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Theorem rngqipring1 21344
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 20347 . 2 (𝜑 → (1r𝑃) = ⟨(1r𝑄), (1r𝐽)⟩)
5 rngqiprngfu.e . . . . . . . . 9 (𝜑𝐸 ∈ (1r𝑄))
65adantr 480 . . . . . . . 8 ((𝜑𝑥𝐵) → 𝐸 ∈ (1r𝑄))
7 eleq2 2828 . . . . . . . . . . 11 ((1r𝑄) = [𝑥] → (𝐸 ∈ (1r𝑄) ↔ 𝐸 ∈ [𝑥] ))
87adantl 481 . . . . . . . . . 10 (((𝜑𝑥𝐵) ∧ (1r𝑄) = [𝑥] ) → (𝐸 ∈ (1r𝑄) ↔ 𝐸 ∈ [𝑥] ))
9 elecg 8788 . . . . . . . . . . . . 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 20299 . . . . . . . . . . . . . . . . . . . . . 22 (𝐽 ∈ Ring → 𝐽 ∈ Rng)
153, 14syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐽 ∈ Rng)
1613, 15eqeltrrid 2844 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑅s 𝐼) ∈ Rng)
1711, 12, 16rng2idlnsg 21294 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐼 ∈ (NrmSGrp‘𝑅))
18 nsgsubg 19189 . . . . . . . . . . . . . . . . . . 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 19209 . . . . . . . . . . . . . . . . 17 (𝐼 ∈ (SubGrp‘𝑅) → Er 𝐵)
2420, 23syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐵) → Er 𝐵)
25 simpr 484 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐵) → 𝑥𝐵)
2624, 25erth 8795 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐵) → (𝑥 𝐸 ↔ [𝑥] = [𝐸] ))
2726biimpa 476 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐵) ∧ 𝑥 𝐸) → [𝑥] = [𝐸] )
2827eqcomd 2741 . . . . . . . . . . . . 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 2778 . . . . 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 21339 . . . . 5 (𝜑 → ∃𝑥𝐵 (1r𝑄) = [𝑥] )
4237, 41r19.29a 3160 . . . 4 (𝜑 → [𝐸] = (1r𝑄))
4342eqcomd 2741 . . 3 (𝜑 → (1r𝑄) = [𝐸] )
4439eqcomi 2744 . . . 4 (1r𝐽) = 1
4544a1i 11 . . 3 (𝜑 → (1r𝐽) = 1 )
4643, 45opeq12d 4886 . 2 (𝜑 → ⟨(1r𝑄), (1r𝐽)⟩ = ⟨[𝐸] , 1 ⟩)
474, 46eqtrd 2775 1 (𝜑 → (1r𝑃) = ⟨[𝐸] , 1 ⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  cop 4637   class class class wbr 5148  cfv 6563  (class class class)co 7431   Er wer 8741  [cec 8742  Basecbs 17245  s cress 17274  +gcplusg 17298  .rcmulr 17299   /s cqus 17552   ×s cxps 17553  -gcsg 18966  SubGrpcsubg 19151  NrmSGrpcnsg 19152   ~QG cqg 19153  Rngcrng 20170  1rcur 20199  Ringcrg 20251  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-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-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-subg 19154  df-nsg 19155  df-eqg 19156  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-subrng 20563  df-lss 20948  df-sra 21190  df-rgmod 21191  df-lidl 21236  df-2idl 21278
This theorem is referenced by:  rngqiprngu  21346
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