Theorem rngqiprngu 21376

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 20368	. . 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 21362	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩) ∈ (𝑅 RngIso (𝑄 ×s 𝐽)))
16		rngimcnv 20492	. . . 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 20504	. . 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 1389	. 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 21375	. . . 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 21374	. . . 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 20490	. . . . 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 21371	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝐵)
31		eqid 2761	. . . . . 6 ⊢ (1r‘(𝑄 ×s 𝐽)) = (1r‘(𝑄 ×s 𝐽))
32	27, 31	ringidcl 20302	. . . . 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 7258	. . . 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 1389	. . 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 234	. 2 ⊢ (𝜑 → (◡(𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)‘(1r‘(𝑄 ×s 𝐽))) = 𝑈)
37	19, 36	eqtrd 2796	1 ⊢ (𝜑 → (1r‘𝑅) = 𝑈)

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1559   ∈ wcel 2141  ⟨cop 4585   ↦ cmpt 5178  ^◡ccnv 5642  –1-1-onto→wf1o 6515  ‘cfv 6516  (class class class)co 7391  [cec 8670  Basecbs 17236   ↾_s cress 17257  +_gcplusg 17277  ._rcmulr 17278   /_s cqus 17526   ×_s cxps 17527  -_gcsg 18968   ~_QG cqg 19155  Rngcrng 20189  1_rcur 20218  Ringcrg 20270   RngIso crngim 20471  2Idealc2idl 21307

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713  ax-cnex 11123  ax-resscn 11124  ax-1cn 11125  ax-icn 11126  ax-addcl 11127  ax-addrcl 11128  ax-mulcl 11129  ax-mulrcl 11130  ax-mulcom 11131  ax-addass 11132  ax-mulass 11133  ax-distr 11134  ax-i2m1 11135  ax-1ne0 11136  ax-1rid 11137  ax-rnegex 11138  ax-rrecex 11139  ax-cnre 11140  ax-pre-lttri 11141  ax-pre-lttrn 11142  ax-pre-ltadd 11143  ax-pre-mulgt0 11144

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6283  df-ord 6344  df-on 6345  df-lim 6346  df-suc 6347  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7348  df-ov 7394  df-oprab 7395  df-mpo 7396  df-om 7842  df-1st 7965  df-2nd 7966  df-tpos 8200  df-frecs 8256  df-wrecs 8287  df-recs 8336  df-rdg 8375  df-1o 8431  df-2o 8432  df-er 8672  df-ec 8674  df-qs 8678  df-map 8804  df-ixp 8874  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-sup 9382  df-inf 9383  df-pnf 11212  df-mnf 11213  df-xr 11214  df-ltxr 11215  df-le 11216  df-sub 11410  df-neg 11411  df-nn 12205  df-2 12274  df-3 12275  df-4 12276  df-5 12277  df-6 12278  df-7 12279  df-8 12280  df-9 12281  df-n0 12476  df-z 12563  df-dec 12683  df-uz 12834  df-fz 13507  df-struct 17174  df-sets 17191  df-slot 17209  df-ndx 17221  df-base 17237  df-ress 17258  df-plusg 17290  df-mulr 17291  df-sca 17293  df-vsca 17294  df-ip 17295  df-tset 17296  df-ple 17297  df-ds 17299  df-hom 17301  df-cco 17302  df-0g 17461  df-prds 17467  df-imas 17529  df-qus 17530  df-xps 17531  df-mgm 18665  df-mgmhm 18717  df-sgrp 18744  df-mnd 18760  df-grp 18969  df-minusg 18970  df-sbg 18971  df-subg 19156  df-nsg 19157  df-eqg 19158  df-ghm 19245  df-cmn 19813  df-abl 19814  df-mgp 20178  df-rng 20190  df-ur 20219  df-ring 20272  df-oppr 20373  df-dvdsr 20393  df-unit 20394  df-invr 20424  df-rnghm 20472  df-rngim 20473  df-subrng 20583  df-lss 20987  df-sra 21228  df-rgmod 21229  df-lidl 21266  df-2idl 21308

This theorem is referenced by:  ring2idlqus1  21377