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Theorem rngqipring1 21327
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 (𝜑𝑅 ∈ Rng)
rngqiprngfu.i (𝜑𝐼 ∈ (2Ideal‘𝑅))
rngqiprngfu.j 𝐽 = (𝑅s 𝐼)
rngqiprngfu.u (𝜑𝐽 ∈ Ring)
rngqiprngfu.b 𝐵 = (Base‘𝑅)
rngqiprngfu.t · = (.r𝑅)
rngqiprngfu.1 1 = (1r𝐽)
rngqiprngfu.g = (𝑅 ~QG 𝐼)
rngqiprngfu.q 𝑄 = (𝑅 /s )
rngqiprngfu.v (𝜑𝑄 ∈ Ring)
rngqiprngfu.e (𝜑𝐸 ∈ (1r𝑄))
rngqiprngfu.m = (-g𝑅)
rngqiprngfu.a + = (+g𝑅)
rngqiprngfu.n 𝑈 = ((𝐸 ( 1 · 𝐸)) + 1 )
rngqipring1.p 𝑃 = (𝑄 ×s 𝐽)
Assertion
Ref Expression
rngqipring1 (𝜑 → (1r𝑃) = ⟨[𝐸] , 1 ⟩)

Proof of Theorem rngqipring1
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 rngqipring1.p . . 3 𝑃 = (𝑄 ×s 𝐽)
2 rngqiprngfu.v . . 3 (𝜑𝑄 ∈ Ring)
3 rngqiprngfu.u . . 3 (𝜑𝐽 ∈ Ring)
41, 2, 3xpsring1d 20331 . 2 (𝜑 → (1r𝑃) = ⟨(1r𝑄), (1r𝐽)⟩)
5 rngqiprngfu.e . . . . . . . . 9 (𝜑𝐸 ∈ (1r𝑄))
65adantr 480 . . . . . . . 8 ((𝜑𝑥𝐵) → 𝐸 ∈ (1r𝑄))
7 eleq2 2829 . . . . . . . . . . 11 ((1r𝑄) = [𝑥] → (𝐸 ∈ (1r𝑄) ↔ 𝐸 ∈ [𝑥] ))
87adantl 481 . . . . . . . . . 10 (((𝜑𝑥𝐵) ∧ (1r𝑄) = [𝑥] ) → (𝐸 ∈ (1r𝑄) ↔ 𝐸 ∈ [𝑥] ))
9 elecg 8790 . . . . . . . . . . . . 13 ((𝐸 ∈ (1r𝑄) ∧ 𝑥𝐵) → (𝐸 ∈ [𝑥] 𝑥 𝐸))
105, 9sylan 580 . . . . . . . . . . . 12 ((𝜑𝑥𝐵) → (𝐸 ∈ [𝑥] 𝑥 𝐸))
11 rngqiprngfu.r . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑅 ∈ Rng)
12 rngqiprngfu.i . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐼 ∈ (2Ideal‘𝑅))
13 rngqiprngfu.j . . . . . . . . . . . . . . . . . . . . 21 𝐽 = (𝑅s 𝐼)
14 ringrng 20283 . . . . . . . . . . . . . . . . . . . . . 22 (𝐽 ∈ Ring → 𝐽 ∈ Rng)
153, 14syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐽 ∈ Rng)
1613, 15eqeltrrid 2845 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑅s 𝐼) ∈ Rng)
1711, 12, 16rng2idlnsg 21277 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐼 ∈ (NrmSGrp‘𝑅))
18 nsgsubg 19177 . . . . . . . . . . . . . . . . . . 19 (𝐼 ∈ (NrmSGrp‘𝑅) → 𝐼 ∈ (SubGrp‘𝑅))
1917, 18syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑𝐼 ∈ (SubGrp‘𝑅))
2019adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐵) → 𝐼 ∈ (SubGrp‘𝑅))
21 rngqiprngfu.b . . . . . . . . . . . . . . . . . 18 𝐵 = (Base‘𝑅)
22 rngqiprngfu.g . . . . . . . . . . . . . . . . . 18 = (𝑅 ~QG 𝐼)
2321, 22eqger 19197 . . . . . . . . . . . . . . . . 17 (𝐼 ∈ (SubGrp‘𝑅) → Er 𝐵)
2420, 23syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐵) → Er 𝐵)
25 simpr 484 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐵) → 𝑥𝐵)
2624, 25erth 8797 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐵) → (𝑥 𝐸 ↔ [𝑥] = [𝐸] ))
2726biimpa 476 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐵) ∧ 𝑥 𝐸) → [𝑥] = [𝐸] )
2827eqcomd 2742 . . . . . . . . . . . . 13 (((𝜑𝑥𝐵) ∧ 𝑥 𝐸) → [𝐸] = [𝑥] )
2928ex 412 . . . . . . . . . . . 12 ((𝜑𝑥𝐵) → (𝑥 𝐸 → [𝐸] = [𝑥] ))
3010, 29sylbid 240 . . . . . . . . . . 11 ((𝜑𝑥𝐵) → (𝐸 ∈ [𝑥] → [𝐸] = [𝑥] ))
3130adantr 480 . . . . . . . . . 10 (((𝜑𝑥𝐵) ∧ (1r𝑄) = [𝑥] ) → (𝐸 ∈ [𝑥] → [𝐸] = [𝑥] ))
328, 31sylbid 240 . . . . . . . . 9 (((𝜑𝑥𝐵) ∧ (1r𝑄) = [𝑥] ) → (𝐸 ∈ (1r𝑄) → [𝐸] = [𝑥] ))
3332ex 412 . . . . . . . 8 ((𝜑𝑥𝐵) → ((1r𝑄) = [𝑥] → (𝐸 ∈ (1r𝑄) → [𝐸] = [𝑥] )))
346, 33mpid 44 . . . . . . 7 ((𝜑𝑥𝐵) → ((1r𝑄) = [𝑥] → [𝐸] = [𝑥] ))
3534imp 406 . . . . . 6 (((𝜑𝑥𝐵) ∧ (1r𝑄) = [𝑥] ) → [𝐸] = [𝑥] )
36 simpr 484 . . . . . 6 (((𝜑𝑥𝐵) ∧ (1r𝑄) = [𝑥] ) → (1r𝑄) = [𝑥] )
3735, 36eqtr4d 2779 . . . . 5 (((𝜑𝑥𝐵) ∧ (1r𝑄) = [𝑥] ) → [𝐸] = (1r𝑄))
38 rngqiprngfu.t . . . . . 6 · = (.r𝑅)
39 rngqiprngfu.1 . . . . . 6 1 = (1r𝐽)
40 rngqiprngfu.q . . . . . 6 𝑄 = (𝑅 /s )
4111, 12, 13, 3, 21, 38, 39, 22, 40, 2rngqiprngfulem1 21322 . . . . 5 (𝜑 → ∃𝑥𝐵 (1r𝑄) = [𝑥] )
4237, 41r19.29a 3161 . . . 4 (𝜑 → [𝐸] = (1r𝑄))
4342eqcomd 2742 . . 3 (𝜑 → (1r𝑄) = [𝐸] )
4439eqcomi 2745 . . . 4 (1r𝐽) = 1
4544a1i 11 . . 3 (𝜑 → (1r𝐽) = 1 )
4643, 45opeq12d 4880 . 2 (𝜑 → ⟨(1r𝑄), (1r𝐽)⟩ = ⟨[𝐸] , 1 ⟩)
474, 46eqtrd 2776 1 (𝜑 → (1r𝑃) = ⟨[𝐸] , 1 ⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  cop 4631   class class class wbr 5142  cfv 6560  (class class class)co 7432   Er wer 8743  [cec 8744  Basecbs 17248  s cress 17275  +gcplusg 17298  .rcmulr 17299   /s cqus 17551   ×s cxps 17552  -gcsg 18954  SubGrpcsubg 19139  NrmSGrpcnsg 19140   ~QG cqg 19141  Rngcrng 20150  1rcur 20179  Ringcrg 20231  2Idealc2idl 21260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-er 8746  df-ec 8748  df-qs 8752  df-map 8869  df-ixp 8939  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-sup 9483  df-inf 9484  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12529  df-z 12616  df-dec 12736  df-uz 12880  df-fz 13549  df-struct 17185  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17249  df-ress 17276  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 17487  df-prds 17493  df-imas 17554  df-qus 17555  df-xps 17556  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-grp 18955  df-minusg 18956  df-subg 19142  df-nsg 19143  df-eqg 19144  df-cmn 19801  df-abl 19802  df-mgp 20139  df-rng 20151  df-ur 20180  df-ring 20233  df-subrng 20547  df-lss 20931  df-sra 21173  df-rgmod 21174  df-lidl 21219  df-2idl 21261
This theorem is referenced by:  rngqiprngu  21329
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