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Theorem divrngidl 35466
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 2798 . . 3 (GId‘𝐻) = (GId‘𝐻)
61, 2, 3, 4, 5isdrngo2 35396 . 2 (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ ((GId‘𝐻) ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻))))
71, 3idl0cl 35456 . . . . . . . . . . 11 ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → 𝑍𝑖)
87adantr 484 . . . . . . . . . 10 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → 𝑍𝑖)
93fvexi 6659 . . . . . . . . . . . . 13 𝑍 ∈ V
109snss 4679 . . . . . . . . . . . 12 (𝑍𝑖 ↔ {𝑍} ⊆ 𝑖)
11 necom 3040 . . . . . . . . . . . 12 (𝑖 ≠ {𝑍} ↔ {𝑍} ≠ 𝑖)
12 pssdifn0 4279 . . . . . . . . . . . . 13 (({𝑍} ⊆ 𝑖 ∧ {𝑍} ≠ 𝑖) → (𝑖 ∖ {𝑍}) ≠ ∅)
13 n0 4260 . . . . . . . . . . . . 13 ((𝑖 ∖ {𝑍}) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (𝑖 ∖ {𝑍}))
1412, 13sylib 221 . . . . . . . . . . . 12 (({𝑍} ⊆ 𝑖 ∧ {𝑍} ≠ 𝑖) → ∃𝑧 𝑧 ∈ (𝑖 ∖ {𝑍}))
1510, 11, 14syl2anb 600 . . . . . . . . . . 11 ((𝑍𝑖𝑖 ≠ {𝑍}) → ∃𝑧 𝑧 ∈ (𝑖 ∖ {𝑍}))
161, 4idlss 35454 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → 𝑖𝑋)
17 ssdif 4067 . . . . . . . . . . . . . . . . . 18 (𝑖𝑋 → (𝑖 ∖ {𝑍}) ⊆ (𝑋 ∖ {𝑍}))
1817sselda 3915 . . . . . . . . . . . . . . . . 17 ((𝑖𝑋𝑧 ∈ (𝑖 ∖ {𝑍})) → 𝑧 ∈ (𝑋 ∖ {𝑍}))
1916, 18sylan 583 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ (𝑖 ∖ {𝑍})) → 𝑧 ∈ (𝑋 ∖ {𝑍}))
20 oveq2 7143 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑧 → (𝑦𝐻𝑥) = (𝑦𝐻𝑧))
2120eqeq1d 2800 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑧 → ((𝑦𝐻𝑥) = (GId‘𝐻) ↔ (𝑦𝐻𝑧) = (GId‘𝐻)))
2221rexbidv 3256 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻) ↔ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑧) = (GId‘𝐻)))
2322rspcva 3569 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ (𝑋 ∖ {𝑍}) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑧) = (GId‘𝐻))
2419, 23sylan 583 . . . . . . . . . . . . . . 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 4054 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ (𝑖 ∖ {𝑍}) → 𝑧𝑖)
26 eldifi 4054 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (𝑋 ∖ {𝑍}) → 𝑦𝑋)
2725, 26anim12i 615 . . . . . . . . . . . . . . . . . . 19 ((𝑧 ∈ (𝑖 ∖ {𝑍}) ∧ 𝑦 ∈ (𝑋 ∖ {𝑍})) → (𝑧𝑖𝑦𝑋))
281, 2, 4idllmulcl 35458 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑧𝑖𝑦𝑋)) → (𝑦𝐻𝑧) ∈ 𝑖)
291, 2, 4, 51idl 35464 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → ((GId‘𝐻) ∈ 𝑖𝑖 = 𝑋))
3029biimpd 232 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → ((GId‘𝐻) ∈ 𝑖𝑖 = 𝑋))
3130adantr 484 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑧𝑖𝑦𝑋)) → ((GId‘𝐻) ∈ 𝑖𝑖 = 𝑋))
32 eleq1 2877 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦𝐻𝑧) = (GId‘𝐻) → ((𝑦𝐻𝑧) ∈ 𝑖 ↔ (GId‘𝐻) ∈ 𝑖))
3332imbi1d 345 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦𝐻𝑧) = (GId‘𝐻) → (((𝑦𝐻𝑧) ∈ 𝑖𝑖 = 𝑋) ↔ ((GId‘𝐻) ∈ 𝑖𝑖 = 𝑋)))
3431, 33syl5ibrcom 250 . . . . . . . . . . . . . . . . . . . 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 595 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑧 ∈ (𝑖 ∖ {𝑍}) ∧ 𝑦 ∈ (𝑋 ∖ {𝑍}))) → ((𝑦𝐻𝑧) = (GId‘𝐻) → 𝑖 = 𝑋))
3736anassrs 471 . . . . . . . . . . . . . . . . 17 ((((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ (𝑖 ∖ {𝑍})) ∧ 𝑦 ∈ (𝑋 ∖ {𝑍})) → ((𝑦𝐻𝑧) = (GId‘𝐻) → 𝑖 = 𝑋))
3837rexlimdva 3243 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ (𝑖 ∖ {𝑍})) → (∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑧) = (GId‘𝐻) → 𝑖 = 𝑋))
3938imp 410 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ (𝑖 ∖ {𝑍})) ∧ ∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑧) = (GId‘𝐻)) → 𝑖 = 𝑋)
4024, 39syldan 594 . . . . . . . . . . . . . 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 416 . . . . . . . . . . . 12 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → (𝑧 ∈ (𝑖 ∖ {𝑍}) → 𝑖 = 𝑋))
4342exlimdv 1934 . . . . . . . . . . 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 3078 . . . . . . . 8 ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ↔ (𝑖 ≠ {𝑍} → 𝑖 = 𝑋))
4846, 47sylibr 237 . . . . . . 7 (((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) ∧ 𝑖 ∈ (Idl‘𝑅)) → (𝑖 = {𝑍} ∨ 𝑖 = 𝑋))
4948ex 416 . . . . . 6 ((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → (𝑖 ∈ (Idl‘𝑅) → (𝑖 = {𝑍} ∨ 𝑖 = 𝑋)))
501, 30idl 35463 . . . . . . . . 9 (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅))
51 eleq1 2877 . . . . . . . . 9 (𝑖 = {𝑍} → (𝑖 ∈ (Idl‘𝑅) ↔ {𝑍} ∈ (Idl‘𝑅)))
5250, 51syl5ibrcom 250 . . . . . . . 8 (𝑅 ∈ RingOps → (𝑖 = {𝑍} → 𝑖 ∈ (Idl‘𝑅)))
531, 4rngoidl 35462 . . . . . . . . 9 (𝑅 ∈ RingOps → 𝑋 ∈ (Idl‘𝑅))
54 eleq1 2877 . . . . . . . . 9 (𝑖 = 𝑋 → (𝑖 ∈ (Idl‘𝑅) ↔ 𝑋 ∈ (Idl‘𝑅)))
5553, 54syl5ibrcom 250 . . . . . . . 8 (𝑅 ∈ RingOps → (𝑖 = 𝑋𝑖 ∈ (Idl‘𝑅)))
5652, 55jaod 856 . . . . . . 7 (𝑅 ∈ RingOps → ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) → 𝑖 ∈ (Idl‘𝑅)))
5756adantr 484 . . . . . 6 ((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) → 𝑖 ∈ (Idl‘𝑅)))
5849, 57impbid 215 . . . . 5 ((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → (𝑖 ∈ (Idl‘𝑅) ↔ (𝑖 = {𝑍} ∨ 𝑖 = 𝑋)))
59 vex 3444 . . . . . 6 𝑖 ∈ V
6059elpr 4548 . . . . 5 (𝑖 ∈ {{𝑍}, 𝑋} ↔ (𝑖 = {𝑍} ∨ 𝑖 = 𝑋))
6158, 60syl6bbr 292 . . . 4 ((𝑅 ∈ RingOps ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = (GId‘𝐻)) → (𝑖 ∈ (Idl‘𝑅) ↔ 𝑖 ∈ {{𝑍}, 𝑋}))
6261eqrdv 2796 . . 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 220 1 (𝑅 ∈ DivRingOps → (Idl‘𝑅) = {{𝑍}, 𝑋})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wo 844   = wceq 1538  wex 1781  wcel 2111  wne 2987  wral 3106  wrex 3107  cdif 3878  wss 3881  c0 4243  {csn 4525  {cpr 4527  ran crn 5520  cfv 6324  (class class class)co 7135  1st c1st 7669  2nd c2nd 7670  GIdcgi 28273  RingOpscrngo 35332  DivRingOpscdrng 35386  Idlcidl 35445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-om 7561  df-1st 7671  df-2nd 7672  df-1o 8085  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-grpo 28276  df-gid 28277  df-ginv 28278  df-ablo 28328  df-ass 35281  df-exid 35283  df-mgmOLD 35287  df-sgrOLD 35299  df-mndo 35305  df-rngo 35333  df-drngo 35387  df-idl 35448
This theorem is referenced by:  divrngpr  35491  isfldidl  35506
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