Theorem rngqiprngu 21235

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 2730	. . . 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 20248	. . 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 2730	. . . . 5 ⊢ (Base‘𝑄) = (Base‘𝑄)
14		eqid 2730	. . . . 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 21221	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩) ∈ (𝑅 RngIso (𝑄 ×s 𝐽)))
16		rngimcnv 20372	. . . 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 20384	. . 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 21234	. . . 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 21233	. . . 4 ⊢ (𝜑 → (1r‘(𝑄 ×s 𝐽)) = ⟨[𝐸] ∼ , 1 ⟩)
26	24, 25	eqtr4d 2768	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)‘𝑈) = (1r‘(𝑄 ×s 𝐽)))
27		eqid 2730	. . . . . 6 ⊢ (Base‘(𝑄 ×s 𝐽)) = (Base‘(𝑄 ×s 𝐽))
28	8, 27	rngimf1o 20370	. . . . 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 21230	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝐵)
31		eqid 2730	. . . . . 6 ⊢ (1r‘(𝑄 ×s 𝐽)) = (1r‘(𝑄 ×s 𝐽))
32	27, 31	ringidcl 20181	. . . . 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 7257	. . . 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 2765	1 ⊢ (𝜑 → (1r‘𝑅) = 𝑈)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109  ⟨cop 4598   ↦ cmpt 5191  ^◡ccnv 5640  –1-1-onto→wf1o 6513  ‘cfv 6514  (class class class)co 7390  [cec 8672  Basecbs 17186   ↾_s cress 17207  +_gcplusg 17227  ._rcmulr 17228   /_s cqus 17475   ×_s cxps 17476  -_gcsg 18874   ~_QG cqg 19061  Rngcrng 20068  1_rcur 20097  Ringcrg 20149   RngIso crngim 20351  2Idealc2idl 21166

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-tpos 8208  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8674  df-ec 8676  df-qs 8680  df-map 8804  df-ixp 8874  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-sup 9400  df-inf 9401  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-3 12257  df-4 12258  df-5 12259  df-6 12260  df-7 12261  df-8 12262  df-9 12263  df-n0 12450  df-z 12537  df-dec 12657  df-uz 12801  df-fz 13476  df-struct 17124  df-sets 17141  df-slot 17159  df-ndx 17171  df-base 17187  df-ress 17208  df-plusg 17240  df-mulr 17241  df-sca 17243  df-vsca 17244  df-ip 17245  df-tset 17246  df-ple 17247  df-ds 17249  df-hom 17251  df-cco 17252  df-0g 17411  df-prds 17417  df-imas 17478  df-qus 17479  df-xps 17480  df-mgm 18574  df-mgmhm 18626  df-sgrp 18653  df-mnd 18669  df-grp 18875  df-minusg 18876  df-sbg 18877  df-subg 19062  df-nsg 19063  df-eqg 19064  df-ghm 19152  df-cmn 19719  df-abl 19720  df-mgp 20057  df-rng 20069  df-ur 20098  df-ring 20151  df-oppr 20253  df-dvdsr 20273  df-unit 20274  df-invr 20304  df-rnghm 20352  df-rngim 20353  df-subrng 20462  df-lss 20845  df-sra 21087  df-rgmod 21088  df-lidl 21125  df-2idl 21167

This theorem is referenced by:  ring2idlqus1  21236