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

Theorem divrngidl 37436
Description: The only ideals in a division ring are the zero ideal and the unit ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
divrngidl.1 𝐺 = (1st𝑅)
divrngidl.2 𝐻 = (2nd𝑅)
divrngidl.3 𝑋 = ran 𝐺
divrngidl.4 𝑍 = (GId‘𝐺)
Assertion
Ref Expression
divrngidl (𝑅 ∈ DivRingOps → (Idl‘𝑅) = {{𝑍}, 𝑋})

Proof of Theorem divrngidl
Dummy variables 𝑖 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 divrngidl.1 . . 3 𝐺 = (1st𝑅)
2 divrngidl.2 . . 3 𝐻 = (2nd𝑅)
3 divrngidl.4 . . 3 𝑍 = (GId‘𝐺)
4 divrngidl.3 . . 3 𝑋 = ran 𝐺
5 eqid 2727 . . 3 (GId‘𝐻) = (GId‘𝐻)
61, 2, 3, 4, 5isdrngo2 37366 . 2 (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ ((GId‘𝐻) ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻))))
71, 3idl0cl 37426 . . . . . . . . . . 11 ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → 𝑍𝑖)
87adantr 480 . . . . . . . . . 10 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → 𝑍𝑖)
93fvexi 6905 . . . . . . . . . . . . 13 𝑍 ∈ V
109snss 4785 . . . . . . . . . . . 12 (𝑍𝑖 ↔ {𝑍} ⊆ 𝑖)
11 necom 2989 . . . . . . . . . . . 12 (𝑖 ≠ {𝑍} ↔ {𝑍} ≠ 𝑖)
12 pssdifn0 4361 . . . . . . . . . . . . 13 (({𝑍} ⊆ 𝑖 ∧ {𝑍} ≠ 𝑖) → (𝑖 ∖ {𝑍}) ≠ ∅)
13 n0 4342 . . . . . . . . . . . . 13 ((𝑖 ∖ {𝑍}) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (𝑖 ∖ {𝑍}))
1412, 13sylib 217 . . . . . . . . . . . 12 (({𝑍} ⊆ 𝑖 ∧ {𝑍} ≠ 𝑖) → ∃𝑧 𝑧 ∈ (𝑖 ∖ {𝑍}))
1510, 11, 14syl2anb 597 . . . . . . . . . . 11 ((𝑍𝑖𝑖 ≠ {𝑍}) → ∃𝑧 𝑧 ∈ (𝑖 ∖ {𝑍}))
161, 4idlss 37424 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → 𝑖𝑋)
17 ssdif 4135 . . . . . . . . . . . . . . . . . 18 (𝑖𝑋 → (𝑖 ∖ {𝑍}) ⊆ (𝑋 ∖ {𝑍}))
1817sselda 3978 . . . . . . . . . . . . . . . . 17 ((𝑖𝑋𝑧 ∈ (𝑖 ∖ {𝑍})) → 𝑧 ∈ (𝑋 ∖ {𝑍}))
1916, 18sylan 579 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ (𝑖 ∖ {𝑍})) → 𝑧 ∈ (𝑋 ∖ {𝑍}))
20 oveq2 7422 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑧 → (𝑦𝐻𝑥) = (𝑦𝐻𝑧))
2120eqeq1d 2729 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑧 → ((𝑦𝐻𝑥) = (GId‘𝐻) ↔ (𝑦𝐻𝑧) = (GId‘𝐻)))
2221rexbidv 3173 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻) ↔ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑧) = (GId‘𝐻)))
2322rspcva 3605 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ (𝑋 ∖ {𝑍}) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑧) = (GId‘𝐻))
2419, 23sylan 579 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ (𝑖 ∖ {𝑍})) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑧) = (GId‘𝐻))
25 eldifi 4122 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ (𝑖 ∖ {𝑍}) → 𝑧𝑖)
26 eldifi 4122 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (𝑋 ∖ {𝑍}) → 𝑦𝑋)
2725, 26anim12i 612 . . . . . . . . . . . . . . . . . . 19 ((𝑧 ∈ (𝑖 ∖ {𝑍}) ∧ 𝑦 ∈ (𝑋 ∖ {𝑍})) → (𝑧𝑖𝑦𝑋))
281, 2, 4idllmulcl 37428 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑧𝑖𝑦𝑋)) → (𝑦𝐻𝑧) ∈ 𝑖)
291, 2, 4, 51idl 37434 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → ((GId‘𝐻) ∈ 𝑖𝑖 = 𝑋))
3029biimpd 228 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → ((GId‘𝐻) ∈ 𝑖𝑖 = 𝑋))
3130adantr 480 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑧𝑖𝑦𝑋)) → ((GId‘𝐻) ∈ 𝑖𝑖 = 𝑋))
32 eleq1 2816 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦𝐻𝑧) = (GId‘𝐻) → ((𝑦𝐻𝑧) ∈ 𝑖 ↔ (GId‘𝐻) ∈ 𝑖))
3332imbi1d 341 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦𝐻𝑧) = (GId‘𝐻) → (((𝑦𝐻𝑧) ∈ 𝑖𝑖 = 𝑋) ↔ ((GId‘𝐻) ∈ 𝑖𝑖 = 𝑋)))
3431, 33syl5ibrcom 246 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑧𝑖𝑦𝑋)) → ((𝑦𝐻𝑧) = (GId‘𝐻) → ((𝑦𝐻𝑧) ∈ 𝑖𝑖 = 𝑋)))
3528, 34mpid 44 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑧𝑖𝑦𝑋)) → ((𝑦𝐻𝑧) = (GId‘𝐻) → 𝑖 = 𝑋))
3627, 35sylan2 592 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑧 ∈ (𝑖 ∖ {𝑍}) ∧ 𝑦 ∈ (𝑋 ∖ {𝑍}))) → ((𝑦𝐻𝑧) = (GId‘𝐻) → 𝑖 = 𝑋))
3736anassrs 467 . . . . . . . . . . . . . . . . 17 ((((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ (𝑖 ∖ {𝑍})) ∧ 𝑦 ∈ (𝑋 ∖ {𝑍})) → ((𝑦𝐻𝑧) = (GId‘𝐻) → 𝑖 = 𝑋))
3837rexlimdva 3150 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ (𝑖 ∖ {𝑍})) → (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑧) = (GId‘𝐻) → 𝑖 = 𝑋))
3938imp 406 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ (𝑖 ∖ {𝑍})) ∧ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑧) = (GId‘𝐻)) → 𝑖 = 𝑋)
4024, 39syldan 590 . . . . . . . . . . . . . 14 ((((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ (𝑖 ∖ {𝑍})) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → 𝑖 = 𝑋)
4140an32s 651 . . . . . . . . . . . . 13 ((((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) ∧ 𝑧 ∈ (𝑖 ∖ {𝑍})) → 𝑖 = 𝑋)
4241ex 412 . . . . . . . . . . . 12 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → (𝑧 ∈ (𝑖 ∖ {𝑍}) → 𝑖 = 𝑋))
4342exlimdv 1929 . . . . . . . . . . 11 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → (∃𝑧 𝑧 ∈ (𝑖 ∖ {𝑍}) → 𝑖 = 𝑋))
4415, 43syl5 34 . . . . . . . . . 10 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → ((𝑍𝑖𝑖 ≠ {𝑍}) → 𝑖 = 𝑋))
458, 44mpand 694 . . . . . . . . 9 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → (𝑖 ≠ {𝑍} → 𝑖 = 𝑋))
4645an32s 651 . . . . . . . 8 (((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) ∧ 𝑖 ∈ (Idl‘𝑅)) → (𝑖 ≠ {𝑍} → 𝑖 = 𝑋))
47 neor 3029 . . . . . . . 8 ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ↔ (𝑖 ≠ {𝑍} → 𝑖 = 𝑋))
4846, 47sylibr 233 . . . . . . 7 (((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) ∧ 𝑖 ∈ (Idl‘𝑅)) → (𝑖 = {𝑍} ∨ 𝑖 = 𝑋))
4948ex 412 . . . . . 6 ((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → (𝑖 ∈ (Idl‘𝑅) → (𝑖 = {𝑍} ∨ 𝑖 = 𝑋)))
501, 30idl 37433 . . . . . . . . 9 (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅))
51 eleq1 2816 . . . . . . . . 9 (𝑖 = {𝑍} → (𝑖 ∈ (Idl‘𝑅) ↔ {𝑍} ∈ (Idl‘𝑅)))
5250, 51syl5ibrcom 246 . . . . . . . 8 (𝑅 ∈ RingOps → (𝑖 = {𝑍} → 𝑖 ∈ (Idl‘𝑅)))
531, 4rngoidl 37432 . . . . . . . . 9 (𝑅 ∈ RingOps → 𝑋 ∈ (Idl‘𝑅))
54 eleq1 2816 . . . . . . . . 9 (𝑖 = 𝑋 → (𝑖 ∈ (Idl‘𝑅) ↔ 𝑋 ∈ (Idl‘𝑅)))
5553, 54syl5ibrcom 246 . . . . . . . 8 (𝑅 ∈ RingOps → (𝑖 = 𝑋𝑖 ∈ (Idl‘𝑅)))
5652, 55jaod 858 . . . . . . 7 (𝑅 ∈ RingOps → ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) → 𝑖 ∈ (Idl‘𝑅)))
5756adantr 480 . . . . . 6 ((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) → 𝑖 ∈ (Idl‘𝑅)))
5849, 57impbid 211 . . . . 5 ((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → (𝑖 ∈ (Idl‘𝑅) ↔ (𝑖 = {𝑍} ∨ 𝑖 = 𝑋)))
59 vex 3473 . . . . . 6 𝑖 ∈ V
6059elpr 4647 . . . . 5 (𝑖 ∈ {{𝑍}, 𝑋} ↔ (𝑖 = {𝑍} ∨ 𝑖 = 𝑋))
6158, 60bitr4di 289 . . . 4 ((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → (𝑖 ∈ (Idl‘𝑅) ↔ 𝑖 ∈ {{𝑍}, 𝑋}))
6261eqrdv 2725 . . 3 ((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → (Idl‘𝑅) = {{𝑍}, 𝑋})
6362adantrl 715 . 2 ((𝑅 ∈ RingOps ∧ ((GId‘𝐻) ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻))) → (Idl‘𝑅) = {{𝑍}, 𝑋})
646, 63sylbi 216 1 (𝑅 ∈ DivRingOps → (Idl‘𝑅) = {{𝑍}, 𝑋})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 846   = wceq 1534  wex 1774  wcel 2099  wne 2935  wral 3056  wrex 3065  cdif 3941  wss 3944  c0 4318  {csn 4624  {cpr 4626  ran crn 5673  cfv 6542  (class class class)co 7414  1st c1st 7985  2nd c2nd 7986  GIdcgi 30287  RingOpscrngo 37302  DivRingOpscdrng 37356  Idlcidl 37415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-1st 7987  df-2nd 7988  df-1o 8480  df-en 8956  df-grpo 30290  df-gid 30291  df-ginv 30292  df-ablo 30342  df-ass 37251  df-exid 37253  df-mgmOLD 37257  df-sgrOLD 37269  df-mndo 37275  df-rngo 37303  df-drngo 37357  df-idl 37418
This theorem is referenced by:  divrngpr  37461  isfldidl  37476
  Copyright terms: Public domain W3C validator