Theorem drnglring 34091

Description: A division ring is a local ring. (Contributed by Thierry Arnoux, 2-Jun-2026.)

Hypothesis

Ref	Expression
drnglring.1	⊢ (𝜑 → 𝐹 ∈ DivRing)

Assertion

Ref	Expression
drnglring	⊢ (𝜑 → 𝐹 ∈ LRing)

Proof of Theorem drnglring

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		drnglring.1	. . 3 ⊢ (𝜑 → 𝐹 ∈ DivRing)
2		drngnzr 21029	. . 3 ⊢ (𝐹 ∈ DivRing → 𝐹 ∈ NzRing)
3	1, 2	syl 18	. 2 ⊢ (𝜑 → 𝐹 ∈ NzRing)
4	1	ad4antr 745	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐹)) ∧ 𝑦 ∈ (Base‘𝐹)) ∧ (𝑥(+g‘𝐹)𝑦) = (1r‘𝐹)) ∧ ¬ 𝑥 = (0g‘𝐹)) → 𝐹 ∈ DivRing)
5		simp-4r 796	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐹)) ∧ 𝑦 ∈ (Base‘𝐹)) ∧ (𝑥(+g‘𝐹)𝑦) = (1r‘𝐹)) ∧ ¬ 𝑥 = (0g‘𝐹)) → 𝑥 ∈ (Base‘𝐹))
6		neqne 2967	. . . . . . . 8 ⊢ (¬ 𝑥 = (0g‘𝐹) → 𝑥 ≠ (0g‘𝐹))
7	6	adantl 487	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐹)) ∧ 𝑦 ∈ (Base‘𝐹)) ∧ (𝑥(+g‘𝐹)𝑦) = (1r‘𝐹)) ∧ ¬ 𝑥 = (0g‘𝐹)) → 𝑥 ≠ (0g‘𝐹))
8		eqid 2763	. . . . . . . . 9 ⊢ (Base‘𝐹) = (Base‘𝐹)
9		eqid 2763	. . . . . . . . 9 ⊢ (Unit‘𝐹) = (Unit‘𝐹)
10		eqid 2763	. . . . . . . . 9 ⊢ (0g‘𝐹) = (0g‘𝐹)
11	8, 9, 10	drngunit 21012	. . . . . . . 8 ⊢ (𝐹 ∈ DivRing → (𝑥 ∈ (Unit‘𝐹) ↔ (𝑥 ∈ (Base‘𝐹) ∧ 𝑥 ≠ (0g‘𝐹))))
12	11	biimpar 483	. . . . . . 7 ⊢ ((𝐹 ∈ DivRing ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑥 ≠ (0g‘𝐹))) → 𝑥 ∈ (Unit‘𝐹))
13	4, 5, 7, 12	syl12anc 850	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐹)) ∧ 𝑦 ∈ (Base‘𝐹)) ∧ (𝑥(+g‘𝐹)𝑦) = (1r‘𝐹)) ∧ ¬ 𝑥 = (0g‘𝐹)) → 𝑥 ∈ (Unit‘𝐹))
14	1	ad4antr 745	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐹)) ∧ 𝑦 ∈ (Base‘𝐹)) ∧ (𝑥(+g‘𝐹)𝑦) = (1r‘𝐹)) ∧ ¬ 𝑦 = (0g‘𝐹)) → 𝐹 ∈ DivRing)
15		simpllr 788	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐹)) ∧ 𝑦 ∈ (Base‘𝐹)) ∧ (𝑥(+g‘𝐹)𝑦) = (1r‘𝐹)) ∧ ¬ 𝑦 = (0g‘𝐹)) → 𝑦 ∈ (Base‘𝐹))
16		neqne 2967	. . . . . . . 8 ⊢ (¬ 𝑦 = (0g‘𝐹) → 𝑦 ≠ (0g‘𝐹))
17	16	adantl 487	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐹)) ∧ 𝑦 ∈ (Base‘𝐹)) ∧ (𝑥(+g‘𝐹)𝑦) = (1r‘𝐹)) ∧ ¬ 𝑦 = (0g‘𝐹)) → 𝑦 ≠ (0g‘𝐹))
18	8, 9, 10	drngunit 21012	. . . . . . . 8 ⊢ (𝐹 ∈ DivRing → (𝑦 ∈ (Unit‘𝐹) ↔ (𝑦 ∈ (Base‘𝐹) ∧ 𝑦 ≠ (0g‘𝐹))))
19	18	biimpar 483	. . . . . . 7 ⊢ ((𝐹 ∈ DivRing ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑦 ≠ (0g‘𝐹))) → 𝑦 ∈ (Unit‘𝐹))
20	14, 15, 17, 19	syl12anc 850	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐹)) ∧ 𝑦 ∈ (Base‘𝐹)) ∧ (𝑥(+g‘𝐹)𝑦) = (1r‘𝐹)) ∧ ¬ 𝑦 = (0g‘𝐹)) → 𝑦 ∈ (Unit‘𝐹))
21		simplll 787	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐹)) ∧ 𝑦 ∈ (Base‘𝐹)) ∧ (𝑥(+g‘𝐹)𝑦) = (1r‘𝐹)) → 𝜑)
22		simpr 490	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐹)) ∧ 𝑦 ∈ (Base‘𝐹)) ∧ (𝑥(+g‘𝐹)𝑦) = (1r‘𝐹)) → (𝑥(+g‘𝐹)𝑦) = (1r‘𝐹))
23		eqid 2763	. . . . . . . . . . . . 13 ⊢ (1r‘𝐹) = (1r‘𝐹)
24	23, 10	nzrnz 20792	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ NzRing → (1r‘𝐹) ≠ (0g‘𝐹))
25	3, 24	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (1r‘𝐹) ≠ (0g‘𝐹))
26	25	ad3antrrr 743	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐹)) ∧ 𝑦 ∈ (Base‘𝐹)) ∧ (𝑥(+g‘𝐹)𝑦) = (1r‘𝐹)) → (1r‘𝐹) ≠ (0g‘𝐹))
27	22, 26	eqnetrd 3026	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐹)) ∧ 𝑦 ∈ (Base‘𝐹)) ∧ (𝑥(+g‘𝐹)𝑦) = (1r‘𝐹)) → (𝑥(+g‘𝐹)𝑦) ≠ (0g‘𝐹))
28	27	neneqd 2964	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐹)) ∧ 𝑦 ∈ (Base‘𝐹)) ∧ (𝑥(+g‘𝐹)𝑦) = (1r‘𝐹)) → ¬ (𝑥(+g‘𝐹)𝑦) = (0g‘𝐹))
29		oveq12 7449	. . . . . . . . . . 11 ⊢ ((𝑥 = (0g‘𝐹) ∧ 𝑦 = (0g‘𝐹)) → (𝑥(+g‘𝐹)𝑦) = ((0g‘𝐹)(+g‘𝐹)(0g‘𝐹)))
30	29	adantl 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 = (0g‘𝐹) ∧ 𝑦 = (0g‘𝐹))) → (𝑥(+g‘𝐹)𝑦) = ((0g‘𝐹)(+g‘𝐹)(0g‘𝐹)))
31		eqid 2763	. . . . . . . . . . 11 ⊢ (+g‘𝐹) = (+g‘𝐹)
32	1	drnggrpd 21016	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ∈ Grp)
33	32	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 = (0g‘𝐹) ∧ 𝑦 = (0g‘𝐹))) → 𝐹 ∈ Grp)
34	8, 10, 33	grpidcld 33667	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 = (0g‘𝐹) ∧ 𝑦 = (0g‘𝐹))) → (0g‘𝐹) ∈ (Base‘𝐹))
35	8, 31, 10, 33, 34	grplidd 19206	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 = (0g‘𝐹) ∧ 𝑦 = (0g‘𝐹))) → ((0g‘𝐹)(+g‘𝐹)(0g‘𝐹)) = (0g‘𝐹))
36	30, 35	eqtrd 2798	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 = (0g‘𝐹) ∧ 𝑦 = (0g‘𝐹))) → (𝑥(+g‘𝐹)𝑦) = (0g‘𝐹))
37	36	stoic1a 1805	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (𝑥(+g‘𝐹)𝑦) = (0g‘𝐹)) → ¬ (𝑥 = (0g‘𝐹) ∧ 𝑦 = (0g‘𝐹)))
38	21, 28, 37	syl2anc 596	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐹)) ∧ 𝑦 ∈ (Base‘𝐹)) ∧ (𝑥(+g‘𝐹)𝑦) = (1r‘𝐹)) → ¬ (𝑥 = (0g‘𝐹) ∧ 𝑦 = (0g‘𝐹)))
39		ianor 997	. . . . . . 7 ⊢ (¬ (𝑥 = (0g‘𝐹) ∧ 𝑦 = (0g‘𝐹)) ↔ (¬ 𝑥 = (0g‘𝐹) ∨ ¬ 𝑦 = (0g‘𝐹)))
40	38, 39	sylib 221	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐹)) ∧ 𝑦 ∈ (Base‘𝐹)) ∧ (𝑥(+g‘𝐹)𝑦) = (1r‘𝐹)) → (¬ 𝑥 = (0g‘𝐹) ∨ ¬ 𝑦 = (0g‘𝐹)))
41	13, 20, 40	orim12da 980	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐹)) ∧ 𝑦 ∈ (Base‘𝐹)) ∧ (𝑥(+g‘𝐹)𝑦) = (1r‘𝐹)) → (𝑥 ∈ (Unit‘𝐹) ∨ 𝑦 ∈ (Unit‘𝐹)))
42	41	ex 418	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐹)) ∧ 𝑦 ∈ (Base‘𝐹)) → ((𝑥(+g‘𝐹)𝑦) = (1r‘𝐹) → (𝑥 ∈ (Unit‘𝐹) ∨ 𝑦 ∈ (Unit‘𝐹))))
43	42	anasss 472	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → ((𝑥(+g‘𝐹)𝑦) = (1r‘𝐹) → (𝑥 ∈ (Unit‘𝐹) ∨ 𝑦 ∈ (Unit‘𝐹))))
44	43	ralrimivva 3209	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐹)∀𝑦 ∈ (Base‘𝐹)((𝑥(+g‘𝐹)𝑦) = (1r‘𝐹) → (𝑥 ∈ (Unit‘𝐹) ∨ 𝑦 ∈ (Unit‘𝐹))))
45	8, 31, 23, 9	islring 20819	. 2 ⊢ (𝐹 ∈ LRing ↔ (𝐹 ∈ NzRing ∧ ∀𝑥 ∈ (Base‘𝐹)∀𝑦 ∈ (Base‘𝐹)((𝑥(+g‘𝐹)𝑦) = (1r‘𝐹) → (𝑥 ∈ (Unit‘𝐹) ∨ 𝑦 ∈ (Unit‘𝐹)))))
46	3, 44, 45	sylanbrc 595	1 ⊢ (𝜑 → 𝐹 ∈ LRing)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080  ‘cfv 6555  (class class class)co 7440  Basecbs 17413  +_gcplusg 17454  0_gc0g 17636  Grpcgrp 19170  1_rcur 20434  Unitcui 20612  NzRingcnzr 20789  LRingclring 20817  DivRingcdr 21007

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-cnex 11280  ax-resscn 11281  ax-1cn 11282  ax-icn 11283  ax-addcl 11284  ax-addrcl 11285  ax-mulcl 11286  ax-mulrcl 11287  ax-mulcom 11288  ax-addass 11289  ax-mulass 11290  ax-distr 11291  ax-i2m1 11292  ax-1ne0 11293  ax-1rid 11294  ax-rnegex 11295  ax-rrecex 11296  ax-cnre 11297  ax-pre-lttri 11298  ax-pre-lttrn 11299  ax-pre-ltadd 11300  ax-pre-mulgt0 11301

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-2nd 8023  df-tpos 8268  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-er 8742  df-en 8999  df-dom 9000  df-sdom 9001  df-pnf 11369  df-mnf 11370  df-xr 11371  df-ltxr 11372  df-le 11373  df-sub 11567  df-neg 11568  df-nn 12360  df-2 12429  df-3 12430  df-sets 17368  df-slot 17386  df-ndx 17398  df-base 17414  df-plusg 17467  df-mulr 17468  df-0g 17638  df-mgm 18842  df-sgrp 18934  df-mnd 18950  df-grp 19173  df-minusg 19174  df-cmn 20023  df-abl 20024  df-mgp 20388  df-rng 20402  df-ur 20435  df-ring 20488  df-oppr 20594  df-dvdsr 20614  df-unit 20615  df-nzr 20790  df-lring 20818  df-drng 21009

This theorem is used by: (None)