Theorem rngqiprngu 21607

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	⊢ (𝜑 → 𝑅 ∈ Rng)
rngqiprngfu.i	⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))
rngqiprngfu.j	⊢ 𝐽 = (𝑅 ↾s 𝐼)
rngqiprngfu.u	⊢ (𝜑 → 𝐽 ∈ Ring)
rngqiprngfu.b	⊢ 𝐵 = (Base‘𝑅)
rngqiprngfu.t	⊢ · = (.r‘𝑅)
rngqiprngfu.1	⊢ 1 = (1r‘𝐽)
rngqiprngfu.g	⊢ ∼ = (𝑅 ~QG 𝐼)
rngqiprngfu.q	⊢ 𝑄 = (𝑅 /s ∼ )
rngqiprngfu.v	⊢ (𝜑 → 𝑄 ∈ Ring)
rngqiprngfu.e	⊢ (𝜑 → 𝐸 ∈ (1r‘𝑄))
rngqiprngfu.m	⊢ − = (-g‘𝑅)
rngqiprngfu.a	⊢ + = (+g‘𝑅)
rngqiprngfu.n	⊢ 𝑈 = ((𝐸 − ( 1 · 𝐸)) + 1 )

Assertion

Ref	Expression
rngqiprngu	⊢ (𝜑 → (1r‘𝑅) = 𝑈)

Proof of Theorem rngqiprngu

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2761	. . . 4 ⊢ (𝑄 ×s 𝐽) = (𝑄 ×s 𝐽)
2		rngqiprngfu.v	. . . 4 ⊢ (𝜑 → 𝑄 ∈ Ring)
3		rngqiprngfu.u	. . . 4 ⊢ (𝜑 → 𝐽 ∈ Ring)
4	1, 2, 3	xpsringd 20555	. . 3 ⊢ (𝜑 → (𝑄 ×s 𝐽) ∈ Ring)
5		rngqiprngfu.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ Rng)
6		rngqiprngfu.i	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))
7		rngqiprngfu.j	. . . . 5 ⊢ 𝐽 = (𝑅 ↾s 𝐼)
8		rngqiprngfu.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑅)
9		rngqiprngfu.t	. . . . 5 ⊢ · = (.r‘𝑅)
10		rngqiprngfu.1	. . . . 5 ⊢ 1 = (1r‘𝐽)
11		rngqiprngfu.g	. . . . 5 ⊢ ∼ = (𝑅 ~QG 𝐼)
12		rngqiprngfu.q	. . . . 5 ⊢ 𝑄 = (𝑅 /s ∼ )
13		eqid 2761	. . . . 5 ⊢ (Base‘𝑄) = (Base‘𝑄)
14		eqid 2761	. . . . 5 ⊢ (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩) = (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)
15	5, 6, 7, 3, 8, 9, 10, 11, 12, 13, 1, 14	rngqiprngim 21593	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩) ∈ (𝑅 RngIso (𝑄 ×s 𝐽)))
16		rngimcnv 20679	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩) ∈ (𝑅 RngIso (𝑄 ×s 𝐽)) → ◡(𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩) ∈ ((𝑄 ×s 𝐽) RngIso 𝑅))
17	15, 16	syl 18	. . 3 ⊢ (𝜑 → ◡(𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩) ∈ ((𝑄 ×s 𝐽) RngIso 𝑅))
18		rngisomring1 20691	. . 3 ⊢ (((𝑄 ×s 𝐽) ∈ Ring ∧ 𝑅 ∈ Rng ∧ ◡(𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩) ∈ ((𝑄 ×s 𝐽) RngIso 𝑅)) → (1r‘𝑅) = (◡(𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)‘(1r‘(𝑄 ×s 𝐽))))
19	4, 5, 17, 18	syl3anc 1398	. 2 ⊢ (𝜑 → (1r‘𝑅) = (◡(𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)‘(1r‘(𝑄 ×s 𝐽))))
20		rngqiprngfu.e	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ (1r‘𝑄))
21		rngqiprngfu.m	. . . . 5 ⊢ − = (-g‘𝑅)
22		rngqiprngfu.a	. . . . 5 ⊢ + = (+g‘𝑅)
23		rngqiprngfu.n	. . . . 5 ⊢ 𝑈 = ((𝐸 − ( 1 · 𝐸)) + 1 )
24	5, 6, 7, 3, 8, 9, 10, 11, 12, 2, 20, 21, 22, 23, 14	rngqiprngfu 21606	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)‘𝑈) = ⟨[𝐸] ∼ , 1 ⟩)
25	5, 6, 7, 3, 8, 9, 10, 11, 12, 2, 20, 21, 22, 23, 1	rngqipring1 21605	. . . 4 ⊢ (𝜑 → (1r‘(𝑄 ×s 𝐽)) = ⟨[𝐸] ∼ , 1 ⟩)
26	24, 25	eqtr4d 2799	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)‘𝑈) = (1r‘(𝑄 ×s 𝐽)))
27		eqid 2761	. . . . . 6 ⊢ (Base‘(𝑄 ×s 𝐽)) = (Base‘(𝑄 ×s 𝐽))
28	8, 27	rngimf1o 20677	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩) ∈ (𝑅 RngIso (𝑄 ×s 𝐽)) → (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩):𝐵–1-1-onto→(Base‘(𝑄 ×s 𝐽)))
29	15, 28	syl 18	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩):𝐵–1-1-onto→(Base‘(𝑄 ×s 𝐽)))
30	5, 6, 7, 3, 8, 9, 10, 11, 12, 2, 20, 21, 22, 23	rngqiprngfulem3 21602	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝐵)
31		eqid 2761	. . . . . 6 ⊢ (1r‘(𝑄 ×s 𝐽)) = (1r‘(𝑄 ×s 𝐽))
32	27, 31	ringidcl 20487	. . . . 5 ⊢ ((𝑄 ×s 𝐽) ∈ Ring → (1r‘(𝑄 ×s 𝐽)) ∈ (Base‘(𝑄 ×s 𝐽)))
33	4, 32	syl 18	. . . 4 ⊢ (𝜑 → (1r‘(𝑄 ×s 𝐽)) ∈ (Base‘(𝑄 ×s 𝐽)))
34		f1ocnvfvb 7285	. . . 4 ⊢ (((𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩):𝐵–1-1-onto→(Base‘(𝑄 ×s 𝐽)) ∧ 𝑈 ∈ 𝐵 ∧ (1r‘(𝑄 ×s 𝐽)) ∈ (Base‘(𝑄 ×s 𝐽))) → (((𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)‘𝑈) = (1r‘(𝑄 ×s 𝐽)) ↔ (◡(𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)‘(1r‘(𝑄 ×s 𝐽))) = 𝑈))
35	29, 30, 33, 34	syl3anc 1398	. . 3 ⊢ (𝜑 → (((𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)‘𝑈) = (1r‘(𝑄 ×s 𝐽)) ↔ (◡(𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)‘(1r‘(𝑄 ×s 𝐽))) = 𝑈))
36	26, 35	mpbid 235	. 2 ⊢ (𝜑 → (◡(𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)‘(1r‘(𝑄 ×s 𝐽))) = 𝑈)
37	19, 36	eqtrd 2796	1 ⊢ (𝜑 → (1r‘𝑅) = 𝑈)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2145  ⟨cop 4590   ↦ cmpt 5186  ^◡ccnv 5650  –1-1-onto→wf1o 6536  ‘cfv 6537  (class class class)co 7418  [cec 8708  Basecbs 17380   ↾_s cress 17401  +_gcplusg 17421  ._rcmulr 17422   /_s cqus 17670   ×_s cxps 17671  -_gcsg 19139   ~_QG cqg 19325  Rngcrng 20367  1_rcur 20400  Ringcrg 20452   RngIso crngim 20658  2Idealc2idl 21535

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-1st 7999  df-2nd 8000  df-tpos 8236  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-2o 8470  df-er 8710  df-ec 8712  df-qs 8716  df-map 8842  df-ixp 8919  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-sup 9427  df-inf 9428  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-nn 12329  df-2 12398  df-3 12399  df-4 12400  df-5 12401  df-6 12402  df-7 12403  df-8 12404  df-9 12405  df-n0 12600  df-z 12687  df-dec 12808  df-uz 12959  df-fz 13633  df-struct 17318  df-sets 17335  df-slot 17353  df-ndx 17365  df-base 17381  df-ress 17402  df-plusg 17434  df-mulr 17435  df-sca 17437  df-vsca 17438  df-ip 17439  df-tset 17440  df-ple 17441  df-ds 17443  df-hom 17445  df-cco 17446  df-0g 17605  df-prds 17611  df-imas 17673  df-qus 17674  df-xps 17675  df-mgm 18809  df-mgmhm 18874  df-sgrp 18901  df-mnd 18917  df-grp 19140  df-minusg 19141  df-sbg 19142  df-subg 19326  df-nsg 19327  df-eqg 19328  df-ghm 19421  df-cmn 19989  df-abl 19990  df-mgp 20354  df-rng 20368  df-ur 20401  df-ring 20454  df-oppr 20560  df-dvdsr 20580  df-unit 20581  df-invr 20611  df-rnghm 20659  df-rngim 20660  df-subrng 20791  df-lss 21200  df-sra 21441  df-rgmod 21442  df-lidl 21479  df-2idl 21536

This theorem is used by:  ring2idlqus1  21608