Theorem drnglring 33584

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 20721	. . 3 ⊢ (𝐹 ∈ DivRing → 𝐹 ∈ NzRing)
3	1, 2	syl 17	. 2 ⊢ (𝜑 → 𝐹 ∈ NzRing)
4	1	ad4antr 738	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐹)) ∧ 𝑦 ∈ (Base‘𝐹)) ∧ (𝑥(+g‘𝐹)𝑦) = (1r‘𝐹)) ∧ ¬ 𝑥 = (0g‘𝐹)) → 𝐹 ∈ DivRing)
5		simp-4r 789	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐹)) ∧ 𝑦 ∈ (Base‘𝐹)) ∧ (𝑥(+g‘𝐹)𝑦) = (1r‘𝐹)) ∧ ¬ 𝑥 = (0g‘𝐹)) → 𝑥 ∈ (Base‘𝐹))
6		neqne 2942	. . . . . . . 8 ⊢ (¬ 𝑥 = (0g‘𝐹) → 𝑥 ≠ (0g‘𝐹))
7	6	adantl 482	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐹)) ∧ 𝑦 ∈ (Base‘𝐹)) ∧ (𝑥(+g‘𝐹)𝑦) = (1r‘𝐹)) ∧ ¬ 𝑥 = (0g‘𝐹)) → 𝑥 ≠ (0g‘𝐹))
8		eqid 2739	. . . . . . . . 9 ⊢ (Base‘𝐹) = (Base‘𝐹)
9		eqid 2739	. . . . . . . . 9 ⊢ (Unit‘𝐹) = (Unit‘𝐹)
10		eqid 2739	. . . . . . . . 9 ⊢ (0g‘𝐹) = (0g‘𝐹)
11	8, 9, 10	drngunit 20707	. . . . . . . 8 ⊢ (𝐹 ∈ DivRing → (𝑥 ∈ (Unit‘𝐹) ↔ (𝑥 ∈ (Base‘𝐹) ∧ 𝑥 ≠ (0g‘𝐹))))
12	11	biimpar 478	. . . . . . 7 ⊢ ((𝐹 ∈ DivRing ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑥 ≠ (0g‘𝐹))) → 𝑥 ∈ (Unit‘𝐹))
13	4, 5, 7, 12	syl12anc 842	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐹)) ∧ 𝑦 ∈ (Base‘𝐹)) ∧ (𝑥(+g‘𝐹)𝑦) = (1r‘𝐹)) ∧ ¬ 𝑥 = (0g‘𝐹)) → 𝑥 ∈ (Unit‘𝐹))
14	1	ad4antr 738	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐹)) ∧ 𝑦 ∈ (Base‘𝐹)) ∧ (𝑥(+g‘𝐹)𝑦) = (1r‘𝐹)) ∧ ¬ 𝑦 = (0g‘𝐹)) → 𝐹 ∈ DivRing)
15		simpllr 781	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐹)) ∧ 𝑦 ∈ (Base‘𝐹)) ∧ (𝑥(+g‘𝐹)𝑦) = (1r‘𝐹)) ∧ ¬ 𝑦 = (0g‘𝐹)) → 𝑦 ∈ (Base‘𝐹))
16		neqne 2942	. . . . . . . 8 ⊢ (¬ 𝑦 = (0g‘𝐹) → 𝑦 ≠ (0g‘𝐹))
17	16	adantl 482	. . . . . . 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 20707	. . . . . . . 8 ⊢ (𝐹 ∈ DivRing → (𝑦 ∈ (Unit‘𝐹) ↔ (𝑦 ∈ (Base‘𝐹) ∧ 𝑦 ≠ (0g‘𝐹))))
19	18	biimpar 478	. . . . . . 7 ⊢ ((𝐹 ∈ DivRing ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑦 ≠ (0g‘𝐹))) → 𝑦 ∈ (Unit‘𝐹))
20	14, 15, 17, 19	syl12anc 842	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐹)) ∧ 𝑦 ∈ (Base‘𝐹)) ∧ (𝑥(+g‘𝐹)𝑦) = (1r‘𝐹)) ∧ ¬ 𝑦 = (0g‘𝐹)) → 𝑦 ∈ (Unit‘𝐹))
21		simplll 780	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐹)) ∧ 𝑦 ∈ (Base‘𝐹)) ∧ (𝑥(+g‘𝐹)𝑦) = (1r‘𝐹)) → 𝜑)
22		simpr 485	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐹)) ∧ 𝑦 ∈ (Base‘𝐹)) ∧ (𝑥(+g‘𝐹)𝑦) = (1r‘𝐹)) → (𝑥(+g‘𝐹)𝑦) = (1r‘𝐹))
23		eqid 2739	. . . . . . . . . . . . 13 ⊢ (1r‘𝐹) = (1r‘𝐹)
24	23, 10	nzrnz 20488	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ NzRing → (1r‘𝐹) ≠ (0g‘𝐹))
25	3, 24	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (1r‘𝐹) ≠ (0g‘𝐹))
26	25	ad3antrrr 736	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐹)) ∧ 𝑦 ∈ (Base‘𝐹)) ∧ (𝑥(+g‘𝐹)𝑦) = (1r‘𝐹)) → (1r‘𝐹) ≠ (0g‘𝐹))
27	22, 26	eqnetrd 3001	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐹)) ∧ 𝑦 ∈ (Base‘𝐹)) ∧ (𝑥(+g‘𝐹)𝑦) = (1r‘𝐹)) → (𝑥(+g‘𝐹)𝑦) ≠ (0g‘𝐹))
28	27	neneqd 2939	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐹)) ∧ 𝑦 ∈ (Base‘𝐹)) ∧ (𝑥(+g‘𝐹)𝑦) = (1r‘𝐹)) → ¬ (𝑥(+g‘𝐹)𝑦) = (0g‘𝐹))
29		oveq12 7366	. . . . . . . . . . 11 ⊢ ((𝑥 = (0g‘𝐹) ∧ 𝑦 = (0g‘𝐹)) → (𝑥(+g‘𝐹)𝑦) = ((0g‘𝐹)(+g‘𝐹)(0g‘𝐹)))
30	29	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 = (0g‘𝐹) ∧ 𝑦 = (0g‘𝐹))) → (𝑥(+g‘𝐹)𝑦) = ((0g‘𝐹)(+g‘𝐹)(0g‘𝐹)))
31		eqid 2739	. . . . . . . . . . 11 ⊢ (+g‘𝐹) = (+g‘𝐹)
32	1	drnggrpd 20711	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ∈ Grp)
33	32	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 = (0g‘𝐹) ∧ 𝑦 = (0g‘𝐹))) → 𝐹 ∈ Grp)
34	8, 10, 33	grpidcld 33120	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 = (0g‘𝐹) ∧ 𝑦 = (0g‘𝐹))) → (0g‘𝐹) ∈ (Base‘𝐹))
35	8, 31, 10, 33, 34	grplidd 18937	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 = (0g‘𝐹) ∧ 𝑦 = (0g‘𝐹))) → ((0g‘𝐹)(+g‘𝐹)(0g‘𝐹)) = (0g‘𝐹))
36	30, 35	eqtrd 2774	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 = (0g‘𝐹) ∧ 𝑦 = (0g‘𝐹))) → (𝑥(+g‘𝐹)𝑦) = (0g‘𝐹))
37	36	stoic1a 1779	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (𝑥(+g‘𝐹)𝑦) = (0g‘𝐹)) → ¬ (𝑥 = (0g‘𝐹) ∧ 𝑦 = (0g‘𝐹)))
38	21, 28, 37	syl2anc 590	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐹)) ∧ 𝑦 ∈ (Base‘𝐹)) ∧ (𝑥(+g‘𝐹)𝑦) = (1r‘𝐹)) → ¬ (𝑥 = (0g‘𝐹) ∧ 𝑦 = (0g‘𝐹)))
39		ianor 989	. . . . . . 7 ⊢ (¬ (𝑥 = (0g‘𝐹) ∧ 𝑦 = (0g‘𝐹)) ↔ (¬ 𝑥 = (0g‘𝐹) ∨ ¬ 𝑦 = (0g‘𝐹)))
40	38, 39	sylib 219	. . . . . 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 32546	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐹)) ∧ 𝑦 ∈ (Base‘𝐹)) ∧ (𝑥(+g‘𝐹)𝑦) = (1r‘𝐹)) → (𝑥 ∈ (Unit‘𝐹) ∨ 𝑦 ∈ (Unit‘𝐹)))
42	41	ex 413	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐹)) ∧ 𝑦 ∈ (Base‘𝐹)) → ((𝑥(+g‘𝐹)𝑦) = (1r‘𝐹) → (𝑥 ∈ (Unit‘𝐹) ∨ 𝑦 ∈ (Unit‘𝐹))))
43	42	anasss 467	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → ((𝑥(+g‘𝐹)𝑦) = (1r‘𝐹) → (𝑥 ∈ (Unit‘𝐹) ∨ 𝑦 ∈ (Unit‘𝐹))))
44	43	ralrimivva 3182	. 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 20513	. 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 589	1 ⊢ (𝜑 → 𝐹 ∈ LRing)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∨ wo 853   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  ∀wral 3053  ‘cfv 6486  (class class class)co 7357  Basecbs 17171  +_gcplusg 17212  0_gc0g 17394  Grpcgrp 18901  1_rcur 20154  Unitcui 20327  NzRingcnzr 20485  LRingclring 20511  DivRingcdr 20702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5200  ax-sep 5219  ax-nul 5229  ax-pow 5295  ax-pr 5363  ax-un 7679  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-iun 4924  df-br 5074  df-opab 5136  df-mpt 5155  df-tr 5181  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7314  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7808  df-2nd 7933  df-tpos 8167  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-er 8634  df-en 8885  df-dom 8886  df-sdom 8887  df-pnf 11173  df-mnf 11174  df-xr 11175  df-ltxr 11176  df-le 11177  df-sub 11371  df-neg 11372  df-nn 12167  df-2 12236  df-3 12237  df-sets 17126  df-slot 17144  df-ndx 17156  df-base 17172  df-plusg 17225  df-mulr 17226  df-0g 17396  df-mgm 18600  df-sgrp 18679  df-mnd 18695  df-grp 18904  df-minusg 18905  df-cmn 19749  df-abl 19750  df-mgp 20114  df-rng 20126  df-ur 20155  df-ring 20208  df-oppr 20309  df-dvdsr 20329  df-unit 20330  df-nzr 20486  df-lring 20512  df-drng 20704

This theorem is referenced by: (None)