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Theorem lringuplu 20544
Description: If the sum of two elements of a local ring is invertible, then at least one of the summands must be invertible. (Contributed by Jim Kingdon, 18-Feb-2025.) (Revised by SN, 23-Feb-2025.)
Hypotheses
Ref Expression
lring.b (𝜑𝐵 = (Base‘𝑅))
lring.u (𝜑𝑈 = (Unit‘𝑅))
lring.p (𝜑+ = (+g𝑅))
lring.l (𝜑𝑅 ∈ LRing)
lring.s (𝜑 → (𝑋 + 𝑌) ∈ 𝑈)
lring.x (𝜑𝑋𝐵)
lring.y (𝜑𝑌𝐵)
Assertion
Ref Expression
lringuplu (𝜑 → (𝑋𝑈𝑌𝑈))

Proof of Theorem lringuplu
Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lring.l . . . . . . . 8 (𝜑𝑅 ∈ LRing)
2 lringring 20542 . . . . . . . 8 (𝑅 ∈ LRing → 𝑅 ∈ Ring)
31, 2syl 17 . . . . . . 7 (𝜑𝑅 ∈ Ring)
4 lring.x . . . . . . . 8 (𝜑𝑋𝐵)
5 lring.b . . . . . . . 8 (𝜑𝐵 = (Base‘𝑅))
64, 5eleqtrd 2843 . . . . . . 7 (𝜑𝑋 ∈ (Base‘𝑅))
7 lring.s . . . . . . . 8 (𝜑 → (𝑋 + 𝑌) ∈ 𝑈)
8 lring.u . . . . . . . 8 (𝜑𝑈 = (Unit‘𝑅))
97, 8eleqtrd 2843 . . . . . . 7 (𝜑 → (𝑋 + 𝑌) ∈ (Unit‘𝑅))
10 eqid 2737 . . . . . . . 8 (Base‘𝑅) = (Base‘𝑅)
11 eqid 2737 . . . . . . . 8 (Unit‘𝑅) = (Unit‘𝑅)
12 eqid 2737 . . . . . . . 8 (/r𝑅) = (/r𝑅)
13 eqid 2737 . . . . . . . 8 (.r𝑅) = (.r𝑅)
1410, 11, 12, 13dvrcan1 20409 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑋 ∈ (Base‘𝑅) ∧ (𝑋 + 𝑌) ∈ (Unit‘𝑅)) → ((𝑋(/r𝑅)(𝑋 + 𝑌))(.r𝑅)(𝑋 + 𝑌)) = 𝑋)
153, 6, 9, 14syl3anc 1373 . . . . . 6 (𝜑 → ((𝑋(/r𝑅)(𝑋 + 𝑌))(.r𝑅)(𝑋 + 𝑌)) = 𝑋)
1615adantr 480 . . . . 5 ((𝜑 ∧ (𝑋(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → ((𝑋(/r𝑅)(𝑋 + 𝑌))(.r𝑅)(𝑋 + 𝑌)) = 𝑋)
173adantr 480 . . . . . 6 ((𝜑 ∧ (𝑋(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → 𝑅 ∈ Ring)
18 simpr 484 . . . . . 6 ((𝜑 ∧ (𝑋(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → (𝑋(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅))
199adantr 480 . . . . . 6 ((𝜑 ∧ (𝑋(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → (𝑋 + 𝑌) ∈ (Unit‘𝑅))
2011, 13unitmulcl 20380 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝑋(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅) ∧ (𝑋 + 𝑌) ∈ (Unit‘𝑅)) → ((𝑋(/r𝑅)(𝑋 + 𝑌))(.r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅))
2117, 18, 19, 20syl3anc 1373 . . . . 5 ((𝜑 ∧ (𝑋(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → ((𝑋(/r𝑅)(𝑋 + 𝑌))(.r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅))
2216, 21eqeltrrd 2842 . . . 4 ((𝜑 ∧ (𝑋(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → 𝑋 ∈ (Unit‘𝑅))
238adantr 480 . . . 4 ((𝜑 ∧ (𝑋(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → 𝑈 = (Unit‘𝑅))
2422, 23eleqtrrd 2844 . . 3 ((𝜑 ∧ (𝑋(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → 𝑋𝑈)
2524orcd 874 . 2 ((𝜑 ∧ (𝑋(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → (𝑋𝑈𝑌𝑈))
26 lring.y . . . . . . . 8 (𝜑𝑌𝐵)
2726, 5eleqtrd 2843 . . . . . . 7 (𝜑𝑌 ∈ (Base‘𝑅))
2810, 11, 12, 13dvrcan1 20409 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑌 ∈ (Base‘𝑅) ∧ (𝑋 + 𝑌) ∈ (Unit‘𝑅)) → ((𝑌(/r𝑅)(𝑋 + 𝑌))(.r𝑅)(𝑋 + 𝑌)) = 𝑌)
293, 27, 9, 28syl3anc 1373 . . . . . 6 (𝜑 → ((𝑌(/r𝑅)(𝑋 + 𝑌))(.r𝑅)(𝑋 + 𝑌)) = 𝑌)
3029adantr 480 . . . . 5 ((𝜑 ∧ (𝑌(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → ((𝑌(/r𝑅)(𝑋 + 𝑌))(.r𝑅)(𝑋 + 𝑌)) = 𝑌)
313adantr 480 . . . . . 6 ((𝜑 ∧ (𝑌(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → 𝑅 ∈ Ring)
32 simpr 484 . . . . . 6 ((𝜑 ∧ (𝑌(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → (𝑌(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅))
339adantr 480 . . . . . 6 ((𝜑 ∧ (𝑌(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → (𝑋 + 𝑌) ∈ (Unit‘𝑅))
3411, 13unitmulcl 20380 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝑌(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅) ∧ (𝑋 + 𝑌) ∈ (Unit‘𝑅)) → ((𝑌(/r𝑅)(𝑋 + 𝑌))(.r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅))
3531, 32, 33, 34syl3anc 1373 . . . . 5 ((𝜑 ∧ (𝑌(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → ((𝑌(/r𝑅)(𝑋 + 𝑌))(.r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅))
3630, 35eqeltrrd 2842 . . . 4 ((𝜑 ∧ (𝑌(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → 𝑌 ∈ (Unit‘𝑅))
378adantr 480 . . . 4 ((𝜑 ∧ (𝑌(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → 𝑈 = (Unit‘𝑅))
3836, 37eleqtrrd 2844 . . 3 ((𝜑 ∧ (𝑌(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → 𝑌𝑈)
3938olcd 875 . 2 ((𝜑 ∧ (𝑌(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → (𝑋𝑈𝑌𝑈))
40 eqid 2737 . . . . . 6 (+g𝑅) = (+g𝑅)
4110, 11, 40, 12dvrdir 20412 . . . . 5 ((𝑅 ∈ Ring ∧ (𝑋 ∈ (Base‘𝑅) ∧ 𝑌 ∈ (Base‘𝑅) ∧ (𝑋 + 𝑌) ∈ (Unit‘𝑅))) → ((𝑋(+g𝑅)𝑌)(/r𝑅)(𝑋 + 𝑌)) = ((𝑋(/r𝑅)(𝑋 + 𝑌))(+g𝑅)(𝑌(/r𝑅)(𝑋 + 𝑌))))
423, 6, 27, 9, 41syl13anc 1374 . . . 4 (𝜑 → ((𝑋(+g𝑅)𝑌)(/r𝑅)(𝑋 + 𝑌)) = ((𝑋(/r𝑅)(𝑋 + 𝑌))(+g𝑅)(𝑌(/r𝑅)(𝑋 + 𝑌))))
43 lring.p . . . . . . 7 (𝜑+ = (+g𝑅))
4443eqcomd 2743 . . . . . 6 (𝜑 → (+g𝑅) = + )
4544oveqd 7448 . . . . 5 (𝜑 → (𝑋(+g𝑅)𝑌) = (𝑋 + 𝑌))
463ringgrpd 20239 . . . . . . 7 (𝜑𝑅 ∈ Grp)
4710, 40, 46, 6, 27grpcld 18965 . . . . . 6 (𝜑 → (𝑋(+g𝑅)𝑌) ∈ (Base‘𝑅))
48 eqid 2737 . . . . . . 7 (1r𝑅) = (1r𝑅)
4910, 11, 12, 48dvreq1 20411 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝑋(+g𝑅)𝑌) ∈ (Base‘𝑅) ∧ (𝑋 + 𝑌) ∈ (Unit‘𝑅)) → (((𝑋(+g𝑅)𝑌)(/r𝑅)(𝑋 + 𝑌)) = (1r𝑅) ↔ (𝑋(+g𝑅)𝑌) = (𝑋 + 𝑌)))
503, 47, 9, 49syl3anc 1373 . . . . 5 (𝜑 → (((𝑋(+g𝑅)𝑌)(/r𝑅)(𝑋 + 𝑌)) = (1r𝑅) ↔ (𝑋(+g𝑅)𝑌) = (𝑋 + 𝑌)))
5145, 50mpbird 257 . . . 4 (𝜑 → ((𝑋(+g𝑅)𝑌)(/r𝑅)(𝑋 + 𝑌)) = (1r𝑅))
5242, 51eqtr3d 2779 . . 3 (𝜑 → ((𝑋(/r𝑅)(𝑋 + 𝑌))(+g𝑅)(𝑌(/r𝑅)(𝑋 + 𝑌))) = (1r𝑅))
53 oveq2 7439 . . . . . 6 (𝑣 = (𝑌(/r𝑅)(𝑋 + 𝑌)) → ((𝑋(/r𝑅)(𝑋 + 𝑌))(+g𝑅)𝑣) = ((𝑋(/r𝑅)(𝑋 + 𝑌))(+g𝑅)(𝑌(/r𝑅)(𝑋 + 𝑌))))
5453eqeq1d 2739 . . . . 5 (𝑣 = (𝑌(/r𝑅)(𝑋 + 𝑌)) → (((𝑋(/r𝑅)(𝑋 + 𝑌))(+g𝑅)𝑣) = (1r𝑅) ↔ ((𝑋(/r𝑅)(𝑋 + 𝑌))(+g𝑅)(𝑌(/r𝑅)(𝑋 + 𝑌))) = (1r𝑅)))
55 eleq1 2829 . . . . . 6 (𝑣 = (𝑌(/r𝑅)(𝑋 + 𝑌)) → (𝑣 ∈ (Unit‘𝑅) ↔ (𝑌(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)))
5655orbi2d 916 . . . . 5 (𝑣 = (𝑌(/r𝑅)(𝑋 + 𝑌)) → (((𝑋(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅) ∨ 𝑣 ∈ (Unit‘𝑅)) ↔ ((𝑋(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅) ∨ (𝑌(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅))))
5754, 56imbi12d 344 . . . 4 (𝑣 = (𝑌(/r𝑅)(𝑋 + 𝑌)) → ((((𝑋(/r𝑅)(𝑋 + 𝑌))(+g𝑅)𝑣) = (1r𝑅) → ((𝑋(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅) ∨ 𝑣 ∈ (Unit‘𝑅))) ↔ (((𝑋(/r𝑅)(𝑋 + 𝑌))(+g𝑅)(𝑌(/r𝑅)(𝑋 + 𝑌))) = (1r𝑅) → ((𝑋(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅) ∨ (𝑌(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)))))
58 oveq1 7438 . . . . . . . 8 (𝑢 = (𝑋(/r𝑅)(𝑋 + 𝑌)) → (𝑢(+g𝑅)𝑣) = ((𝑋(/r𝑅)(𝑋 + 𝑌))(+g𝑅)𝑣))
5958eqeq1d 2739 . . . . . . 7 (𝑢 = (𝑋(/r𝑅)(𝑋 + 𝑌)) → ((𝑢(+g𝑅)𝑣) = (1r𝑅) ↔ ((𝑋(/r𝑅)(𝑋 + 𝑌))(+g𝑅)𝑣) = (1r𝑅)))
60 eleq1 2829 . . . . . . . 8 (𝑢 = (𝑋(/r𝑅)(𝑋 + 𝑌)) → (𝑢 ∈ (Unit‘𝑅) ↔ (𝑋(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)))
6160orbi1d 917 . . . . . . 7 (𝑢 = (𝑋(/r𝑅)(𝑋 + 𝑌)) → ((𝑢 ∈ (Unit‘𝑅) ∨ 𝑣 ∈ (Unit‘𝑅)) ↔ ((𝑋(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅) ∨ 𝑣 ∈ (Unit‘𝑅))))
6259, 61imbi12d 344 . . . . . 6 (𝑢 = (𝑋(/r𝑅)(𝑋 + 𝑌)) → (((𝑢(+g𝑅)𝑣) = (1r𝑅) → (𝑢 ∈ (Unit‘𝑅) ∨ 𝑣 ∈ (Unit‘𝑅))) ↔ (((𝑋(/r𝑅)(𝑋 + 𝑌))(+g𝑅)𝑣) = (1r𝑅) → ((𝑋(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅) ∨ 𝑣 ∈ (Unit‘𝑅)))))
6362ralbidv 3178 . . . . 5 (𝑢 = (𝑋(/r𝑅)(𝑋 + 𝑌)) → (∀𝑣 ∈ (Base‘𝑅)((𝑢(+g𝑅)𝑣) = (1r𝑅) → (𝑢 ∈ (Unit‘𝑅) ∨ 𝑣 ∈ (Unit‘𝑅))) ↔ ∀𝑣 ∈ (Base‘𝑅)(((𝑋(/r𝑅)(𝑋 + 𝑌))(+g𝑅)𝑣) = (1r𝑅) → ((𝑋(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅) ∨ 𝑣 ∈ (Unit‘𝑅)))))
6410, 40, 48, 11islring 20540 . . . . . . 7 (𝑅 ∈ LRing ↔ (𝑅 ∈ NzRing ∧ ∀𝑢 ∈ (Base‘𝑅)∀𝑣 ∈ (Base‘𝑅)((𝑢(+g𝑅)𝑣) = (1r𝑅) → (𝑢 ∈ (Unit‘𝑅) ∨ 𝑣 ∈ (Unit‘𝑅)))))
651, 64sylib 218 . . . . . 6 (𝜑 → (𝑅 ∈ NzRing ∧ ∀𝑢 ∈ (Base‘𝑅)∀𝑣 ∈ (Base‘𝑅)((𝑢(+g𝑅)𝑣) = (1r𝑅) → (𝑢 ∈ (Unit‘𝑅) ∨ 𝑣 ∈ (Unit‘𝑅)))))
6665simprd 495 . . . . 5 (𝜑 → ∀𝑢 ∈ (Base‘𝑅)∀𝑣 ∈ (Base‘𝑅)((𝑢(+g𝑅)𝑣) = (1r𝑅) → (𝑢 ∈ (Unit‘𝑅) ∨ 𝑣 ∈ (Unit‘𝑅))))
6710, 11, 12dvrcl 20404 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑋 ∈ (Base‘𝑅) ∧ (𝑋 + 𝑌) ∈ (Unit‘𝑅)) → (𝑋(/r𝑅)(𝑋 + 𝑌)) ∈ (Base‘𝑅))
683, 6, 9, 67syl3anc 1373 . . . . 5 (𝜑 → (𝑋(/r𝑅)(𝑋 + 𝑌)) ∈ (Base‘𝑅))
6963, 66, 68rspcdva 3623 . . . 4 (𝜑 → ∀𝑣 ∈ (Base‘𝑅)(((𝑋(/r𝑅)(𝑋 + 𝑌))(+g𝑅)𝑣) = (1r𝑅) → ((𝑋(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅) ∨ 𝑣 ∈ (Unit‘𝑅))))
7010, 11, 12dvrcl 20404 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑌 ∈ (Base‘𝑅) ∧ (𝑋 + 𝑌) ∈ (Unit‘𝑅)) → (𝑌(/r𝑅)(𝑋 + 𝑌)) ∈ (Base‘𝑅))
713, 27, 9, 70syl3anc 1373 . . . 4 (𝜑 → (𝑌(/r𝑅)(𝑋 + 𝑌)) ∈ (Base‘𝑅))
7257, 69, 71rspcdva 3623 . . 3 (𝜑 → (((𝑋(/r𝑅)(𝑋 + 𝑌))(+g𝑅)(𝑌(/r𝑅)(𝑋 + 𝑌))) = (1r𝑅) → ((𝑋(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅) ∨ (𝑌(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅))))
7352, 72mpd 15 . 2 (𝜑 → ((𝑋(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅) ∨ (𝑌(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)))
7425, 39, 73mpjaodan 961 1 (𝜑 → (𝑋𝑈𝑌𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 848   = wceq 1540  wcel 2108  wral 3061  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  .rcmulr 17298  1rcur 20178  Ringcrg 20230  Unitcui 20355  /rcdvr 20400  NzRingcnzr 20512  LRingclring 20538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-tpos 8251  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-oppr 20334  df-dvdsr 20357  df-unit 20358  df-invr 20388  df-dvr 20401  df-nzr 20513  df-lring 20539
This theorem is referenced by: (None)
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