Theorem rngqiprngu 21346

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 2735	. . . 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 20346	. . 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 2735	. . . . 5 ⊢ (Base‘𝑄) = (Base‘𝑄)
14		eqid 2735	. . . . 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 21332	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩) ∈ (𝑅 RngIso (𝑄 ×s 𝐽)))
16		rngimcnv 20473	. . . 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 20485	. . 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 1370	. 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 21345	. . . 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 21344	. . . 4 ⊢ (𝜑 → (1r‘(𝑄 ×s 𝐽)) = ⟨[𝐸] ∼ , 1 ⟩)
26	24, 25	eqtr4d 2778	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)‘𝑈) = (1r‘(𝑄 ×s 𝐽)))
27		eqid 2735	. . . . . 6 ⊢ (Base‘(𝑄 ×s 𝐽)) = (Base‘(𝑄 ×s 𝐽))
28	8, 27	rngimf1o 20471	. . . . 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 21341	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝐵)
31		eqid 2735	. . . . . 6 ⊢ (1r‘(𝑄 ×s 𝐽)) = (1r‘(𝑄 ×s 𝐽))
32	27, 31	ringidcl 20280	. . . . 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 7299	. . . 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 1370	. . 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 2775	1 ⊢ (𝜑 → (1r‘𝑅) = 𝑈)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537   ∈ wcel 2106  ⟨cop 4637   ↦ cmpt 5231  ^◡ccnv 5688  –1-1-onto→wf1o 6562  ‘cfv 6563  (class class class)co 7431  [cec 8742  Basecbs 17245   ↾_s cress 17274  +_gcplusg 17298  ._rcmulr 17299   /_s cqus 17552   ×_s cxps 17553  -_gcsg 18966   ~_QG cqg 19153  Rngcrng 20170  1_rcur 20199  Ringcrg 20251   RngIso crngim 20452  2Idealc2idl 21277

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-tpos 8250  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-ec 8746  df-qs 8750  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-hom 17322  df-cco 17323  df-0g 17488  df-prds 17494  df-imas 17555  df-qus 17556  df-xps 17557  df-mgm 18666  df-mgmhm 18718  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-sbg 18969  df-subg 19154  df-nsg 19155  df-eqg 19156  df-ghm 19244  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-oppr 20351  df-dvdsr 20374  df-unit 20375  df-invr 20405  df-rnghm 20453  df-rngim 20454  df-subrng 20563  df-lss 20948  df-sra 21190  df-rgmod 21191  df-lidl 21236  df-2idl 21278

This theorem is referenced by:  ring2idlqus1  21347