Theorem rngqipring1 21627

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 20578	. 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 2853	. . . . . . . . . . 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 8776	. . . . . . . . . . . . 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 20529	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐽 ∈ Ring → 𝐽 ∈ Rng)
15	3, 14	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐽 ∈ Rng)
16	13, 15	eqeltrrid 2869	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑅 ↾s 𝐼) ∈ Rng)
17	11, 12, 16	rng2idlnsg 21575	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))
18		nsgsubg 19383	. . . . . . . . . . . . . . . . . . 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 19405	. . . . . . . . . . . . . . . . 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 8786	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 ∼ 𝐸 ↔ [𝑥] ∼ = [𝐸] ∼ ))
27	26	biimpa 482	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑥 ∼ 𝐸) → [𝑥] ∼ = [𝐸] ∼ )
28	27	eqcomd 2769	. . . . . . . . . . . . 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 2801	. . . . 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 21622	. . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 (1r‘𝑄) = [𝑥] ∼ )
42	37, 41	r19.29a 3174	. . . 4 ⊢ (𝜑 → [𝐸] ∼ = (1r‘𝑄))
43	42	eqcomd 2769	. . 3 ⊢ (𝜑 → (1r‘𝑄) = [𝐸] ∼ )
44	39	eqcomi 2772	. . . 4 ⊢ (1r‘𝐽) = 1
45	44	a1i 11	. . 3 ⊢ (𝜑 → (1r‘𝐽) = 1 )
46	43, 45	opeq12d 4851	. 2 ⊢ (𝜑 → ⟨(1r‘𝑄), (1r‘𝐽)⟩ = ⟨[𝐸] ∼ , 1 ⟩)
47	4, 46	eqtrd 2798	1 ⊢ (𝜑 → (1r‘𝑃) = ⟨[𝐸] ∼ , 1 ⟩)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2146  ⟨cop 4838   class class class wbr 5106  ‘cfv 6547  (class class class)co 7430   Er wer 8728  [cec 8729  Basecbs 17402   ↾_s cress 17423  +_gcplusg 17443  ._rcmulr 17444   /_s cqus 17692   ×_s cxps 17693  -_gcsg 19161  SubGrpcsubg 19345  NrmSGrpcnsg 19346   ~_QG cqg 19347  Rngcrng 20389  1_rcur 20422  Ringcrg 20474  2Idealc2idl 21557

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7761  ax-cnex 11269  ax-resscn 11270  ax-1cn 11271  ax-icn 11272  ax-addcl 11273  ax-addrcl 11274  ax-mulcl 11275  ax-mulrcl 11276  ax-mulcom 11277  ax-addass 11278  ax-mulass 11279  ax-distr 11280  ax-i2m1 11281  ax-1ne0 11282  ax-1rid 11283  ax-rnegex 11284  ax-rrecex 11285  ax-cnre 11286  ax-pre-lttri 11287  ax-pre-lttrn 11288  ax-pre-ltadd 11289  ax-pre-mulgt0 11290

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6311  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7888  df-1st 8011  df-2nd 8012  df-frecs 8313  df-wrecs 8344  df-recs 8393  df-rdg 8432  df-1o 8490  df-2o 8491  df-er 8731  df-ec 8733  df-qs 8737  df-map 8863  df-ixp 8940  df-en 8988  df-dom 8989  df-sdom 8990  df-fin 8991  df-sup 9447  df-inf 9448  df-pnf 11358  df-mnf 11359  df-xr 11360  df-ltxr 11361  df-le 11362  df-sub 11556  df-neg 11557  df-nn 12349  df-2 12418  df-3 12419  df-4 12420  df-5 12421  df-6 12422  df-7 12423  df-8 12424  df-9 12425  df-n0 12620  df-z 12707  df-dec 12828  df-uz 12979  df-fz 13653  df-struct 17340  df-sets 17357  df-slot 17375  df-ndx 17387  df-base 17403  df-ress 17424  df-plusg 17456  df-mulr 17457  df-sca 17459  df-vsca 17460  df-ip 17461  df-tset 17462  df-ple 17463  df-ds 17465  df-hom 17467  df-cco 17468  df-0g 17627  df-prds 17633  df-imas 17695  df-qus 17696  df-xps 17697  df-mgm 18831  df-sgrp 18923  df-mnd 18939  df-grp 19162  df-minusg 19163  df-subg 19348  df-nsg 19349  df-eqg 19350  df-cmn 20011  df-abl 20012  df-mgp 20376  df-rng 20390  df-ur 20423  df-ring 20476  df-subrng 20813  df-lss 21222  df-sra 21463  df-rgmod 21464  df-lidl 21501  df-2idl 21558

This theorem is used by:  rngqiprngu  21629