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Theorem rngqiprngu 21328
Description: If a non-unital ring has a (two-sided) ideal which is unital, and the quotient of the ring and the ideal is also unital, then the ring is also unital with a ring unity which can be constructed from the ring unity of the ideal and a representative of the ring unity of the quotient. (Contributed by AV, 17-Mar-2025.)
Hypotheses
Ref Expression
rngqiprngfu.r (𝜑𝑅 ∈ Rng)
rngqiprngfu.i (𝜑𝐼 ∈ (2Ideal‘𝑅))
rngqiprngfu.j 𝐽 = (𝑅s 𝐼)
rngqiprngfu.u (𝜑𝐽 ∈ Ring)
rngqiprngfu.b 𝐵 = (Base‘𝑅)
rngqiprngfu.t · = (.r𝑅)
rngqiprngfu.1 1 = (1r𝐽)
rngqiprngfu.g = (𝑅 ~QG 𝐼)
rngqiprngfu.q 𝑄 = (𝑅 /s )
rngqiprngfu.v (𝜑𝑄 ∈ Ring)
rngqiprngfu.e (𝜑𝐸 ∈ (1r𝑄))
rngqiprngfu.m = (-g𝑅)
rngqiprngfu.a + = (+g𝑅)
rngqiprngfu.n 𝑈 = ((𝐸 ( 1 · 𝐸)) + 1 )
Assertion
Ref Expression
rngqiprngu (𝜑 → (1r𝑅) = 𝑈)

Proof of Theorem rngqiprngu
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . . . 4 (𝑄 ×s 𝐽) = (𝑄 ×s 𝐽)
2 rngqiprngfu.v . . . 4 (𝜑𝑄 ∈ Ring)
3 rngqiprngfu.u . . . 4 (𝜑𝐽 ∈ Ring)
41, 2, 3xpsringd 20329 . . 3 (𝜑 → (𝑄 ×s 𝐽) ∈ Ring)
5 rngqiprngfu.r . . 3 (𝜑𝑅 ∈ Rng)
6 rngqiprngfu.i . . . . 5 (𝜑𝐼 ∈ (2Ideal‘𝑅))
7 rngqiprngfu.j . . . . 5 𝐽 = (𝑅s 𝐼)
8 rngqiprngfu.b . . . . 5 𝐵 = (Base‘𝑅)
9 rngqiprngfu.t . . . . 5 · = (.r𝑅)
10 rngqiprngfu.1 . . . . 5 1 = (1r𝐽)
11 rngqiprngfu.g . . . . 5 = (𝑅 ~QG 𝐼)
12 rngqiprngfu.q . . . . 5 𝑄 = (𝑅 /s )
13 eqid 2737 . . . . 5 (Base‘𝑄) = (Base‘𝑄)
14 eqid 2737 . . . . 5 (𝑥𝐵 ↦ ⟨[𝑥] , ( 1 · 𝑥)⟩) = (𝑥𝐵 ↦ ⟨[𝑥] , ( 1 · 𝑥)⟩)
155, 6, 7, 3, 8, 9, 10, 11, 12, 13, 1, 14rngqiprngim 21314 . . . 4 (𝜑 → (𝑥𝐵 ↦ ⟨[𝑥] , ( 1 · 𝑥)⟩) ∈ (𝑅 RngIso (𝑄 ×s 𝐽)))
16 rngimcnv 20456 . . . 4 ((𝑥𝐵 ↦ ⟨[𝑥] , ( 1 · 𝑥)⟩) ∈ (𝑅 RngIso (𝑄 ×s 𝐽)) → (𝑥𝐵 ↦ ⟨[𝑥] , ( 1 · 𝑥)⟩) ∈ ((𝑄 ×s 𝐽) RngIso 𝑅))
1715, 16syl 17 . . 3 (𝜑(𝑥𝐵 ↦ ⟨[𝑥] , ( 1 · 𝑥)⟩) ∈ ((𝑄 ×s 𝐽) RngIso 𝑅))
18 rngisomring1 20468 . . 3 (((𝑄 ×s 𝐽) ∈ Ring ∧ 𝑅 ∈ Rng ∧ (𝑥𝐵 ↦ ⟨[𝑥] , ( 1 · 𝑥)⟩) ∈ ((𝑄 ×s 𝐽) RngIso 𝑅)) → (1r𝑅) = ((𝑥𝐵 ↦ ⟨[𝑥] , ( 1 · 𝑥)⟩)‘(1r‘(𝑄 ×s 𝐽))))
194, 5, 17, 18syl3anc 1373 . 2 (𝜑 → (1r𝑅) = ((𝑥𝐵 ↦ ⟨[𝑥] , ( 1 · 𝑥)⟩)‘(1r‘(𝑄 ×s 𝐽))))
20 rngqiprngfu.e . . . . 5 (𝜑𝐸 ∈ (1r𝑄))
21 rngqiprngfu.m . . . . 5 = (-g𝑅)
22 rngqiprngfu.a . . . . 5 + = (+g𝑅)
23 rngqiprngfu.n . . . . 5 𝑈 = ((𝐸 ( 1 · 𝐸)) + 1 )
245, 6, 7, 3, 8, 9, 10, 11, 12, 2, 20, 21, 22, 23, 14rngqiprngfu 21327 . . . 4 (𝜑 → ((𝑥𝐵 ↦ ⟨[𝑥] , ( 1 · 𝑥)⟩)‘𝑈) = ⟨[𝐸] , 1 ⟩)
255, 6, 7, 3, 8, 9, 10, 11, 12, 2, 20, 21, 22, 23, 1rngqipring1 21326 . . . 4 (𝜑 → (1r‘(𝑄 ×s 𝐽)) = ⟨[𝐸] , 1 ⟩)
2624, 25eqtr4d 2780 . . 3 (𝜑 → ((𝑥𝐵 ↦ ⟨[𝑥] , ( 1 · 𝑥)⟩)‘𝑈) = (1r‘(𝑄 ×s 𝐽)))
27 eqid 2737 . . . . . 6 (Base‘(𝑄 ×s 𝐽)) = (Base‘(𝑄 ×s 𝐽))
288, 27rngimf1o 20454 . . . . 5 ((𝑥𝐵 ↦ ⟨[𝑥] , ( 1 · 𝑥)⟩) ∈ (𝑅 RngIso (𝑄 ×s 𝐽)) → (𝑥𝐵 ↦ ⟨[𝑥] , ( 1 · 𝑥)⟩):𝐵1-1-onto→(Base‘(𝑄 ×s 𝐽)))
2915, 28syl 17 . . . 4 (𝜑 → (𝑥𝐵 ↦ ⟨[𝑥] , ( 1 · 𝑥)⟩):𝐵1-1-onto→(Base‘(𝑄 ×s 𝐽)))
305, 6, 7, 3, 8, 9, 10, 11, 12, 2, 20, 21, 22, 23rngqiprngfulem3 21323 . . . 4 (𝜑𝑈𝐵)
31 eqid 2737 . . . . . 6 (1r‘(𝑄 ×s 𝐽)) = (1r‘(𝑄 ×s 𝐽))
3227, 31ringidcl 20262 . . . . 5 ((𝑄 ×s 𝐽) ∈ Ring → (1r‘(𝑄 ×s 𝐽)) ∈ (Base‘(𝑄 ×s 𝐽)))
334, 32syl 17 . . . 4 (𝜑 → (1r‘(𝑄 ×s 𝐽)) ∈ (Base‘(𝑄 ×s 𝐽)))
34 f1ocnvfvb 7299 . . . 4 (((𝑥𝐵 ↦ ⟨[𝑥] , ( 1 · 𝑥)⟩):𝐵1-1-onto→(Base‘(𝑄 ×s 𝐽)) ∧ 𝑈𝐵 ∧ (1r‘(𝑄 ×s 𝐽)) ∈ (Base‘(𝑄 ×s 𝐽))) → (((𝑥𝐵 ↦ ⟨[𝑥] , ( 1 · 𝑥)⟩)‘𝑈) = (1r‘(𝑄 ×s 𝐽)) ↔ ((𝑥𝐵 ↦ ⟨[𝑥] , ( 1 · 𝑥)⟩)‘(1r‘(𝑄 ×s 𝐽))) = 𝑈))
3529, 30, 33, 34syl3anc 1373 . . 3 (𝜑 → (((𝑥𝐵 ↦ ⟨[𝑥] , ( 1 · 𝑥)⟩)‘𝑈) = (1r‘(𝑄 ×s 𝐽)) ↔ ((𝑥𝐵 ↦ ⟨[𝑥] , ( 1 · 𝑥)⟩)‘(1r‘(𝑄 ×s 𝐽))) = 𝑈))
3626, 35mpbid 232 . 2 (𝜑 → ((𝑥𝐵 ↦ ⟨[𝑥] , ( 1 · 𝑥)⟩)‘(1r‘(𝑄 ×s 𝐽))) = 𝑈)
3719, 36eqtrd 2777 1 (𝜑 → (1r𝑅) = 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540  wcel 2108  cop 4632  cmpt 5225  ccnv 5684  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  [cec 8743  Basecbs 17247  s cress 17274  +gcplusg 17297  .rcmulr 17298   /s cqus 17550   ×s cxps 17551  -gcsg 18953   ~QG cqg 19140  Rngcrng 20149  1rcur 20178  Ringcrg 20230   RngIso crngim 20435  2Idealc2idl 21259
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-tp 4631  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-1o 8506  df-2o 8507  df-er 8745  df-ec 8747  df-qs 8751  df-map 8868  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  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-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-hom 17321  df-cco 17322  df-0g 17486  df-prds 17492  df-imas 17553  df-qus 17554  df-xps 17555  df-mgm 18653  df-mgmhm 18705  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-sbg 18956  df-subg 19141  df-nsg 19142  df-eqg 19143  df-ghm 19231  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-rnghm 20436  df-rngim 20437  df-subrng 20546  df-lss 20930  df-sra 21172  df-rgmod 21173  df-lidl 21218  df-2idl 21260
This theorem is referenced by:  ring2idlqus1  21329
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