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Theorem lringuplu 20429
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 20427 . . . . . . . 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 2827 . . . . . . 7 (𝜑𝑋 ∈ (Base‘𝑅))
7 lring.s . . . . . . . 8 (𝜑 → (𝑋 + 𝑌) ∈ 𝑈)
8 lring.u . . . . . . . 8 (𝜑𝑈 = (Unit‘𝑅))
97, 8eleqtrd 2827 . . . . . . 7 (𝜑 → (𝑋 + 𝑌) ∈ (Unit‘𝑅))
10 eqid 2724 . . . . . . . 8 (Base‘𝑅) = (Base‘𝑅)
11 eqid 2724 . . . . . . . 8 (Unit‘𝑅) = (Unit‘𝑅)
12 eqid 2724 . . . . . . . 8 (/r𝑅) = (/r𝑅)
13 eqid 2724 . . . . . . . 8 (.r𝑅) = (.r𝑅)
1410, 11, 12, 13dvrcan1 20296 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑋 ∈ (Base‘𝑅) ∧ (𝑋 + 𝑌) ∈ (Unit‘𝑅)) → ((𝑋(/r𝑅)(𝑋 + 𝑌))(.r𝑅)(𝑋 + 𝑌)) = 𝑋)
153, 6, 9, 14syl3anc 1368 . . . . . 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 20267 . . . . . 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 1368 . . . . 5 ((𝜑 ∧ (𝑋(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → ((𝑋(/r𝑅)(𝑋 + 𝑌))(.r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅))
2216, 21eqeltrrd 2826 . . . 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 2828 . . 3 ((𝜑 ∧ (𝑋(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → 𝑋𝑈)
2524orcd 870 . 2 ((𝜑 ∧ (𝑋(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → (𝑋𝑈𝑌𝑈))
26 lring.y . . . . . . . 8 (𝜑𝑌𝐵)
2726, 5eleqtrd 2827 . . . . . . 7 (𝜑𝑌 ∈ (Base‘𝑅))
2810, 11, 12, 13dvrcan1 20296 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑌 ∈ (Base‘𝑅) ∧ (𝑋 + 𝑌) ∈ (Unit‘𝑅)) → ((𝑌(/r𝑅)(𝑋 + 𝑌))(.r𝑅)(𝑋 + 𝑌)) = 𝑌)
293, 27, 9, 28syl3anc 1368 . . . . . 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 20267 . . . . . 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 1368 . . . . 5 ((𝜑 ∧ (𝑌(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → ((𝑌(/r𝑅)(𝑋 + 𝑌))(.r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅))
3630, 35eqeltrrd 2826 . . . 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 2828 . . 3 ((𝜑 ∧ (𝑌(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → 𝑌𝑈)
3938olcd 871 . 2 ((𝜑 ∧ (𝑌(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → (𝑋𝑈𝑌𝑈))
40 eqid 2724 . . . . . 6 (+g𝑅) = (+g𝑅)
4110, 11, 40, 12dvrdir 20299 . . . . 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 1369 . . . 4 (𝜑 → ((𝑋(+g𝑅)𝑌)(/r𝑅)(𝑋 + 𝑌)) = ((𝑋(/r𝑅)(𝑋 + 𝑌))(+g𝑅)(𝑌(/r𝑅)(𝑋 + 𝑌))))
43 lring.p . . . . . . 7 (𝜑+ = (+g𝑅))
4443eqcomd 2730 . . . . . 6 (𝜑 → (+g𝑅) = + )
4544oveqd 7418 . . . . 5 (𝜑 → (𝑋(+g𝑅)𝑌) = (𝑋 + 𝑌))
463ringgrpd 20132 . . . . . . 7 (𝜑𝑅 ∈ Grp)
4710, 40, 46, 6, 27grpcld 18864 . . . . . 6 (𝜑 → (𝑋(+g𝑅)𝑌) ∈ (Base‘𝑅))
48 eqid 2724 . . . . . . 7 (1r𝑅) = (1r𝑅)
4910, 11, 12, 48dvreq1 20298 . . . . . 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 1368 . . . . 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 2766 . . 3 (𝜑 → ((𝑋(/r𝑅)(𝑋 + 𝑌))(+g𝑅)(𝑌(/r𝑅)(𝑋 + 𝑌))) = (1r𝑅))
53 oveq2 7409 . . . . . 6 (𝑣 = (𝑌(/r𝑅)(𝑋 + 𝑌)) → ((𝑋(/r𝑅)(𝑋 + 𝑌))(+g𝑅)𝑣) = ((𝑋(/r𝑅)(𝑋 + 𝑌))(+g𝑅)(𝑌(/r𝑅)(𝑋 + 𝑌))))
5453eqeq1d 2726 . . . . 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 2813 . . . . . 6 (𝑣 = (𝑌(/r𝑅)(𝑋 + 𝑌)) → (𝑣 ∈ (Unit‘𝑅) ↔ (𝑌(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)))
5655orbi2d 912 . . . . 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 7408 . . . . . . . 8 (𝑢 = (𝑋(/r𝑅)(𝑋 + 𝑌)) → (𝑢(+g𝑅)𝑣) = ((𝑋(/r𝑅)(𝑋 + 𝑌))(+g𝑅)𝑣))
5958eqeq1d 2726 . . . . . . 7 (𝑢 = (𝑋(/r𝑅)(𝑋 + 𝑌)) → ((𝑢(+g𝑅)𝑣) = (1r𝑅) ↔ ((𝑋(/r𝑅)(𝑋 + 𝑌))(+g𝑅)𝑣) = (1r𝑅)))
60 eleq1 2813 . . . . . . . 8 (𝑢 = (𝑋(/r𝑅)(𝑋 + 𝑌)) → (𝑢 ∈ (Unit‘𝑅) ↔ (𝑋(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)))
6160orbi1d 913 . . . . . . 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 3169 . . . . 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 20425 . . . . . . 7 (𝑅 ∈ LRing ↔ (𝑅 ∈ NzRing ∧ ∀𝑢 ∈ (Base‘𝑅)∀𝑣 ∈ (Base‘𝑅)((𝑢(+g𝑅)𝑣) = (1r𝑅) → (𝑢 ∈ (Unit‘𝑅) ∨ 𝑣 ∈ (Unit‘𝑅)))))
651, 64sylib 217 . . . . . 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 20291 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑋 ∈ (Base‘𝑅) ∧ (𝑋 + 𝑌) ∈ (Unit‘𝑅)) → (𝑋(/r𝑅)(𝑋 + 𝑌)) ∈ (Base‘𝑅))
683, 6, 9, 67syl3anc 1368 . . . . 5 (𝜑 → (𝑋(/r𝑅)(𝑋 + 𝑌)) ∈ (Base‘𝑅))
6963, 66, 68rspcdva 3605 . . . 4 (𝜑 → ∀𝑣 ∈ (Base‘𝑅)(((𝑋(/r𝑅)(𝑋 + 𝑌))(+g𝑅)𝑣) = (1r𝑅) → ((𝑋(/r𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅) ∨ 𝑣 ∈ (Unit‘𝑅))))
7010, 11, 12dvrcl 20291 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑌 ∈ (Base‘𝑅) ∧ (𝑋 + 𝑌) ∈ (Unit‘𝑅)) → (𝑌(/r𝑅)(𝑋 + 𝑌)) ∈ (Base‘𝑅))
713, 27, 9, 70syl3anc 1368 . . . 4 (𝜑 → (𝑌(/r𝑅)(𝑋 + 𝑌)) ∈ (Base‘𝑅))
7257, 69, 71rspcdva 3605 . . 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 955 1 (𝜑 → (𝑋𝑈𝑌𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wo 844   = wceq 1533  wcel 2098  wral 3053  cfv 6533  (class class class)co 7401  Basecbs 17140  +gcplusg 17193  .rcmulr 17194  1rcur 20071  Ringcrg 20123  Unitcui 20242  /rcdvr 20287  NzRingcnzr 20399  LRingclring 20423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11161  ax-resscn 11162  ax-1cn 11163  ax-icn 11164  ax-addcl 11165  ax-addrcl 11166  ax-mulcl 11167  ax-mulrcl 11168  ax-mulcom 11169  ax-addass 11170  ax-mulass 11171  ax-distr 11172  ax-i2m1 11173  ax-1ne0 11174  ax-1rid 11175  ax-rnegex 11176  ax-rrecex 11177  ax-cnre 11178  ax-pre-lttri 11179  ax-pre-lttrn 11180  ax-pre-ltadd 11181  ax-pre-mulgt0 11182
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-tpos 8206  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-er 8698  df-en 8935  df-dom 8936  df-sdom 8937  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-ress 17170  df-plusg 17206  df-mulr 17207  df-0g 17383  df-mgm 18560  df-sgrp 18639  df-mnd 18655  df-grp 18853  df-minusg 18854  df-cmn 19687  df-abl 19688  df-mgp 20025  df-rng 20043  df-ur 20072  df-ring 20125  df-oppr 20221  df-dvdsr 20244  df-unit 20245  df-invr 20275  df-dvr 20288  df-nzr 20400  df-lring 20424
This theorem is referenced by: (None)
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