) → ∃𝑧 ∈ (𝑋 ∖ {𝑍})(𝑧(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑢) = 𝑈)
166165adantlrl 758 . . . . 5 (((𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) ∧ 𝑢 ∈ (𝑋 ∖ {𝑍})) → ∃𝑧 ∈ (𝑋 ∖ {𝑍})(𝑧(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑢) = 𝑈)
16779, 120, 142, 145, 152, 166isgrpda 34085 . . . 4 ((𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) → (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp)
16873, 167impbida 913 . . 3 (𝑅 ∈ RingOps → ((𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp ↔ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)))
169168pm5.32i 672 . 2 ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ↔ (𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)))
1705, 169bitri 264 1 (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932  wral 3050  wrex 3051  Vcvv 3340  cdif 3712  cin 3714  wss 3715  {csn 4321   × cxp 5264  dom cdm 5266  ran crn 5267  cres 5268   Fn wfn 6044  wf 6045  cfv 6049  (class class class)co 6814  1st c1st 7332  2nd c2nd 7333  GrpOpcgr 27673  GIdcgi 27674  invcgn 27675   ExId cexid 33974  Magmacmagm 33978  MndOpcmndo 33996  RingOpscrngo 34024  DivRingOpscdrng 34078
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 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  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-rmo 3058  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-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  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-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  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-riota 6775  df-ov 6817  df-om 7232  df-1st 7334  df-2nd 7335  df-1o 7730  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-grpo 27677  df-gid 27678  df-ginv 27679  df-ablo 27729  df-ass 33973  df-exid 33975  df-mgmOLD 33979  df-sgrOLD 33991  df-mndo 33997  df-rngo 34025  df-drngo 34079
This theorem is referenced by:  isdrngo3  34089  divrngidl  34158
  Copyright terms: Public domain W3C validator