Theorem rngqipring1 21305

Description: The ring unity of the product of the quotient with a two-sided ideal and the two-sided ideal, which both are rings. (Contributed by AV, 16-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 )
rngqipring1.p	⊢ 𝑃 = (𝑄 ×s 𝐽)

Assertion

Ref	Expression
rngqipring1	⊢ (𝜑 → (1r‘𝑃) = ⟨[𝐸] ∼ , 1 ⟩)

Proof of Theorem rngqipring1

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rngqipring1.p	. . 3 ⊢ 𝑃 = (𝑄 ×s 𝐽)
2		rngqiprngfu.v	. . 3 ⊢ (𝜑 → 𝑄 ∈ Ring)
3		rngqiprngfu.u	. . 3 ⊢ (𝜑 → 𝐽 ∈ Ring)
4	1, 2, 3	xpsring1d 20312	. 2 ⊢ (𝜑 → (1r‘𝑃) = ⟨(1r‘𝑄), (1r‘𝐽)⟩)
5		rngqiprngfu.e	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ (1r‘𝑄))
6	5	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐸 ∈ (1r‘𝑄))
7		eleq2 2815	. . . . . . . . . . 11 ⊢ ((1r‘𝑄) = [𝑥] ∼ → (𝐸 ∈ (1r‘𝑄) ↔ 𝐸 ∈ [𝑥] ∼ ))
8	7	adantl 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (1r‘𝑄) = [𝑥] ∼ ) → (𝐸 ∈ (1r‘𝑄) ↔ 𝐸 ∈ [𝑥] ∼ ))
9		elecg 8778	. . . . . . . . . . . . 13 ⊢ ((𝐸 ∈ (1r‘𝑄) ∧ 𝑥 ∈ 𝐵) → (𝐸 ∈ [𝑥] ∼ ↔ 𝑥 ∼ 𝐸))
10	5, 9	sylan 578	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐸 ∈ [𝑥] ∼ ↔ 𝑥 ∼ 𝐸))
11		rngqiprngfu.r	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑅 ∈ Rng)
12		rngqiprngfu.i	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))
13		rngqiprngfu.j	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐽 = (𝑅 ↾s 𝐼)
14		ringrng 20264	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐽 ∈ Ring → 𝐽 ∈ Rng)
15	3, 14	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐽 ∈ Rng)
16	13, 15	eqeltrrid 2831	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑅 ↾s 𝐼) ∈ Rng)
17	11, 12, 16	rng2idlnsg 21255	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))
18		nsgsubg 19152	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐼 ∈ (NrmSGrp‘𝑅) → 𝐼 ∈ (SubGrp‘𝑅))
19	17, 18	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐼 ∈ (SubGrp‘𝑅))
20	19	adantr 479	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐼 ∈ (SubGrp‘𝑅))
21		rngqiprngfu.b	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐵 = (Base‘𝑅)
22		rngqiprngfu.g	. . . . . . . . . . . . . . . . . 18 ⊢ ∼ = (𝑅 ~QG 𝐼)
23	21, 22	eqger 19172	. . . . . . . . . . . . . . . . 17 ⊢ (𝐼 ∈ (SubGrp‘𝑅) → ∼ Er 𝐵)
24	20, 23	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∼ Er 𝐵)
25		simpr 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
26	24, 25	erth 8785	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 ∼ 𝐸 ↔ [𝑥] ∼ = [𝐸] ∼ ))
27	26	biimpa 475	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑥 ∼ 𝐸) → [𝑥] ∼ = [𝐸] ∼ )
28	27	eqcomd 2732	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑥 ∼ 𝐸) → [𝐸] ∼ = [𝑥] ∼ )
29	28	ex 411	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 ∼ 𝐸 → [𝐸] ∼ = [𝑥] ∼ ))
30	10, 29	sylbid 239	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐸 ∈ [𝑥] ∼ → [𝐸] ∼ = [𝑥] ∼ ))
31	30	adantr 479	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (1r‘𝑄) = [𝑥] ∼ ) → (𝐸 ∈ [𝑥] ∼ → [𝐸] ∼ = [𝑥] ∼ ))
32	8, 31	sylbid 239	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (1r‘𝑄) = [𝑥] ∼ ) → (𝐸 ∈ (1r‘𝑄) → [𝐸] ∼ = [𝑥] ∼ ))
33	32	ex 411	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((1r‘𝑄) = [𝑥] ∼ → (𝐸 ∈ (1r‘𝑄) → [𝐸] ∼ = [𝑥] ∼ )))
34	6, 33	mpid 44	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((1r‘𝑄) = [𝑥] ∼ → [𝐸] ∼ = [𝑥] ∼ ))
35	34	imp 405	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (1r‘𝑄) = [𝑥] ∼ ) → [𝐸] ∼ = [𝑥] ∼ )
36		simpr 483	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (1r‘𝑄) = [𝑥] ∼ ) → (1r‘𝑄) = [𝑥] ∼ )
37	35, 36	eqtr4d 2769	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (1r‘𝑄) = [𝑥] ∼ ) → [𝐸] ∼ = (1r‘𝑄))
38		rngqiprngfu.t	. . . . . 6 ⊢ · = (.r‘𝑅)
39		rngqiprngfu.1	. . . . . 6 ⊢ 1 = (1r‘𝐽)
40		rngqiprngfu.q	. . . . . 6 ⊢ 𝑄 = (𝑅 /s ∼ )
41	11, 12, 13, 3, 21, 38, 39, 22, 40, 2	rngqiprngfulem1 21300	. . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 (1r‘𝑄) = [𝑥] ∼ )
42	37, 41	r19.29a 3152	. . . 4 ⊢ (𝜑 → [𝐸] ∼ = (1r‘𝑄))
43	42	eqcomd 2732	. . 3 ⊢ (𝜑 → (1r‘𝑄) = [𝐸] ∼ )
44	39	eqcomi 2735	. . . 4 ⊢ (1r‘𝐽) = 1
45	44	a1i 11	. . 3 ⊢ (𝜑 → (1r‘𝐽) = 1 )
46	43, 45	opeq12d 4887	. 2 ⊢ (𝜑 → ⟨(1r‘𝑄), (1r‘𝐽)⟩ = ⟨[𝐸] ∼ , 1 ⟩)
47	4, 46	eqtrd 2766	1 ⊢ (𝜑 → (1r‘𝑃) = ⟨[𝐸] ∼ , 1 ⟩)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1534   ∈ wcel 2099  ⟨cop 4639   class class class wbr 5153  ‘cfv 6554  (class class class)co 7424   Er wer 8731  [cec 8732  Basecbs 17213   ↾_s cress 17242  +_gcplusg 17266  ._rcmulr 17267   /_s cqus 17520   ×_s cxps 17521  -_gcsg 18930  SubGrpcsubg 19114  NrmSGrpcnsg 19115   ~_QG cqg 19116  Rngcrng 20135  1_rcur 20164  Ringcrg 20216  2Idealc2idl 21238

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-tp 4638  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-2o 8497  df-er 8734  df-ec 8736  df-qs 8740  df-map 8857  df-ixp 8927  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-sup 9485  df-inf 9486  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  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 12611  df-dec 12730  df-uz 12875  df-fz 13539  df-struct 17149  df-sets 17166  df-slot 17184  df-ndx 17196  df-base 17214  df-ress 17243  df-plusg 17279  df-mulr 17280  df-sca 17282  df-vsca 17283  df-ip 17284  df-tset 17285  df-ple 17286  df-ds 17288  df-hom 17290  df-cco 17291  df-0g 17456  df-prds 17462  df-imas 17523  df-qus 17524  df-xps 17525  df-mgm 18633  df-sgrp 18712  df-mnd 18728  df-grp 18931  df-minusg 18932  df-subg 19117  df-nsg 19118  df-eqg 19119  df-cmn 19780  df-abl 19781  df-mgp 20118  df-rng 20136  df-ur 20165  df-ring 20218  df-subrng 20528  df-lss 20909  df-sra 21151  df-rgmod 21152  df-lidl 21197  df-2idl 21239

This theorem is referenced by:  rngqiprngu  21307