Theorem rngqiprngu 21261

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 2731	. . . 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 20256	. . 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 2731	. . . . 5 ⊢ (Base‘𝑄) = (Base‘𝑄)
14		eqid 2731	. . . . 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 21247	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩) ∈ (𝑅 RngIso (𝑄 ×s 𝐽)))
16		rngimcnv 20380	. . . 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 17	. . 3 ⊢ (𝜑 → ◡(𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩) ∈ ((𝑄 ×s 𝐽) RngIso 𝑅))
18		rngisomring1 20392	. . 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 1373	. 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 21260	. . . 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 21259	. . . 4 ⊢ (𝜑 → (1r‘(𝑄 ×s 𝐽)) = ⟨[𝐸] ∼ , 1 ⟩)
26	24, 25	eqtr4d 2769	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)‘𝑈) = (1r‘(𝑄 ×s 𝐽)))
27		eqid 2731	. . . . . 6 ⊢ (Base‘(𝑄 ×s 𝐽)) = (Base‘(𝑄 ×s 𝐽))
28	8, 27	rngimf1o 20378	. . . . 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 17	. . . 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 21256	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝐵)
31		eqid 2731	. . . . . 6 ⊢ (1r‘(𝑄 ×s 𝐽)) = (1r‘(𝑄 ×s 𝐽))
32	27, 31	ringidcl 20189	. . . . 5 ⊢ ((𝑄 ×s 𝐽) ∈ Ring → (1r‘(𝑄 ×s 𝐽)) ∈ (Base‘(𝑄 ×s 𝐽)))
33	4, 32	syl 17	. . . 4 ⊢ (𝜑 → (1r‘(𝑄 ×s 𝐽)) ∈ (Base‘(𝑄 ×s 𝐽)))
34		f1ocnvfvb 7219	. . . 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 1373	. . 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 232	. 2 ⊢ (𝜑 → (◡(𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)‘(1r‘(𝑄 ×s 𝐽))) = 𝑈)
37	19, 36	eqtrd 2766	1 ⊢ (𝜑 → (1r‘𝑅) = 𝑈)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2111  ⟨cop 4581   ↦ cmpt 5174  ^◡ccnv 5618  –1-1-onto→wf1o 6486  ‘cfv 6487  (class class class)co 7352  [cec 8626  Basecbs 17126   ↾_s cress 17147  +_gcplusg 17167  ._rcmulr 17168   /_s cqus 17415   ×_s cxps 17416  -_gcsg 18854   ~_QG cqg 19041  Rngcrng 20076  1_rcur 20105  Ringcrg 20157   RngIso crngim 20359  2Idealc2idl 21192

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-cnex 11068  ax-resscn 11069  ax-1cn 11070  ax-icn 11071  ax-addcl 11072  ax-addrcl 11073  ax-mulcl 11074  ax-mulrcl 11075  ax-mulcom 11076  ax-addass 11077  ax-mulass 11078  ax-distr 11079  ax-i2m1 11080  ax-1ne0 11081  ax-1rid 11082  ax-rnegex 11083  ax-rrecex 11084  ax-cnre 11085  ax-pre-lttri 11086  ax-pre-lttrn 11087  ax-pre-ltadd 11088  ax-pre-mulgt0 11089

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-tp 4580  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-1st 7927  df-2nd 7928  df-tpos 8162  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-2o 8392  df-er 8628  df-ec 8630  df-qs 8634  df-map 8758  df-ixp 8828  df-en 8876  df-dom 8877  df-sdom 8878  df-fin 8879  df-sup 9332  df-inf 9333  df-pnf 11154  df-mnf 11155  df-xr 11156  df-ltxr 11157  df-le 11158  df-sub 11352  df-neg 11353  df-nn 12132  df-2 12194  df-3 12195  df-4 12196  df-5 12197  df-6 12198  df-7 12199  df-8 12200  df-9 12201  df-n0 12388  df-z 12475  df-dec 12595  df-uz 12739  df-fz 13414  df-struct 17064  df-sets 17081  df-slot 17099  df-ndx 17111  df-base 17127  df-ress 17148  df-plusg 17180  df-mulr 17181  df-sca 17183  df-vsca 17184  df-ip 17185  df-tset 17186  df-ple 17187  df-ds 17189  df-hom 17191  df-cco 17192  df-0g 17351  df-prds 17357  df-imas 17418  df-qus 17419  df-xps 17420  df-mgm 18554  df-mgmhm 18606  df-sgrp 18633  df-mnd 18649  df-grp 18855  df-minusg 18856  df-sbg 18857  df-subg 19042  df-nsg 19043  df-eqg 19044  df-ghm 19131  df-cmn 19700  df-abl 19701  df-mgp 20065  df-rng 20077  df-ur 20106  df-ring 20159  df-oppr 20261  df-dvdsr 20281  df-unit 20282  df-invr 20312  df-rnghm 20360  df-rngim 20361  df-subrng 20467  df-lss 20871  df-sra 21113  df-rgmod 21114  df-lidl 21151  df-2idl 21193

This theorem is referenced by:  ring2idlqus1  21262