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Theorem divrngidl 36490
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 2737 . . 3 (GId‘𝐻) = (GId‘𝐻)
61, 2, 3, 4, 5isdrngo2 36420 . 2 (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ ((GId‘𝐻) ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻))))
71, 3idl0cl 36480 . . . . . . . . . . 11 ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → 𝑍𝑖)
87adantr 482 . . . . . . . . . 10 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → 𝑍𝑖)
93fvexi 6857 . . . . . . . . . . . . 13 𝑍 ∈ V
109snss 4747 . . . . . . . . . . . 12 (𝑍𝑖 ↔ {𝑍} ⊆ 𝑖)
11 necom 2998 . . . . . . . . . . . 12 (𝑖 ≠ {𝑍} ↔ {𝑍} ≠ 𝑖)
12 pssdifn0 4326 . . . . . . . . . . . . 13 (({𝑍} ⊆ 𝑖 ∧ {𝑍} ≠ 𝑖) → (𝑖 ∖ {𝑍}) ≠ ∅)
13 n0 4307 . . . . . . . . . . . . 13 ((𝑖 ∖ {𝑍}) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (𝑖 ∖ {𝑍}))
1412, 13sylib 217 . . . . . . . . . . . 12 (({𝑍} ⊆ 𝑖 ∧ {𝑍} ≠ 𝑖) → ∃𝑧 𝑧 ∈ (𝑖 ∖ {𝑍}))
1510, 11, 14syl2anb 599 . . . . . . . . . . 11 ((𝑍𝑖𝑖 ≠ {𝑍}) → ∃𝑧 𝑧 ∈ (𝑖 ∖ {𝑍}))
161, 4idlss 36478 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → 𝑖𝑋)
17 ssdif 4100 . . . . . . . . . . . . . . . . . 18 (𝑖𝑋 → (𝑖 ∖ {𝑍}) ⊆ (𝑋 ∖ {𝑍}))
1817sselda 3945 . . . . . . . . . . . . . . . . 17 ((𝑖𝑋𝑧 ∈ (𝑖 ∖ {𝑍})) → 𝑧 ∈ (𝑋 ∖ {𝑍}))
1916, 18sylan 581 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ (𝑖 ∖ {𝑍})) → 𝑧 ∈ (𝑋 ∖ {𝑍}))
20 oveq2 7366 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑧 → (𝑦𝐻𝑥) = (𝑦𝐻𝑧))
2120eqeq1d 2739 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑧 → ((𝑦𝐻𝑥) = (GId‘𝐻) ↔ (𝑦𝐻𝑧) = (GId‘𝐻)))
2221rexbidv 3176 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻) ↔ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑧) = (GId‘𝐻)))
2322rspcva 3580 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ (𝑋 ∖ {𝑍}) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑧) = (GId‘𝐻))
2419, 23sylan 581 . . . . . . . . . . . . . . 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 4087 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ (𝑖 ∖ {𝑍}) → 𝑧𝑖)
26 eldifi 4087 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (𝑋 ∖ {𝑍}) → 𝑦𝑋)
2725, 26anim12i 614 . . . . . . . . . . . . . . . . . . 19 ((𝑧 ∈ (𝑖 ∖ {𝑍}) ∧ 𝑦 ∈ (𝑋 ∖ {𝑍})) → (𝑧𝑖𝑦𝑋))
281, 2, 4idllmulcl 36482 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑧𝑖𝑦𝑋)) → (𝑦𝐻𝑧) ∈ 𝑖)
291, 2, 4, 51idl 36488 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → ((GId‘𝐻) ∈ 𝑖𝑖 = 𝑋))
3029biimpd 228 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → ((GId‘𝐻) ∈ 𝑖𝑖 = 𝑋))
3130adantr 482 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑧𝑖𝑦𝑋)) → ((GId‘𝐻) ∈ 𝑖𝑖 = 𝑋))
32 eleq1 2826 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦𝐻𝑧) = (GId‘𝐻) → ((𝑦𝐻𝑧) ∈ 𝑖 ↔ (GId‘𝐻) ∈ 𝑖))
3332imbi1d 342 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦𝐻𝑧) = (GId‘𝐻) → (((𝑦𝐻𝑧) ∈ 𝑖𝑖 = 𝑋) ↔ ((GId‘𝐻) ∈ 𝑖𝑖 = 𝑋)))
3431, 33syl5ibrcom 247 . . . . . . . . . . . . . . . . . . . 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 594 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑧 ∈ (𝑖 ∖ {𝑍}) ∧ 𝑦 ∈ (𝑋 ∖ {𝑍}))) → ((𝑦𝐻𝑧) = (GId‘𝐻) → 𝑖 = 𝑋))
3736anassrs 469 . . . . . . . . . . . . . . . . 17 ((((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ (𝑖 ∖ {𝑍})) ∧ 𝑦 ∈ (𝑋 ∖ {𝑍})) → ((𝑦𝐻𝑧) = (GId‘𝐻) → 𝑖 = 𝑋))
3837rexlimdva 3153 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ (𝑖 ∖ {𝑍})) → (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑧) = (GId‘𝐻) → 𝑖 = 𝑋))
3938imp 408 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ (𝑖 ∖ {𝑍})) ∧ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑧) = (GId‘𝐻)) → 𝑖 = 𝑋)
4024, 39syldan 592 . . . . . . . . . . . . . 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 414 . . . . . . . . . . . 12 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → (𝑧 ∈ (𝑖 ∖ {𝑍}) → 𝑖 = 𝑋))
4342exlimdv 1937 . . . . . . . . . . 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 3037 . . . . . . . 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 414 . . . . . 6 ((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → (𝑖 ∈ (Idl‘𝑅) → (𝑖 = {𝑍} ∨ 𝑖 = 𝑋)))
501, 30idl 36487 . . . . . . . . 9 (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅))
51 eleq1 2826 . . . . . . . . 9 (𝑖 = {𝑍} → (𝑖 ∈ (Idl‘𝑅) ↔ {𝑍} ∈ (Idl‘𝑅)))
5250, 51syl5ibrcom 247 . . . . . . . 8 (𝑅 ∈ RingOps → (𝑖 = {𝑍} → 𝑖 ∈ (Idl‘𝑅)))
531, 4rngoidl 36486 . . . . . . . . 9 (𝑅 ∈ RingOps → 𝑋 ∈ (Idl‘𝑅))
54 eleq1 2826 . . . . . . . . 9 (𝑖 = 𝑋 → (𝑖 ∈ (Idl‘𝑅) ↔ 𝑋 ∈ (Idl‘𝑅)))
5553, 54syl5ibrcom 247 . . . . . . . 8 (𝑅 ∈ RingOps → (𝑖 = 𝑋𝑖 ∈ (Idl‘𝑅)))
5652, 55jaod 858 . . . . . . 7 (𝑅 ∈ RingOps → ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) → 𝑖 ∈ (Idl‘𝑅)))
5756adantr 482 . . . . . 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 3450 . . . . . 6 𝑖 ∈ V
6059elpr 4610 . . . . 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 2735 . . 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 397  wo 846   = wceq 1542  wex 1782  wcel 2107  wne 2944  wral 3065  wrex 3074  cdif 3908  wss 3911  c0 4283  {csn 4587  {cpr 4589  ran crn 5635  cfv 6497  (class class class)co 7358  1st c1st 7920  2nd c2nd 7921  GIdcgi 29435  RingOpscrngo 36356  DivRingOpscdrng 36410  Idlcidl 36469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-1st 7922  df-2nd 7923  df-1o 8413  df-en 8885  df-grpo 29438  df-gid 29439  df-ginv 29440  df-ablo 29490  df-ass 36305  df-exid 36307  df-mgmOLD 36311  df-sgrOLD 36323  df-mndo 36329  df-rngo 36357  df-drngo 36411  df-idl 36472
This theorem is referenced by:  divrngpr  36515  isfldidl  36530
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