Step | Hyp | Ref
| Expression |
1 | | isdivrng1.1 |
. . 3
⊢ 𝐺 = (1st ‘𝑅) |
2 | | isdivrng1.2 |
. . 3
⊢ 𝐻 = (2nd ‘𝑅) |
3 | | isdivrng1.3 |
. . 3
⊢ 𝑍 = (GId‘𝐺) |
4 | | isdivrng1.4 |
. . 3
⊢ 𝑋 = ran 𝐺 |
5 | 1, 2, 3, 4 | isdrngo1 32925 |
. 2
⊢ (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp)) |
6 | | isdivrng2.5 |
. . . . . . 7
⊢ 𝑈 = (GId‘𝐻) |
7 | 1, 2, 4, 3, 6 | dvrunz 32923 |
. . . . . 6
⊢ (𝑅 ∈ DivRingOps → 𝑈 ≠ 𝑍) |
8 | 5, 7 | sylbir 224 |
. . . . 5
⊢ ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → 𝑈 ≠ 𝑍) |
9 | | grporndm 26748 |
. . . . . . . . . . . 12
⊢ ((𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp → ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = dom dom (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))) |
10 | 9 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = dom dom (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))) |
11 | | difss 3699 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑋 ∖ {𝑍}) ⊆ 𝑋 |
12 | | xpss12 5148 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑋 ∖ {𝑍}) ⊆ 𝑋 ∧ (𝑋 ∖ {𝑍}) ⊆ 𝑋) → ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})) ⊆ (𝑋 × 𝑋)) |
13 | 11, 11, 12 | mp2an 704 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})) ⊆ (𝑋 × 𝑋) |
14 | 1, 2, 4 | rngosm 32869 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑅 ∈ RingOps → 𝐻:(𝑋 × 𝑋)⟶𝑋) |
15 | | fdm 5964 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐻:(𝑋 × 𝑋)⟶𝑋 → dom 𝐻 = (𝑋 × 𝑋)) |
16 | 14, 15 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝑅 ∈ RingOps → dom 𝐻 = (𝑋 × 𝑋)) |
17 | 13, 16 | syl5sseqr 3617 |
. . . . . . . . . . . . . . 15
⊢ (𝑅 ∈ RingOps → ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})) ⊆ dom 𝐻) |
18 | 17 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})) ⊆ dom 𝐻) |
19 | | ssdmres 5340 |
. . . . . . . . . . . . . 14
⊢ (((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})) ⊆ dom 𝐻 ↔ dom (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) |
20 | 18, 19 | sylib 207 |
. . . . . . . . . . . . 13
⊢ ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → dom (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) |
21 | 20 | dmeqd 5248 |
. . . . . . . . . . . 12
⊢ ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → dom dom (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = dom ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) |
22 | | dmxpid 5266 |
. . . . . . . . . . . 12
⊢ dom
((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})) = (𝑋 ∖ {𝑍}) |
23 | 21, 22 | syl6eq 2660 |
. . . . . . . . . . 11
⊢ ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → dom dom (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = (𝑋 ∖ {𝑍})) |
24 | 10, 23 | eqtrd 2644 |
. . . . . . . . . 10
⊢ ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = (𝑋 ∖ {𝑍})) |
25 | 24 | eleq2d 2673 |
. . . . . . . . 9
⊢ ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → (𝑥 ∈ ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ↔ 𝑥 ∈ (𝑋 ∖ {𝑍}))) |
26 | 25 | biimpar 501 |
. . . . . . . 8
⊢ (((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → 𝑥 ∈ ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))) |
27 | | eqid 2610 |
. . . . . . . . . . 11
⊢ ran
(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) |
28 | | eqid 2610 |
. . . . . . . . . . 11
⊢
(inv‘(𝐻
↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))) = (inv‘(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))) |
29 | 27, 28 | grpoinvcl 26762 |
. . . . . . . . . 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})))) |
30 | 29 | adantll 746 |
. . . . . . . . 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 2610 |
. . . . . . . . . . . 12
⊢
(GId‘(𝐻
↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))) = (GId‘(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))) |
32 | 27, 31, 28 | grpolinv 26764 |
. . . . . . . . . . 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}))))) |
33 | 32 | adantll 746 |
. . . . . . . . . 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}))))) |
34 | 2 | rngomndo 32904 |
. . . . . . . . . . . . . 14
⊢ (𝑅 ∈ RingOps → 𝐻 ∈ MndOp) |
35 | | mndomgmid 32840 |
. . . . . . . . . . . . . 14
⊢ (𝐻 ∈ MndOp → 𝐻 ∈ (Magma ∩ ExId
)) |
36 | 34, 35 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝑅 ∈ RingOps → 𝐻 ∈ (Magma ∩ ExId
)) |
37 | 36 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → 𝐻 ∈ (Magma ∩ ExId
)) |
38 | 11, 4 | sseqtri 3600 |
. . . . . . . . . . . . . 14
⊢ (𝑋 ∖ {𝑍}) ⊆ ran 𝐺 |
39 | 2, 1 | rngorn1eq 32903 |
. . . . . . . . . . . . . 14
⊢ (𝑅 ∈ RingOps → ran 𝐺 = ran 𝐻) |
40 | 38, 39 | syl5sseq 3616 |
. . . . . . . . . . . . 13
⊢ (𝑅 ∈ RingOps → (𝑋 ∖ {𝑍}) ⊆ ran 𝐻) |
41 | 40 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → (𝑋 ∖ {𝑍}) ⊆ ran 𝐻) |
42 | 1 | rneqi 5273 |
. . . . . . . . . . . . . . . 16
⊢ ran 𝐺 = ran (1st
‘𝑅) |
43 | 4, 42 | eqtri 2632 |
. . . . . . . . . . . . . . 15
⊢ 𝑋 = ran (1st
‘𝑅) |
44 | 43, 2, 6 | rngo1cl 32908 |
. . . . . . . . . . . . . 14
⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋) |
45 | 44 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → 𝑈 ∈ 𝑋) |
46 | | eldifsn 4260 |
. . . . . . . . . . . . 13
⊢ (𝑈 ∈ (𝑋 ∖ {𝑍}) ↔ (𝑈 ∈ 𝑋 ∧ 𝑈 ≠ 𝑍)) |
47 | 45, 8, 46 | sylanbrc 695 |
. . . . . . . . . . . 12
⊢ ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → 𝑈 ∈ (𝑋 ∖ {𝑍})) |
48 | | grpomndo 32844 |
. . . . . . . . . . . . . 14
⊢ ((𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp → (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ MndOp) |
49 | | mndoismgmOLD 32839 |
. . . . . . . . . . . . . 14
⊢ ((𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ MndOp → (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ Magma) |
50 | 48, 49 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp → (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ Magma) |
51 | 50 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ Magma) |
52 | | eqid 2610 |
. . . . . . . . . . . . 13
⊢ ran 𝐻 = ran 𝐻 |
53 | | eqid 2610 |
. . . . . . . . . . . . 13
⊢ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) |
54 | 52, 6, 53 | exidresid 32848 |
. . . . . . . . . . . 12
⊢ (((𝐻 ∈ (Magma ∩ ExId )
∧ (𝑋 ∖ {𝑍}) ⊆ ran 𝐻 ∧ 𝑈 ∈ (𝑋 ∖ {𝑍})) ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ Magma) → (GId‘(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))) = 𝑈) |
55 | 37, 41, 47, 51, 54 | syl31anc 1321 |
. . . . . . . . . . 11
⊢ ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → (GId‘(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))) = 𝑈) |
56 | 55 | adantr 480 |
. . . . . . . . . 10
⊢ (((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ∧ 𝑥 ∈ ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))) → (GId‘(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))) = 𝑈) |
57 | 33, 56 | eqtrd 2644 |
. . . . . . . . 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 6556 |
. . . . . . . . . . 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)) |
59 | 58 | eqeq1d 2612 |
. . . . . . . . . 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)) |
60 | 59 | rspcev 3282 |
. . . . . . . . 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) |
61 | 30, 57, 60 | syl2anc 691 |
. . . . . . . 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) |
62 | 26, 61 | syldan 486 |
. . . . . . 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) |
63 | 24 | adantr 480 |
. . . . . . . . 9
⊢ (((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → ran (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = (𝑋 ∖ {𝑍})) |
64 | 63 | rexeqdv 3122 |
. . . . . . . 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 6698 |
. . . . . . . . . . . 12
⊢ ((𝑦 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (𝑦(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑥) = (𝑦𝐻𝑥)) |
66 | 65 | ancoms 468 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑦 ∈ (𝑋 ∖ {𝑍})) → (𝑦(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑥) = (𝑦𝐻𝑥)) |
67 | 66 | eqeq1d 2612 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑦 ∈ (𝑋 ∖ {𝑍})) → ((𝑦(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑥) = 𝑈 ↔ (𝑦𝐻𝑥) = 𝑈)) |
68 | 67 | rexbidva 3031 |
. . . . . . . . 9
⊢ (𝑥 ∈ (𝑋 ∖ {𝑍}) → (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑥) = 𝑈 ↔ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) |
69 | 68 | adantl 481 |
. . . . . . . 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)) |
70 | 64, 69 | bitrd 267 |
. . . . . . 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)) |
71 | 62, 70 | mpbid 221 |
. . . . . 6
⊢ (((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈) |
72 | 71 | ralrimiva 2949 |
. . . . 5
⊢ ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈) |
73 | 8, 72 | jca 553 |
. . . 4
⊢ ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → (𝑈 ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) |
74 | | fvex 6113 |
. . . . . . . . 9
⊢
(1st ‘𝑅) ∈ V |
75 | 1, 74 | eqeltri 2684 |
. . . . . . . 8
⊢ 𝐺 ∈ V |
76 | 75 | rnex 6992 |
. . . . . . 7
⊢ ran 𝐺 ∈ V |
77 | 4, 76 | eqeltri 2684 |
. . . . . 6
⊢ 𝑋 ∈ V |
78 | | difexg 4735 |
. . . . . 6
⊢ (𝑋 ∈ V → (𝑋 ∖ {𝑍}) ∈ V) |
79 | 77, 78 | mp1i 13 |
. . . . 5
⊢ ((𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) → (𝑋 ∖ {𝑍}) ∈ V) |
80 | | ffn 5958 |
. . . . . . . . 9
⊢ (𝐻:(𝑋 × 𝑋)⟶𝑋 → 𝐻 Fn (𝑋 × 𝑋)) |
81 | 14, 80 | syl 17 |
. . . . . . . 8
⊢ (𝑅 ∈ RingOps → 𝐻 Fn (𝑋 × 𝑋)) |
82 | 81 | adantr 480 |
. . . . . . 7
⊢ ((𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) → 𝐻 Fn (𝑋 × 𝑋)) |
83 | | fnssres 5918 |
. . . . . . 7
⊢ ((𝐻 Fn (𝑋 × 𝑋) ∧ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})) ⊆ (𝑋 × 𝑋)) → (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) Fn ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) |
84 | 82, 13, 83 | sylancl 693 |
. . . . . 6
⊢ ((𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) → (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) Fn ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) |
85 | | ovres 6698 |
. . . . . . . . 9
⊢ ((𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍})) → (𝑢(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑣) = (𝑢𝐻𝑣)) |
86 | 85 | adantl 481 |
. . . . . . . 8
⊢ (((𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}))) → (𝑢(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑣) = (𝑢𝐻𝑣)) |
87 | | eldifi 3694 |
. . . . . . . . . . . 12
⊢ (𝑢 ∈ (𝑋 ∖ {𝑍}) → 𝑢 ∈ 𝑋) |
88 | | eldifi 3694 |
. . . . . . . . . . . 12
⊢ (𝑣 ∈ (𝑋 ∖ {𝑍}) → 𝑣 ∈ 𝑋) |
89 | 87, 88 | anim12i 588 |
. . . . . . . . . . 11
⊢ ((𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍})) → (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) |
90 | 1, 2, 4 | rngocl 32870 |
. . . . . . . . . . . 12
⊢ ((𝑅 ∈ RingOps ∧ 𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) → (𝑢𝐻𝑣) ∈ 𝑋) |
91 | 90 | 3expb 1258 |
. . . . . . . . . . 11
⊢ ((𝑅 ∈ RingOps ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝑢𝐻𝑣) ∈ 𝑋) |
92 | 89, 91 | sylan2 490 |
. . . . . . . . . 10
⊢ ((𝑅 ∈ RingOps ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}))) → (𝑢𝐻𝑣) ∈ 𝑋) |
93 | 92 | adantlr 747 |
. . . . . . . . 9
⊢ (((𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}))) → (𝑢𝐻𝑣) ∈ 𝑋) |
94 | | oveq2 6557 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = 𝑢 → (𝑦𝐻𝑥) = (𝑦𝐻𝑢)) |
95 | 94 | eqeq1d 2612 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑢 → ((𝑦𝐻𝑥) = 𝑈 ↔ (𝑦𝐻𝑢) = 𝑈)) |
96 | 95 | rexbidv 3034 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑢 → (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈 ↔ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈)) |
97 | 96 | rspcv 3278 |
. . . . . . . . . . . . 13
⊢ (𝑢 ∈ (𝑋 ∖ {𝑍}) → (∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈 → ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈)) |
98 | 97 | imdistanri 723 |
. . . . . . . . . . . 12
⊢
((∀𝑥 ∈
(𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈 ∧ 𝑢 ∈ (𝑋 ∖ {𝑍})) → (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈 ∧ 𝑢 ∈ (𝑋 ∖ {𝑍}))) |
99 | | eldifsn 4260 |
. . . . . . . . . . . . . . 15
⊢ (𝑣 ∈ (𝑋 ∖ {𝑍}) ↔ (𝑣 ∈ 𝑋 ∧ 𝑣 ≠ 𝑍)) |
100 | | ssrexv 3630 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑋 ∖ {𝑍}) ⊆ 𝑋 → (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈 → ∃𝑦 ∈ 𝑋 (𝑦𝐻𝑢) = 𝑈)) |
101 | 11, 100 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(∃𝑦 ∈
(𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈 → ∃𝑦 ∈ 𝑋 (𝑦𝐻𝑢) = 𝑈) |
102 | 1, 2, 3, 4, 6 | zerdivemp1x 32916 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑅 ∈ RingOps ∧ 𝑢 ∈ 𝑋 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐻𝑢) = 𝑈) → (𝑣 ∈ 𝑋 → ((𝑢𝐻𝑣) = 𝑍 → 𝑣 = 𝑍))) |
103 | 101, 102 | syl3an3 1353 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑅 ∈ RingOps ∧ 𝑢 ∈ 𝑋 ∧ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈) → (𝑣 ∈ 𝑋 → ((𝑢𝐻𝑣) = 𝑍 → 𝑣 = 𝑍))) |
104 | 87, 103 | syl3an2 1352 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑅 ∈ RingOps ∧ 𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈) → (𝑣 ∈ 𝑋 → ((𝑢𝐻𝑣) = 𝑍 → 𝑣 = 𝑍))) |
105 | 104 | 3expb 1258 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑅 ∈ RingOps ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈)) → (𝑣 ∈ 𝑋 → ((𝑢𝐻𝑣) = 𝑍 → 𝑣 = 𝑍))) |
106 | 105 | imp 444 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑅 ∈ RingOps ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈)) ∧ 𝑣 ∈ 𝑋) → ((𝑢𝐻𝑣) = 𝑍 → 𝑣 = 𝑍)) |
107 | 106 | necon3d 2803 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑅 ∈ RingOps ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈)) ∧ 𝑣 ∈ 𝑋) → (𝑣 ≠ 𝑍 → (𝑢𝐻𝑣) ≠ 𝑍)) |
108 | 107 | impr 647 |
. . . . . . . . . . . . . . 15
⊢ (((𝑅 ∈ RingOps ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈)) ∧ (𝑣 ∈ 𝑋 ∧ 𝑣 ≠ 𝑍)) → (𝑢𝐻𝑣) ≠ 𝑍) |
109 | 99, 108 | sylan2b 491 |
. . . . . . . . . . . . . 14
⊢ (((𝑅 ∈ RingOps ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈)) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍})) → (𝑢𝐻𝑣) ≠ 𝑍) |
110 | 109 | an32s 842 |
. . . . . . . . . . . . 13
⊢ (((𝑅 ∈ RingOps ∧ 𝑣 ∈ (𝑋 ∖ {𝑍})) ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈)) → (𝑢𝐻𝑣) ≠ 𝑍) |
111 | 110 | ancom2s 840 |
. . . . . . . . . . . 12
⊢ (((𝑅 ∈ RingOps ∧ 𝑣 ∈ (𝑋 ∖ {𝑍})) ∧ (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈 ∧ 𝑢 ∈ (𝑋 ∖ {𝑍}))) → (𝑢𝐻𝑣) ≠ 𝑍) |
112 | 98, 111 | sylan2 490 |
. . . . . . . . . . 11
⊢ (((𝑅 ∈ RingOps ∧ 𝑣 ∈ (𝑋 ∖ {𝑍})) ∧ (∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈 ∧ 𝑢 ∈ (𝑋 ∖ {𝑍}))) → (𝑢𝐻𝑣) ≠ 𝑍) |
113 | 112 | an42s 866 |
. . . . . . . . . 10
⊢ (((𝑅 ∈ RingOps ∧
∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈) ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}))) → (𝑢𝐻𝑣) ≠ 𝑍) |
114 | 113 | adantlrl 752 |
. . . . . . . . 9
⊢ (((𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}))) → (𝑢𝐻𝑣) ≠ 𝑍) |
115 | | eldifsn 4260 |
. . . . . . . . 9
⊢ ((𝑢𝐻𝑣) ∈ (𝑋 ∖ {𝑍}) ↔ ((𝑢𝐻𝑣) ∈ 𝑋 ∧ (𝑢𝐻𝑣) ≠ 𝑍)) |
116 | 93, 114, 115 | sylanbrc 695 |
. . . . . . . 8
⊢ (((𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}))) → (𝑢𝐻𝑣) ∈ (𝑋 ∖ {𝑍})) |
117 | 86, 116 | eqeltrd 2688 |
. . . . . . 7
⊢ (((𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}))) → (𝑢(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑣) ∈ (𝑋 ∖ {𝑍})) |
118 | 117 | ralrimivva 2954 |
. . . . . 6
⊢ ((𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀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 6662 |
. . . . . 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}))) |
120 | 84, 118, 119 | sylanbrc 695 |
. . . . 5
⊢ ((𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) → (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))):((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))⟶(𝑋 ∖ {𝑍})) |
121 | 116 | 3adantr3 1215 |
. . . . . . 7
⊢ (((𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑤 ∈ (𝑋 ∖ {𝑍}))) → (𝑢𝐻𝑣) ∈ (𝑋 ∖ {𝑍})) |
122 | | simpr3 1062 |
. . . . . . 7
⊢ (((𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑤 ∈ (𝑋 ∖ {𝑍}))) → 𝑤 ∈ (𝑋 ∖ {𝑍})) |
123 | 121, 122 | ovresd 6699 |
. . . . . 6
⊢ (((𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀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)) |
124 | 85 | 3adant3 1074 |
. . . . . . . 8
⊢ ((𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑤 ∈ (𝑋 ∖ {𝑍})) → (𝑢(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑣) = (𝑢𝐻𝑣)) |
125 | 124 | adantl 481 |
. . . . . . 7
⊢ (((𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀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)) |
126 | 125 | oveq1d 6564 |
. . . . . 6
⊢ (((𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀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 6698 |
. . . . . . . . . 10
⊢ ((𝑣 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑤 ∈ (𝑋 ∖ {𝑍})) → (𝑣(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑤) = (𝑣𝐻𝑤)) |
128 | 127 | 3adant1 1072 |
. . . . . . . . 9
⊢ ((𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑤 ∈ (𝑋 ∖ {𝑍})) → (𝑣(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑤) = (𝑣𝐻𝑤)) |
129 | 128 | adantl 481 |
. . . . . . . 8
⊢ (((𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀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)) |
130 | 129 | oveq2d 6565 |
. . . . . . 7
⊢ (((𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀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 1060 |
. . . . . . . 8
⊢ (((𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑤 ∈ (𝑋 ∖ {𝑍}))) → 𝑢 ∈ (𝑋 ∖ {𝑍})) |
132 | | fovrn 6702 |
. . . . . . . . . 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})) |
133 | 132 | 3adant3r1 1266 |
. . . . . . . . 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})) |
134 | 120, 133 | sylan 487 |
. . . . . . . 8
⊢ (((𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀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})) |
135 | 131, 134 | ovresd 6699 |
. . . . . . 7
⊢ (((𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀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 3694 |
. . . . . . . . . 10
⊢ (𝑤 ∈ (𝑋 ∖ {𝑍}) → 𝑤 ∈ 𝑋) |
137 | 87, 88, 136 | 3anim123i 1240 |
. . . . . . . . 9
⊢ ((𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑤 ∈ (𝑋 ∖ {𝑍})) → (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) |
138 | 1, 2, 4 | rngoass 32875 |
. . . . . . . . 9
⊢ ((𝑅 ∈ RingOps ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝑢𝐻𝑣)𝐻𝑤) = (𝑢𝐻(𝑣𝐻𝑤))) |
139 | 137, 138 | sylan2 490 |
. . . . . . . 8
⊢ ((𝑅 ∈ RingOps ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑤 ∈ (𝑋 ∖ {𝑍}))) → ((𝑢𝐻𝑣)𝐻𝑤) = (𝑢𝐻(𝑣𝐻𝑤))) |
140 | 139 | adantlr 747 |
. . . . . . 7
⊢ (((𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) ∧ (𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑣 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑤 ∈ (𝑋 ∖ {𝑍}))) → ((𝑢𝐻𝑣)𝐻𝑤) = (𝑢𝐻(𝑣𝐻𝑤))) |
141 | 130, 135,
140 | 3eqtr4d 2654 |
. . . . . 6
⊢ (((𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀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)) |
142 | 123, 126,
141 | 3eqtr4d 2654 |
. . . . 5
⊢ (((𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀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))) |
143 | 44 | anim1i 590 |
. . . . . . 7
⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) → (𝑈 ∈ 𝑋 ∧ 𝑈 ≠ 𝑍)) |
144 | 143, 46 | sylibr 223 |
. . . . . 6
⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) → 𝑈 ∈ (𝑋 ∖ {𝑍})) |
145 | 144 | adantrr 749 |
. . . . 5
⊢ ((𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) → 𝑈 ∈ (𝑋 ∖ {𝑍})) |
146 | | ovres 6698 |
. . . . . . . 8
⊢ ((𝑈 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑢 ∈ (𝑋 ∖ {𝑍})) → (𝑈(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑢) = (𝑈𝐻𝑢)) |
147 | 144, 146 | sylan 487 |
. . . . . . 7
⊢ (((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) ∧ 𝑢 ∈ (𝑋 ∖ {𝑍})) → (𝑈(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑢) = (𝑈𝐻𝑢)) |
148 | 2, 43, 6 | rngolidm 32906 |
. . . . . . . . 9
⊢ ((𝑅 ∈ RingOps ∧ 𝑢 ∈ 𝑋) → (𝑈𝐻𝑢) = 𝑢) |
149 | 87, 148 | sylan2 490 |
. . . . . . . 8
⊢ ((𝑅 ∈ RingOps ∧ 𝑢 ∈ (𝑋 ∖ {𝑍})) → (𝑈𝐻𝑢) = 𝑢) |
150 | 149 | adantlr 747 |
. . . . . . 7
⊢ (((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) ∧ 𝑢 ∈ (𝑋 ∖ {𝑍})) → (𝑈𝐻𝑢) = 𝑢) |
151 | 147, 150 | eqtrd 2644 |
. . . . . 6
⊢ (((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) ∧ 𝑢 ∈ (𝑋 ∖ {𝑍})) → (𝑈(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑢) = 𝑢) |
152 | 151 | adantlrr 753 |
. . . . 5
⊢ (((𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) ∧ 𝑢 ∈ (𝑋 ∖ {𝑍})) → (𝑈(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑢) = 𝑢) |
153 | 96 | rspcva 3280 |
. . . . . . . . 9
⊢ ((𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈) → ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈) |
154 | | oveq1 6556 |
. . . . . . . . . . . 12
⊢ (𝑦 = 𝑧 → (𝑦𝐻𝑢) = (𝑧𝐻𝑢)) |
155 | 154 | eqeq1d 2612 |
. . . . . . . . . . 11
⊢ (𝑦 = 𝑧 → ((𝑦𝐻𝑢) = 𝑈 ↔ (𝑧𝐻𝑢) = 𝑈)) |
156 | 155 | cbvrexv 3148 |
. . . . . . . . . 10
⊢
(∃𝑦 ∈
(𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈 ↔ ∃𝑧 ∈ (𝑋 ∖ {𝑍})(𝑧𝐻𝑢) = 𝑈) |
157 | | ovres 6698 |
. . . . . . . . . . . . . 14
⊢ ((𝑧 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑢 ∈ (𝑋 ∖ {𝑍})) → (𝑧(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑢) = (𝑧𝐻𝑢)) |
158 | 157 | eqeq1d 2612 |
. . . . . . . . . . . . 13
⊢ ((𝑧 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑢 ∈ (𝑋 ∖ {𝑍})) → ((𝑧(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑢) = 𝑈 ↔ (𝑧𝐻𝑢) = 𝑈)) |
159 | 158 | ancoms 468 |
. . . . . . . . . . . 12
⊢ ((𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ 𝑧 ∈ (𝑋 ∖ {𝑍})) → ((𝑧(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑢) = 𝑈 ↔ (𝑧𝐻𝑢) = 𝑈)) |
160 | 159 | rexbidva 3031 |
. . . . . . . . . . 11
⊢ (𝑢 ∈ (𝑋 ∖ {𝑍}) → (∃𝑧 ∈ (𝑋 ∖ {𝑍})(𝑧(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑢) = 𝑈 ↔ ∃𝑧 ∈ (𝑋 ∖ {𝑍})(𝑧𝐻𝑢) = 𝑈)) |
161 | 160 | biimpar 501 |
. . . . . . . . . 10
⊢ ((𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ ∃𝑧 ∈ (𝑋 ∖ {𝑍})(𝑧𝐻𝑢) = 𝑈) → ∃𝑧 ∈ (𝑋 ∖ {𝑍})(𝑧(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑢) = 𝑈) |
162 | 156, 161 | sylan2b 491 |
. . . . . . . . 9
⊢ ((𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑢) = 𝑈) → ∃𝑧 ∈ (𝑋 ∖ {𝑍})(𝑧(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑢) = 𝑈) |
163 | 153, 162 | syldan 486 |
. . . . . . . 8
⊢ ((𝑢 ∈ (𝑋 ∖ {𝑍}) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈) → ∃𝑧 ∈ (𝑋 ∖ {𝑍})(𝑧(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑢) = 𝑈) |
164 | 163 | ancoms 468 |
. . . . . . 7
⊢
((∀𝑥 ∈
(𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈 ∧ 𝑢 ∈ (𝑋 ∖ {𝑍})) → ∃𝑧 ∈ (𝑋 ∖ {𝑍})(𝑧(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑢) = 𝑈) |
165 | 164 | adantll 746 |
. . . . . 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) |
166 | 165 | adantlrl 752 |
. . . . 5
⊢ (((𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) ∧ 𝑢 ∈ (𝑋 ∖ {𝑍})) → ∃𝑧 ∈ (𝑋 ∖ {𝑍})(𝑧(𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))𝑢) = 𝑈) |
167 | 79, 120, 142, 145, 152, 166 | isgrpda 32924 |
. . . 4
⊢ ((𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)) → (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) |
168 | 73, 167 | impbida 873 |
. . 3
⊢ (𝑅 ∈ RingOps → ((𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp ↔ (𝑈 ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈))) |
169 | 168 | pm5.32i 667 |
. 2
⊢ ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ↔ (𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈))) |
170 | 5, 169 | bitri 263 |
1
⊢ (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈))) |