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Theorem divrngidl 33957
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 2651 . . 3 (GId‘𝐻) = (GId‘𝐻)
61, 2, 3, 4, 5isdrngo2 33887 . 2 (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ ((GId‘𝐻) ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻))))
71, 3idl0cl 33947 . . . . . . . . . . 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)
9 fvex 6239 . . . . . . . . . . . . . 14 (GId‘𝐺) ∈ V
103, 9eqeltri 2726 . . . . . . . . . . . . 13 𝑍 ∈ V
1110snss 4348 . . . . . . . . . . . 12 (𝑍𝑖 ↔ {𝑍} ⊆ 𝑖)
12 necom 2876 . . . . . . . . . . . 12 (𝑖 ≠ {𝑍} ↔ {𝑍} ≠ 𝑖)
13 pssdifn0 3977 . . . . . . . . . . . . 13 (({𝑍} ⊆ 𝑖 ∧ {𝑍} ≠ 𝑖) → (𝑖 ∖ {𝑍}) ≠ ∅)
14 n0 3964 . . . . . . . . . . . . 13 ((𝑖 ∖ {𝑍}) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (𝑖 ∖ {𝑍}))
1513, 14sylib 208 . . . . . . . . . . . 12 (({𝑍} ⊆ 𝑖 ∧ {𝑍} ≠ 𝑖) → ∃𝑧 𝑧 ∈ (𝑖 ∖ {𝑍}))
1611, 12, 15syl2anb 495 . . . . . . . . . . 11 ((𝑍𝑖𝑖 ≠ {𝑍}) → ∃𝑧 𝑧 ∈ (𝑖 ∖ {𝑍}))
171, 4idlss 33945 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → 𝑖𝑋)
18 ssdif 3778 . . . . . . . . . . . . . . . . . 18 (𝑖𝑋 → (𝑖 ∖ {𝑍}) ⊆ (𝑋 ∖ {𝑍}))
1918sselda 3636 . . . . . . . . . . . . . . . . 17 ((𝑖𝑋𝑧 ∈ (𝑖 ∖ {𝑍})) → 𝑧 ∈ (𝑋 ∖ {𝑍}))
2017, 19sylan 487 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ (𝑖 ∖ {𝑍})) → 𝑧 ∈ (𝑋 ∖ {𝑍}))
21 oveq2 6698 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑧 → (𝑦𝐻𝑥) = (𝑦𝐻𝑧))
2221eqeq1d 2653 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑧 → ((𝑦𝐻𝑥) = (GId‘𝐻) ↔ (𝑦𝐻𝑧) = (GId‘𝐻)))
2322rexbidv 3081 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻) ↔ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑧) = (GId‘𝐻)))
2423rspcva 3338 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ (𝑋 ∖ {𝑍}) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑧) = (GId‘𝐻))
2520, 24sylan 487 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ (𝑖 ∖ {𝑍})) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑧) = (GId‘𝐻))
26 eldifi 3765 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ (𝑖 ∖ {𝑍}) → 𝑧𝑖)
27 eldifi 3765 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (𝑋 ∖ {𝑍}) → 𝑦𝑋)
2826, 27anim12i 589 . . . . . . . . . . . . . . . . . . 19 ((𝑧 ∈ (𝑖 ∖ {𝑍}) ∧ 𝑦 ∈ (𝑋 ∖ {𝑍})) → (𝑧𝑖𝑦𝑋))
291, 2, 4idllmulcl 33949 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑧𝑖𝑦𝑋)) → (𝑦𝐻𝑧) ∈ 𝑖)
301, 2, 4, 51idl 33955 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → ((GId‘𝐻) ∈ 𝑖𝑖 = 𝑋))
3130biimpd 219 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → ((GId‘𝐻) ∈ 𝑖𝑖 = 𝑋))
3231adantr 480 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑧𝑖𝑦𝑋)) → ((GId‘𝐻) ∈ 𝑖𝑖 = 𝑋))
33 eleq1 2718 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦𝐻𝑧) = (GId‘𝐻) → ((𝑦𝐻𝑧) ∈ 𝑖 ↔ (GId‘𝐻) ∈ 𝑖))
3433imbi1d 330 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦𝐻𝑧) = (GId‘𝐻) → (((𝑦𝐻𝑧) ∈ 𝑖𝑖 = 𝑋) ↔ ((GId‘𝐻) ∈ 𝑖𝑖 = 𝑋)))
3532, 34syl5ibrcom 237 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑧𝑖𝑦𝑋)) → ((𝑦𝐻𝑧) = (GId‘𝐻) → ((𝑦𝐻𝑧) ∈ 𝑖𝑖 = 𝑋)))
3629, 35mpid 44 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑧𝑖𝑦𝑋)) → ((𝑦𝐻𝑧) = (GId‘𝐻) → 𝑖 = 𝑋))
3728, 36sylan2 490 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑧 ∈ (𝑖 ∖ {𝑍}) ∧ 𝑦 ∈ (𝑋 ∖ {𝑍}))) → ((𝑦𝐻𝑧) = (GId‘𝐻) → 𝑖 = 𝑋))
3837anassrs 681 . . . . . . . . . . . . . . . . 17 ((((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ (𝑖 ∖ {𝑍})) ∧ 𝑦 ∈ (𝑋 ∖ {𝑍})) → ((𝑦𝐻𝑧) = (GId‘𝐻) → 𝑖 = 𝑋))
3938rexlimdva 3060 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ (𝑖 ∖ {𝑍})) → (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑧) = (GId‘𝐻) → 𝑖 = 𝑋))
4039imp 444 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ (𝑖 ∖ {𝑍})) ∧ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑧) = (GId‘𝐻)) → 𝑖 = 𝑋)
4125, 40syldan 486 . . . . . . . . . . . . . 14 ((((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ (𝑖 ∖ {𝑍})) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → 𝑖 = 𝑋)
4241an32s 863 . . . . . . . . . . . . 13 ((((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) ∧ 𝑧 ∈ (𝑖 ∖ {𝑍})) → 𝑖 = 𝑋)
4342ex 449 . . . . . . . . . . . 12 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → (𝑧 ∈ (𝑖 ∖ {𝑍}) → 𝑖 = 𝑋))
4443exlimdv 1901 . . . . . . . . . . 11 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → (∃𝑧 𝑧 ∈ (𝑖 ∖ {𝑍}) → 𝑖 = 𝑋))
4516, 44syl5 34 . . . . . . . . . 10 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → ((𝑍𝑖𝑖 ≠ {𝑍}) → 𝑖 = 𝑋))
468, 45mpand 711 . . . . . . . . 9 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → (𝑖 ≠ {𝑍} → 𝑖 = 𝑋))
4746an32s 863 . . . . . . . 8 (((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) ∧ 𝑖 ∈ (Idl‘𝑅)) → (𝑖 ≠ {𝑍} → 𝑖 = 𝑋))
48 neor 2914 . . . . . . . 8 ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ↔ (𝑖 ≠ {𝑍} → 𝑖 = 𝑋))
4947, 48sylibr 224 . . . . . . 7 (((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) ∧ 𝑖 ∈ (Idl‘𝑅)) → (𝑖 = {𝑍} ∨ 𝑖 = 𝑋))
5049ex 449 . . . . . 6 ((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → (𝑖 ∈ (Idl‘𝑅) → (𝑖 = {𝑍} ∨ 𝑖 = 𝑋)))
511, 30idl 33954 . . . . . . . . 9 (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅))
52 eleq1 2718 . . . . . . . . 9 (𝑖 = {𝑍} → (𝑖 ∈ (Idl‘𝑅) ↔ {𝑍} ∈ (Idl‘𝑅)))
5351, 52syl5ibrcom 237 . . . . . . . 8 (𝑅 ∈ RingOps → (𝑖 = {𝑍} → 𝑖 ∈ (Idl‘𝑅)))
541, 4rngoidl 33953 . . . . . . . . 9 (𝑅 ∈ RingOps → 𝑋 ∈ (Idl‘𝑅))
55 eleq1 2718 . . . . . . . . 9 (𝑖 = 𝑋 → (𝑖 ∈ (Idl‘𝑅) ↔ 𝑋 ∈ (Idl‘𝑅)))
5654, 55syl5ibrcom 237 . . . . . . . 8 (𝑅 ∈ RingOps → (𝑖 = 𝑋𝑖 ∈ (Idl‘𝑅)))
5753, 56jaod 394 . . . . . . 7 (𝑅 ∈ RingOps → ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) → 𝑖 ∈ (Idl‘𝑅)))
5857adantr 480 . . . . . 6 ((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) → 𝑖 ∈ (Idl‘𝑅)))
5950, 58impbid 202 . . . . 5 ((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → (𝑖 ∈ (Idl‘𝑅) ↔ (𝑖 = {𝑍} ∨ 𝑖 = 𝑋)))
60 vex 3234 . . . . . 6 𝑖 ∈ V
6160elpr 4231 . . . . 5 (𝑖 ∈ {{𝑍}, 𝑋} ↔ (𝑖 = {𝑍} ∨ 𝑖 = 𝑋))
6259, 61syl6bbr 278 . . . 4 ((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → (𝑖 ∈ (Idl‘𝑅) ↔ 𝑖 ∈ {{𝑍}, 𝑋}))
6362eqrdv 2649 . . 3 ((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → (Idl‘𝑅) = {{𝑍}, 𝑋})
6463adantrl 752 . 2 ((𝑅 ∈ RingOps ∧ ((GId‘𝐻) ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻))) → (Idl‘𝑅) = {{𝑍}, 𝑋})
656, 64sylbi 207 1 (𝑅 ∈ DivRingOps → (Idl‘𝑅) = {{𝑍}, 𝑋})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382  wa 383   = wceq 1523  wex 1744  wcel 2030  wne 2823  wral 2941  wrex 2942  Vcvv 3231  cdif 3604  wss 3607  c0 3948  {csn 4210  {cpr 4212  ran crn 5144  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  GIdcgi 27472  RingOpscrngo 33823  DivRingOpscdrng 33877  Idlcidl 33936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-om 7108  df-1st 7210  df-2nd 7211  df-1o 7605  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-grpo 27475  df-gid 27476  df-ginv 27477  df-ablo 27527  df-ass 33772  df-exid 33774  df-mgmOLD 33778  df-sgrOLD 33790  df-mndo 33796  df-rngo 33824  df-drngo 33878  df-idl 33939
This theorem is referenced by:  divrngpr  33982  isfldidl  33997
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