Theorem rngqipring1 21612

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 20563	. 2 ⊢ (𝜑 → (1r‘𝑃) = ⟨(1r‘𝑄), (1r‘𝐽)⟩)
5		rngqiprngfu.e	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ (1r‘𝑄))
6	5	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐸 ∈ (1r‘𝑄))
7		eleq2 2850	. . . . . . . . . . 11 ⊢ ((1r‘𝑄) = [𝑥] ∼ → (𝐸 ∈ (1r‘𝑄) ↔ 𝐸 ∈ [𝑥] ∼ ))
8	7	adantl 487	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (1r‘𝑄) = [𝑥] ∼ ) → (𝐸 ∈ (1r‘𝑄) ↔ 𝐸 ∈ [𝑥] ∼ ))
9		elecg 8762	. . . . . . . . . . . . 13 ⊢ ((𝐸 ∈ (1r‘𝑄) ∧ 𝑥 ∈ 𝐵) → (𝐸 ∈ [𝑥] ∼ ↔ 𝑥 ∼ 𝐸))
10	5, 9	sylan 592	. . . . . . . . . . . 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 20514	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐽 ∈ Ring → 𝐽 ∈ Rng)
15	3, 14	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐽 ∈ Rng)
16	13, 15	eqeltrrid 2866	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑅 ↾s 𝐼) ∈ Rng)
17	11, 12, 16	rng2idlnsg 21560	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))
18		nsgsubg 19368	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐼 ∈ (NrmSGrp‘𝑅) → 𝐼 ∈ (SubGrp‘𝑅))
19	17, 18	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐼 ∈ (SubGrp‘𝑅))
20	19	adantr 486	. . . . . . . . . . . . . . . . 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 19390	. . . . . . . . . . . . . . . . 17 ⊢ (𝐼 ∈ (SubGrp‘𝑅) → ∼ Er 𝐵)
24	20, 23	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∼ Er 𝐵)
25		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
26	24, 25	erth 8772	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 ∼ 𝐸 ↔ [𝑥] ∼ = [𝐸] ∼ ))
27	26	biimpa 482	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑥 ∼ 𝐸) → [𝑥] ∼ = [𝐸] ∼ )
28	27	eqcomd 2767	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑥 ∼ 𝐸) → [𝐸] ∼ = [𝑥] ∼ )
29	28	ex 418	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 ∼ 𝐸 → [𝐸] ∼ = [𝑥] ∼ ))
30	10, 29	sylbid 243	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐸 ∈ [𝑥] ∼ → [𝐸] ∼ = [𝑥] ∼ ))
31	30	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (1r‘𝑄) = [𝑥] ∼ ) → (𝐸 ∈ [𝑥] ∼ → [𝐸] ∼ = [𝑥] ∼ ))
32	8, 31	sylbid 243	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (1r‘𝑄) = [𝑥] ∼ ) → (𝐸 ∈ (1r‘𝑄) → [𝐸] ∼ = [𝑥] ∼ ))
33	32	ex 418	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((1r‘𝑄) = [𝑥] ∼ → (𝐸 ∈ (1r‘𝑄) → [𝐸] ∼ = [𝑥] ∼ )))
34	6, 33	mpid 45	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((1r‘𝑄) = [𝑥] ∼ → [𝐸] ∼ = [𝑥] ∼ ))
35	34	imp 412	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (1r‘𝑄) = [𝑥] ∼ ) → [𝐸] ∼ = [𝑥] ∼ )
36		simpr 490	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (1r‘𝑄) = [𝑥] ∼ ) → (1r‘𝑄) = [𝑥] ∼ )
37	35, 36	eqtr4d 2799	. . . . 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 21607	. . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 (1r‘𝑄) = [𝑥] ∼ )
42	37, 41	r19.29a 3171	. . . 4 ⊢ (𝜑 → [𝐸] ∼ = (1r‘𝑄))
43	42	eqcomd 2767	. . 3 ⊢ (𝜑 → (1r‘𝑄) = [𝐸] ∼ )
44	39	eqcomi 2770	. . . 4 ⊢ (1r‘𝐽) = 1
45	44	a1i 11	. . 3 ⊢ (𝜑 → (1r‘𝐽) = 1 )
46	43, 45	opeq12d 4841	. 2 ⊢ (𝜑 → ⟨(1r‘𝑄), (1r‘𝐽)⟩ = ⟨[𝐸] ∼ , 1 ⟩)
47	4, 46	eqtrd 2796	1 ⊢ (𝜑 → (1r‘𝑃) = ⟨[𝐸] ∼ , 1 ⟩)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ⟨cop 4590   class class class wbr 5103  ‘cfv 6538  (class class class)co 7420   Er wer 8714  [cec 8715  Basecbs 17387   ↾_s cress 17408  +_gcplusg 17428  ._rcmulr 17429   /_s cqus 17677   ×_s cxps 17678  -_gcsg 19146  SubGrpcsubg 19330  NrmSGrpcnsg 19331   ~_QG cqg 19332  Rngcrng 20374  1_rcur 20407  Ringcrg 20459  2Idealc2idl 21542

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-er 8717  df-ec 8719  df-qs 8723  df-map 8849  df-ixp 8926  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-sup 9434  df-inf 9435  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-7 12410  df-8 12411  df-9 12412  df-n0 12607  df-z 12694  df-dec 12815  df-uz 12966  df-fz 13640  df-struct 17325  df-sets 17342  df-slot 17360  df-ndx 17372  df-base 17388  df-ress 17409  df-plusg 17441  df-mulr 17442  df-sca 17444  df-vsca 17445  df-ip 17446  df-tset 17447  df-ple 17448  df-ds 17450  df-hom 17452  df-cco 17453  df-0g 17612  df-prds 17618  df-imas 17680  df-qus 17681  df-xps 17682  df-mgm 18816  df-sgrp 18908  df-mnd 18924  df-grp 19147  df-minusg 19148  df-subg 19333  df-nsg 19334  df-eqg 19335  df-cmn 19996  df-abl 19997  df-mgp 20361  df-rng 20375  df-ur 20408  df-ring 20461  df-subrng 20798  df-lss 21207  df-sra 21448  df-rgmod 21449  df-lidl 21486  df-2idl 21543

This theorem is used by:  rngqiprngu  21614