Theorem drnglring 33689

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 20799	. . 3 ⊢ (𝐹 ∈ DivRing → 𝐹 ∈ NzRing)
3	1, 2	syl 17	. 2 ⊢ (𝜑 → 𝐹 ∈ NzRing)
4	1	ad4antr 742	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐹)) ∧ 𝑦 ∈ (Base‘𝐹)) ∧ (𝑥(+g‘𝐹)𝑦) = (1r‘𝐹)) ∧ ¬ 𝑥 = (0g‘𝐹)) → 𝐹 ∈ DivRing)
5		simp-4r 793	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐹)) ∧ 𝑦 ∈ (Base‘𝐹)) ∧ (𝑥(+g‘𝐹)𝑦) = (1r‘𝐹)) ∧ ¬ 𝑥 = (0g‘𝐹)) → 𝑥 ∈ (Base‘𝐹))
6		neqne 2966	. . . . . . . 8 ⊢ (¬ 𝑥 = (0g‘𝐹) → 𝑥 ≠ (0g‘𝐹))
7	6	adantl 485	. . . . . . 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 20785	. . . . . . . 8 ⊢ (𝐹 ∈ DivRing → (𝑥 ∈ (Unit‘𝐹) ↔ (𝑥 ∈ (Base‘𝐹) ∧ 𝑥 ≠ (0g‘𝐹))))
12	11	biimpar 481	. . . . . . 7 ⊢ ((𝐹 ∈ DivRing ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑥 ≠ (0g‘𝐹))) → 𝑥 ∈ (Unit‘𝐹))
13	4, 5, 7, 12	syl12anc 847	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐹)) ∧ 𝑦 ∈ (Base‘𝐹)) ∧ (𝑥(+g‘𝐹)𝑦) = (1r‘𝐹)) ∧ ¬ 𝑥 = (0g‘𝐹)) → 𝑥 ∈ (Unit‘𝐹))
14	1	ad4antr 742	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐹)) ∧ 𝑦 ∈ (Base‘𝐹)) ∧ (𝑥(+g‘𝐹)𝑦) = (1r‘𝐹)) ∧ ¬ 𝑦 = (0g‘𝐹)) → 𝐹 ∈ DivRing)
15		simpllr 785	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐹)) ∧ 𝑦 ∈ (Base‘𝐹)) ∧ (𝑥(+g‘𝐹)𝑦) = (1r‘𝐹)) ∧ ¬ 𝑦 = (0g‘𝐹)) → 𝑦 ∈ (Base‘𝐹))
16		neqne 2966	. . . . . . . 8 ⊢ (¬ 𝑦 = (0g‘𝐹) → 𝑦 ≠ (0g‘𝐹))
17	16	adantl 485	. . . . . . 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 20785	. . . . . . . 8 ⊢ (𝐹 ∈ DivRing → (𝑦 ∈ (Unit‘𝐹) ↔ (𝑦 ∈ (Base‘𝐹) ∧ 𝑦 ≠ (0g‘𝐹))))
19	18	biimpar 481	. . . . . . 7 ⊢ ((𝐹 ∈ DivRing ∧ (𝑦 ∈ (Base‘𝐹) ∧ 𝑦 ≠ (0g‘𝐹))) → 𝑦 ∈ (Unit‘𝐹))
20	14, 15, 17, 19	syl12anc 847	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐹)) ∧ 𝑦 ∈ (Base‘𝐹)) ∧ (𝑥(+g‘𝐹)𝑦) = (1r‘𝐹)) ∧ ¬ 𝑦 = (0g‘𝐹)) → 𝑦 ∈ (Unit‘𝐹))
21		simplll 784	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐹)) ∧ 𝑦 ∈ (Base‘𝐹)) ∧ (𝑥(+g‘𝐹)𝑦) = (1r‘𝐹)) → 𝜑)
22		simpr 488	. . . . . . . . . 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 20566	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ NzRing → (1r‘𝐹) ≠ (0g‘𝐹))
25	3, 24	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (1r‘𝐹) ≠ (0g‘𝐹))
26	25	ad3antrrr 740	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐹)) ∧ 𝑦 ∈ (Base‘𝐹)) ∧ (𝑥(+g‘𝐹)𝑦) = (1r‘𝐹)) → (1r‘𝐹) ≠ (0g‘𝐹))
27	22, 26	eqnetrd 3025	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐹)) ∧ 𝑦 ∈ (Base‘𝐹)) ∧ (𝑥(+g‘𝐹)𝑦) = (1r‘𝐹)) → (𝑥(+g‘𝐹)𝑦) ≠ (0g‘𝐹))
28	27	neneqd 2963	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐹)) ∧ 𝑦 ∈ (Base‘𝐹)) ∧ (𝑥(+g‘𝐹)𝑦) = (1r‘𝐹)) → ¬ (𝑥(+g‘𝐹)𝑦) = (0g‘𝐹))
29		oveq12 7406	. . . . . . . . . . 11 ⊢ ((𝑥 = (0g‘𝐹) ∧ 𝑦 = (0g‘𝐹)) → (𝑥(+g‘𝐹)𝑦) = ((0g‘𝐹)(+g‘𝐹)(0g‘𝐹)))
30	29	adantl 485	. . . . . . . . . 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 20789	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ∈ Grp)
33	32	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 = (0g‘𝐹) ∧ 𝑦 = (0g‘𝐹))) → 𝐹 ∈ Grp)
34	8, 10, 33	grpidcld 33219	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 = (0g‘𝐹) ∧ 𝑦 = (0g‘𝐹))) → (0g‘𝐹) ∈ (Base‘𝐹))
35	8, 31, 10, 33, 34	grplidd 19012	. . . . . . . . . 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 1793	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (𝑥(+g‘𝐹)𝑦) = (0g‘𝐹)) → ¬ (𝑥 = (0g‘𝐹) ∧ 𝑦 = (0g‘𝐹)))
38	21, 28, 37	syl2anc 593	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐹)) ∧ 𝑦 ∈ (Base‘𝐹)) ∧ (𝑥(+g‘𝐹)𝑦) = (1r‘𝐹)) → ¬ (𝑥 = (0g‘𝐹) ∧ 𝑦 = (0g‘𝐹)))
39		ianor 995	. . . . . . 7 ⊢ (¬ (𝑥 = (0g‘𝐹) ∧ 𝑦 = (0g‘𝐹)) ↔ (¬ 𝑥 = (0g‘𝐹) ∨ ¬ 𝑦 = (0g‘𝐹)))
40	38, 39	sylib 220	. . . . . 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 978	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐹)) ∧ 𝑦 ∈ (Base‘𝐹)) ∧ (𝑥(+g‘𝐹)𝑦) = (1r‘𝐹)) → (𝑥 ∈ (Unit‘𝐹) ∨ 𝑦 ∈ (Unit‘𝐹)))
42	41	ex 416	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐹)) ∧ 𝑦 ∈ (Base‘𝐹)) → ((𝑥(+g‘𝐹)𝑦) = (1r‘𝐹) → (𝑥 ∈ (Unit‘𝐹) ∨ 𝑦 ∈ (Unit‘𝐹))))
43	42	anasss 470	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐹) ∧ 𝑦 ∈ (Base‘𝐹))) → ((𝑥(+g‘𝐹)𝑦) = (1r‘𝐹) → (𝑥 ∈ (Unit‘𝐹) ∨ 𝑦 ∈ (Unit‘𝐹))))
44	43	ralrimivva 3206	. 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 20591	. 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 592	1 ⊢ (𝜑 → 𝐹 ∈ LRing)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 858   = wceq 1561   ∈ wcel 2143   ≠ wne 2958  ∀wral 3077  ‘cfv 6522  (class class class)co 7397  Basecbs 17246  +_gcplusg 17287  0_gc0g 17469  Grpcgrp 18976  1_rcur 20232  Unitcui 20405  NzRingcnzr 20563  LRingclring 20589  DivRingcdr 20780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7719  ax-cnex 11130  ax-resscn 11131  ax-1cn 11132  ax-icn 11133  ax-addcl 11134  ax-addrcl 11135  ax-mulcl 11136  ax-mulrcl 11137  ax-mulcom 11138  ax-addass 11139  ax-mulass 11140  ax-distr 11141  ax-i2m1 11142  ax-1ne0 11143  ax-1rid 11144  ax-rnegex 11145  ax-rrecex 11146  ax-cnre 11147  ax-pre-lttri 11148  ax-pre-lttrn 11149  ax-pre-ltadd 11150  ax-pre-mulgt0 11151

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6289  df-ord 6350  df-on 6351  df-lim 6352  df-suc 6353  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-f1 6527  df-fo 6528  df-f1o 6529  df-fv 6530  df-riota 7354  df-ov 7400  df-oprab 7401  df-mpo 7402  df-om 7848  df-2nd 7972  df-tpos 8207  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8382  df-er 8679  df-en 8929  df-dom 8930  df-sdom 8931  df-pnf 11219  df-mnf 11220  df-xr 11221  df-ltxr 11222  df-le 11223  df-sub 11417  df-neg 11418  df-nn 12212  df-2 12281  df-3 12282  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17247  df-plusg 17300  df-mulr 17301  df-0g 17471  df-mgm 18675  df-sgrp 18754  df-mnd 18770  df-grp 18979  df-minusg 18980  df-cmn 19823  df-abl 19824  df-mgp 20188  df-rng 20200  df-ur 20233  df-ring 20286  df-oppr 20387  df-dvdsr 20407  df-unit 20408  df-nzr 20564  df-lring 20590  df-drng 20782

This theorem is referenced by: (None)