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Theorem isdrngo2 33389
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 33387 . 2 (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))
6 isdivrng2.5 . . . . . . 7 𝑈 = (GId‘𝐻)
71, 2, 4, 3, 6dvrunz 33385 . . . . . 6 (𝑅 ∈ DivRingOps → 𝑈𝑍)
85, 7sylbir 225 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → 𝑈𝑍)
9 grporndm 27213 . . . . . . . . . . . 12 ((𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp → ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = dom dom (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))))
109adantl 482 . . . . . . . . . . 11 ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = dom dom (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))))
11 difss 3715 . . . . . . . . . . . . . . . . 17 (𝑋 ∖ {𝑍}) ⊆ 𝑋
12 xpss12 5186 . . . . . . . . . . . . . . . . 17 (((𝑋 ∖ {𝑍}) ⊆ 𝑋 ∧ (𝑋 ∖ {𝑍}) ⊆ 𝑋) → ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})) ⊆ (𝑋 × 𝑋))
1311, 11, 12mp2an 707 . . . . . . . . . . . . . . . 16 ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})) ⊆ (𝑋 × 𝑋)
141, 2, 4rngosm 33331 . . . . . . . . . . . . . . . . 17 (𝑅 ∈ RingOps → 𝐻:(𝑋 × 𝑋)⟶𝑋)
15 fdm 6008 . . . . . . . . . . . . . . . . 17 (𝐻:(𝑋 × 𝑋)⟶𝑋 → dom 𝐻 = (𝑋 × 𝑋))
1614, 15syl 17 . . . . . . . . . . . . . . . 16 (𝑅 ∈ RingOps → dom 𝐻 = (𝑋 × 𝑋))
1713, 16syl5sseqr 3633 . . . . . . . . . . . . . . 15 (𝑅 ∈ RingOps → ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})) ⊆ dom 𝐻)
1817adantr 481 . . . . . . . . . . . . . 14 ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})) ⊆ dom 𝐻)
19 ssdmres 5379 . . . . . . . . . . . . . 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 5286 . . . . . . . . . . . 12 ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → dom dom (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = dom ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))
22 dmxpid 5305 . . . . . . . . . . . 12 dom ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})) = (𝑋 ∖ {𝑍})
2321, 22syl6eq 2671 . . . . . . . . . . 11 ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → dom dom (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = (𝑋 ∖ {𝑍}))
2410, 23eqtrd 2655 . . . . . . . . . 10 ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = (𝑋 ∖ {𝑍}))
2524eleq2d 2684 . . . . . . . . 9 ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → (𝑥 ∈ ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ↔ 𝑥 ∈ (𝑋 ∖ {𝑍})))
2625biimpar 502 . . . . . . . 8 (((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → 𝑥 ∈ ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))))
27 eqid 2621 . . . . . . . . . . 11 ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))
28 eqid 2621 . . . . . . . . . . 11 (inv‘(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))) = (inv‘(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))))
2927, 28grpoinvcl 27227 . . . . . . . . . 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 749 . . . . . . . . 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 2621 . . . . . . . . . . . 12 (GId‘(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))) = (GId‘(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))))
3227, 31, 28grpolinv 27229 . . . . . . . . . . 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 749 . . . . . . . . . 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 33366 . . . . . . . . . . . . . 14 (𝑅 ∈ RingOps → 𝐻 ∈ MndOp)
35 mndomgmid 33302 . . . . . . . . . . . . . 14 (𝐻 ∈ MndOp → 𝐻 ∈ (Magma ∩ ExId ))
3634, 35syl 17 . . . . . . . . . . . . 13 (𝑅 ∈ RingOps → 𝐻 ∈ (Magma ∩ ExId ))
3736adantr 481 . . . . . . . . . . . 12 ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → 𝐻 ∈ (Magma ∩ ExId ))
3811, 4sseqtri 3616 . . . . . . . . . . . . . 14 (𝑋 ∖ {𝑍}) ⊆ ran 𝐺
392, 1rngorn1eq 33365 . . . . . . . . . . . . . 14 (𝑅 ∈ RingOps → ran 𝐺 = ran 𝐻)
4038, 39syl5sseq 3632 . . . . . . . . . . . . 13 (𝑅 ∈ RingOps → (𝑋 ∖ {𝑍}) ⊆ ran 𝐻)
4140adantr 481 . . . . . . . . . . . 12 ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → (𝑋 ∖ {𝑍}) ⊆ ran 𝐻)
421rneqi 5312 . . . . . . . . . . . . . . . 16 ran 𝐺 = ran (1st𝑅)
434, 42eqtri 2643 . . . . . . . . . . . . . . 15 𝑋 = ran (1st𝑅)
4443, 2, 6rngo1cl 33370 . . . . . . . . . . . . . 14 (𝑅 ∈ RingOps → 𝑈𝑋)
4544adantr 481 . . . . . . . . . . . . 13 ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → 𝑈𝑋)
46 eldifsn 4287 . . . . . . . . . . . . 13 (𝑈 ∈ (𝑋 ∖ {𝑍}) ↔ (𝑈𝑋𝑈𝑍))
4745, 8, 46sylanbrc 697 . . . . . . . . . . . 12 ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → 𝑈 ∈ (𝑋 ∖ {𝑍}))
48 grpomndo 33306 . . . . . . . . . . . . . 14 ((𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp → (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ MndOp)
49 mndoismgmOLD 33301 . . . . . . . . . . . . . 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 482 . . . . . . . . . . . 12 ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ Magma)
52 eqid 2621 . . . . . . . . . . . . 13 ran 𝐻 = ran 𝐻
53 eqid 2621 . . . . . . . . . . . . 13 (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))
5452, 6, 53exidresid 33310 . . . . . . . . . . . 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 1326 . . . . . . . . . . 11 ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → (GId‘(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))) = 𝑈)
5655adantr 481 . . . . . . . . . 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 2655 . . . . . . . . 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 6611 . . . . . . . . . . 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 2623 . . . . . . . . . 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 3295 . . . . . . . . 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 692 . . . . . . . 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 487 . . . . . . 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 481 . . . . . . . . 9 (((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = (𝑋 ∖ {𝑍}))
6463rexeqdv 3134 . . . . . . . 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 6753 . . . . . . . . . . . 12 ((𝑦 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (𝑦(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑥) = (𝑦𝐻𝑥))
6665ancoms 469 . . . . . . . . . . 11 ((𝑥 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑦 ∈ (𝑋 ∖ {𝑍})) → (𝑦(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑥) = (𝑦𝐻𝑥))
6766eqeq1d 2623 . . . . . . . . . 10 ((𝑥 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑦 ∈ (𝑋 ∖ {𝑍})) → ((𝑦(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑥) = 𝑈 ↔ (𝑦𝐻𝑥) = 𝑈))
6867rexbidva 3042 . . . . . . . . 9 (𝑥 ∈ (𝑋 ∖ {𝑍}) → (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑥) = 𝑈 ↔ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈))
6968adantl 482 . . . . . . . 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 2960 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)
738, 72jca 554 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈))
74 fvex 6158 . . . . . . . . 9 (1st𝑅) ∈ V
751, 74eqeltri 2694 . . . . . . . 8 𝐺 ∈ V
7675rnex 7047 . . . . . . 7 ran 𝐺 ∈ V
774, 76eqeltri 2694 . . . . . 6 𝑋 ∈ V
78 difexg 4768 . . . . . 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 6002 . . . . . . . . 9 (𝐻:(𝑋 × 𝑋)⟶𝑋𝐻 Fn (𝑋 × 𝑋))
8114, 80syl 17 . . . . . . . 8 (𝑅 ∈ RingOps → 𝐻 Fn (𝑋 × 𝑋))
8281adantr 481 . . . . . . 7 ((𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) → 𝐻 Fn (𝑋 × 𝑋))
83 fnssres 5962 . . . . . . 7 ((𝐻 Fn (𝑋 × 𝑋) ∧ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})) ⊆ (𝑋 × 𝑋)) → (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) Fn ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))
8482, 13, 83sylancl 693 . . . . . 6 ((𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) → (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) Fn ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))
85 ovres 6753 . . . . . . . . 9 ((𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍})) → (𝑢(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑣) = (𝑢𝐻𝑣))
8685adantl 482 . . . . . . . 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 3710 . . . . . . . . . . . 12 (𝑢 ∈ (𝑋 ∖ {𝑍}) → 𝑢𝑋)
88 eldifi 3710 . . . . . . . . . . . 12 (𝑣 ∈ (𝑋 ∖ {𝑍}) → 𝑣𝑋)
8987, 88anim12i 589 . . . . . . . . . . 11 ((𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍})) → (𝑢𝑋𝑣𝑋))
901, 2, 4rngocl 33332 . . . . . . . . . . . 12 ((𝑅 ∈ RingOps ∧ 𝑢𝑋𝑣𝑋) → (𝑢𝐻𝑣) ∈ 𝑋)
91903expb 1263 . . . . . . . . . . 11 ((𝑅 ∈ RingOps ∧ (𝑢𝑋𝑣𝑋)) → (𝑢𝐻𝑣) ∈ 𝑋)
9289, 91sylan2 491 . . . . . . . . . 10 ((𝑅 ∈ RingOps ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}))) → (𝑢𝐻𝑣) ∈ 𝑋)
9392adantlr 750 . . . . . . . . 9 (((𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}))) → (𝑢𝐻𝑣) ∈ 𝑋)
94 oveq2 6612 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑢 → (𝑦𝐻𝑥) = (𝑦𝐻𝑢))
9594eqeq1d 2623 . . . . . . . . . . . . . . 15 (𝑥 = 𝑢 → ((𝑦𝐻𝑥) = 𝑈 ↔ (𝑦𝐻𝑢) = 𝑈))
9695rexbidv 3045 . . . . . . . . . . . . . 14 (𝑥 = 𝑢 → (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈 ↔ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈))
9796rspcv 3291 . . . . . . . . . . . . 13 (𝑢 ∈ (𝑋 ∖ {𝑍}) → (∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈 → ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈))
9897imdistanri 726 . . . . . . . . . . . 12 ((∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈𝑢 ∈ (𝑋 ∖ {𝑍})) → (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈𝑢 ∈ (𝑋 ∖ {𝑍})))
99 eldifsn 4287 . . . . . . . . . . . . . . 15 (𝑣 ∈ (𝑋 ∖ {𝑍}) ↔ (𝑣𝑋𝑣𝑍))
100 ssrexv 3646 . . . . . . . . . . . . . . . . . . . . . 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 33378 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ RingOps ∧ 𝑢𝑋 ∧ ∃𝑦𝑋 (𝑦𝐻𝑢) = 𝑈) → (𝑣𝑋 → ((𝑢𝐻𝑣) = 𝑍𝑣 = 𝑍)))
103101, 102syl3an3 1358 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ RingOps ∧ 𝑢𝑋 ∧ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈) → (𝑣𝑋 → ((𝑢𝐻𝑣) = 𝑍𝑣 = 𝑍)))
10487, 103syl3an2 1357 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∈ RingOps ∧ 𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈) → (𝑣𝑋 → ((𝑢𝐻𝑣) = 𝑍𝑣 = 𝑍)))
1051043expb 1263 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ RingOps ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈)) → (𝑣𝑋 → ((𝑢𝐻𝑣) = 𝑍𝑣 = 𝑍)))
106105imp 445 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ RingOps ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈)) ∧ 𝑣𝑋) → ((𝑢𝐻𝑣) = 𝑍𝑣 = 𝑍))
107106necon3d 2811 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ RingOps ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈)) ∧ 𝑣𝑋) → (𝑣𝑍 → (𝑢𝐻𝑣) ≠ 𝑍))
108107impr 648 . . . . . . . . . . . . . . 15 (((𝑅 ∈ RingOps ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈)) ∧ (𝑣𝑋𝑣𝑍)) → (𝑢𝐻𝑣) ≠ 𝑍)
10999, 108sylan2b 492 . . . . . . . . . . . . . 14 (((𝑅 ∈ RingOps ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈)) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍})) → (𝑢𝐻𝑣) ≠ 𝑍)
110109an32s 845 . . . . . . . . . . . . 13 (((𝑅 ∈ RingOps ∧ 𝑣 ∈ (𝑋 ∖ {𝑍})) ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈)) → (𝑢𝐻𝑣) ≠ 𝑍)
111110ancom2s 843 . . . . . . . . . . . 12 (((𝑅 ∈ RingOps ∧ 𝑣 ∈ (𝑋 ∖ {𝑍})) ∧ (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈𝑢 ∈ (𝑋 ∖ {𝑍}))) → (𝑢𝐻𝑣) ≠ 𝑍)
11298, 111sylan2 491 . . . . . . . . . . 11 (((𝑅 ∈ RingOps ∧ 𝑣 ∈ (𝑋 ∖ {𝑍})) ∧ (∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈𝑢 ∈ (𝑋 ∖ {𝑍}))) → (𝑢𝐻𝑣) ≠ 𝑍)
113112an42s 869 . . . . . . . . . 10 (((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈) ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}))) → (𝑢𝐻𝑣) ≠ 𝑍)
114113adantlrl 755 . . . . . . . . 9 (((𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}))) → (𝑢𝐻𝑣) ≠ 𝑍)
115 eldifsn 4287 . . . . . . . . 9 ((𝑢𝐻𝑣) ∈ (𝑋 ∖ {𝑍}) ↔ ((𝑢𝐻𝑣) ∈ 𝑋 ∧ (𝑢𝐻𝑣) ≠ 𝑍))
11693, 114, 115sylanbrc 697 . . . . . . . 8 (((𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}))) → (𝑢𝐻𝑣) ∈ (𝑋 ∖ {𝑍}))
11786, 116eqeltrd 2698 . . . . . . 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 2965 . . . . . 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 6717 . . . . . 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 697 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) → (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))):((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))⟶(𝑋 ∖ {𝑍}))
1211163adantr3 1220 . . . . . . 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 1067 . . . . . . 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 6754 . . . . . 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 1079 . . . . . . . 8 ((𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑤 ∈ (𝑋 ∖ {𝑍})) → (𝑢(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑣) = (𝑢𝐻𝑣))
125124adantl 482 . . . . . . 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 6619 . . . . . 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 6753 . . . . . . . . . 10 ((𝑣 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑤 ∈ (𝑋 ∖ {𝑍})) → (𝑣(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑤) = (𝑣𝐻𝑤))
1281273adant1 1077 . . . . . . . . 9 ((𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑤 ∈ (𝑋 ∖ {𝑍})) → (𝑣(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑤) = (𝑣𝐻𝑤))
129128adantl 482 . . . . . . . 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 6620 . . . . . . 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 1065 . . . . . . . 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 6757 . . . . . . . . . 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 1271 . . . . . . . . 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 488 . . . . . . . 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 6754 . . . . . . 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 3710 . . . . . . . . . 10 (𝑤 ∈ (𝑋 ∖ {𝑍}) → 𝑤𝑋)
13787, 88, 1363anim123i 1245 . . . . . . . . 9 ((𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑤 ∈ (𝑋 ∖ {𝑍})) → (𝑢𝑋𝑣𝑋𝑤𝑋))
1381, 2, 4rngoass 33337 . . . . . . . . 9 ((𝑅 ∈ RingOps ∧ (𝑢𝑋𝑣𝑋𝑤𝑋)) → ((𝑢𝐻𝑣)𝐻𝑤) = (𝑢𝐻(𝑣𝐻𝑤)))
139137, 138sylan2 491 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑤 ∈ (𝑋 ∖ {𝑍}))) → ((𝑢𝐻𝑣)𝐻𝑤) = (𝑢𝐻(𝑣𝐻𝑤)))
140139adantlr 750 . . . . . . 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 2665 . . . . . 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 2665 . . . . 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 591 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑈𝑍) → (𝑈𝑋𝑈𝑍))
144143, 46sylibr 224 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑈𝑍) → 𝑈 ∈ (𝑋 ∖ {𝑍}))
145144adantrr 752 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) → 𝑈 ∈ (𝑋 ∖ {𝑍}))
146 ovres 6753 . . . . . . . 8 ((𝑈 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑢 ∈ (𝑋 ∖ {𝑍})) → (𝑈(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑢) = (𝑈𝐻𝑢))
147144, 146sylan 488 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝑈𝑍) ∧ 𝑢 ∈ (𝑋 ∖ {𝑍})) → (𝑈(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑢) = (𝑈𝐻𝑢))
1482, 43, 6rngolidm 33368 . . . . . . . . 9 ((𝑅 ∈ RingOps ∧ 𝑢𝑋) → (𝑈𝐻𝑢) = 𝑢)
14987, 148sylan2 491 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑢 ∈ (𝑋 ∖ {𝑍})) → (𝑈𝐻𝑢) = 𝑢)
150149adantlr 750 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝑈𝑍) ∧ 𝑢 ∈ (𝑋 ∖ {𝑍})) → (𝑈𝐻𝑢) = 𝑢)
151147, 150eqtrd 2655 . . . . . 6 (((𝑅 ∈ RingOps ∧ 𝑈𝑍) ∧ 𝑢 ∈ (𝑋 ∖ {𝑍})) → (𝑈(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑢) = 𝑢)
152151adantlrr 756 . . . . 5 (((𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) ∧ 𝑢 ∈ (𝑋 ∖ {𝑍})) → (𝑈(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑢) = 𝑢)
15396rspcva 3293 . . . . . . . . 9 ((𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈) → ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈)
154 oveq1 6611 . . . . . . . . . . . 12 (𝑦 = 𝑧 → (𝑦𝐻𝑢) = (𝑧𝐻𝑢))
155154eqeq1d 2623 . . . . . . . . . . 11 (𝑦 = 𝑧 → ((𝑦𝐻𝑢) = 𝑈 ↔ (𝑧𝐻𝑢) = 𝑈))
156155cbvrexv 3160 . . . . . . . . . 10 (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈 ↔ ∃𝑧 ∈ (𝑋 ∖ {𝑍})(𝑧𝐻𝑢) = 𝑈)
157 ovres 6753 . . . . . . . . . . . . . 14 ((𝑧 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑢 ∈ (𝑋 ∖ {𝑍})) → (𝑧(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑢) = (𝑧𝐻𝑢))
158157eqeq1d 2623 . . . . . . . . . . . . 13 ((𝑧 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑢 ∈ (𝑋 ∖ {𝑍})) → ((𝑧(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑢) = 𝑈 ↔ (𝑧𝐻𝑢) = 𝑈))
159158ancoms 469 . . . . . . . . . . . 12 ((𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑧 ∈ (𝑋 ∖ {𝑍})) → ((𝑧(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑢) = 𝑈 ↔ (𝑧𝐻𝑢) = 𝑈))
160159rexbidva 3042 . . . . . . . . . . 11 (𝑢 ∈ (𝑋 ∖ {𝑍}) → (∃𝑧 ∈ (𝑋 ∖ {𝑍})(𝑧(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑢) = 𝑈 ↔ ∃𝑧 ∈ (𝑋 ∖ {𝑍})(𝑧𝐻𝑢) = 𝑈))
161160biimpar 502 . . . . . . . . . 10 ((𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ ∃𝑧 ∈ (𝑋 ∖ {𝑍})(𝑧𝐻𝑢) = 𝑈) → ∃𝑧 ∈ (𝑋 ∖ {𝑍})(𝑧(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑢) = 𝑈)
162156, 161sylan2b 492 . . . . . . . . 9 ((𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈) → ∃𝑧 ∈ (𝑋 ∖ {𝑍})(𝑧(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑢) = 𝑈)
163153, 162syldan 487 . . . . . . . 8 ((𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈) → ∃𝑧 ∈ (𝑋 ∖ {𝑍})(𝑧(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑢) = 𝑈)
164163ancoms 469 . . . . . . 7 ((∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈𝑢 ∈ (𝑋 ∖ {𝑍})) → ∃𝑧 ∈ (𝑋 ∖ {𝑍})(𝑧(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑢) = 𝑈)
165164adantll 749 . . . . . 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 755 . . . . 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 33386 . . . 4 ((𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) → (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp)
16873, 167impbida 876 . . 3 (𝑅 ∈ RingOps → ((𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp ↔ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)))
169168pm5.32i 668 . 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 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  Vcvv 3186  cdif 3552  cin 3554  wss 3555  {csn 4148   × cxp 5072  dom cdm 5074  ran crn 5075  cres 5076   Fn wfn 5842  wf 5843  cfv 5847  (class class class)co 6604  1st c1st 7111  2nd c2nd 7112  GrpOpcgr 27192  GIdcgi 27193  invcgn 27194   ExId cexid 33275  Magmacmagm 33279  MndOpcmndo 33297  RingOpscrngo 33325  DivRingOpscdrng 33379
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-om 7013  df-1st 7113  df-2nd 7114  df-1o 7505  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-grpo 27196  df-gid 27197  df-ginv 27198  df-ablo 27248  df-ass 33274  df-exid 33276  df-mgmOLD 33280  df-sgrOLD 33292  df-mndo 33298  df-rngo 33326  df-drngo 33380
This theorem is referenced by:  isdrngo3  33390  divrngidl  33459
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