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Theorem rngqiprngimf1 21327
Description: 𝐹 is a one-to-one function from (the base set of) a non-unital ring to the product of the (base set of the) quotient with a two-sided ideal and the (base set of the) two-sided ideal. (Contributed by AV, 7-Mar-2025.)
Hypotheses
Ref Expression
rng2idlring.r (𝜑𝑅 ∈ Rng)
rng2idlring.i (𝜑𝐼 ∈ (2Ideal‘𝑅))
rng2idlring.j 𝐽 = (𝑅s 𝐼)
rng2idlring.u (𝜑𝐽 ∈ Ring)
rng2idlring.b 𝐵 = (Base‘𝑅)
rng2idlring.t · = (.r𝑅)
rng2idlring.1 1 = (1r𝐽)
rngqiprngim.g = (𝑅 ~QG 𝐼)
rngqiprngim.q 𝑄 = (𝑅 /s )
rngqiprngim.c 𝐶 = (Base‘𝑄)
rngqiprngim.p 𝑃 = (𝑄 ×s 𝐽)
rngqiprngim.f 𝐹 = (𝑥𝐵 ↦ ⟨[𝑥] , ( 1 · 𝑥)⟩)
Assertion
Ref Expression
rngqiprngimf1 (𝜑𝐹:𝐵1-1→(𝐶 × 𝐼))
Distinct variable groups:   𝑥,𝐶   𝑥,𝐼   𝑥,𝐵   𝜑,𝑥   𝑥,   𝑥, 1   𝑥, ·   𝑥,𝑅
Allowed substitution hints:   𝑃(𝑥)   𝑄(𝑥)   𝐹(𝑥)   𝐽(𝑥)

Proof of Theorem rngqiprngimf1
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 rng2idlring.r . . . . . . . . 9 (𝜑𝑅 ∈ Rng)
2 rng2idlring.i . . . . . . . . 9 (𝜑𝐼 ∈ (2Ideal‘𝑅))
3 rng2idlring.j . . . . . . . . . . . 12 𝐽 = (𝑅s 𝐼)
4 rng2idlring.u . . . . . . . . . . . . 13 (𝜑𝐽 ∈ Ring)
5 ringrng 20298 . . . . . . . . . . . . 13 (𝐽 ∈ Ring → 𝐽 ∈ Rng)
64, 5syl 17 . . . . . . . . . . . 12 (𝜑𝐽 ∈ Rng)
73, 6eqeltrrid 2843 . . . . . . . . . . 11 (𝜑 → (𝑅s 𝐼) ∈ Rng)
81, 2, 7rng2idlnsg 21293 . . . . . . . . . 10 (𝜑𝐼 ∈ (NrmSGrp‘𝑅))
9 nsgsubg 19188 . . . . . . . . . 10 (𝐼 ∈ (NrmSGrp‘𝑅) → 𝐼 ∈ (SubGrp‘𝑅))
108, 9syl 17 . . . . . . . . 9 (𝜑𝐼 ∈ (SubGrp‘𝑅))
11 rngqiprngim.q . . . . . . . . . . 11 𝑄 = (𝑅 /s )
12 rngqiprngim.g . . . . . . . . . . . 12 = (𝑅 ~QG 𝐼)
1312oveq2i 7441 . . . . . . . . . . 11 (𝑅 /s ) = (𝑅 /s (𝑅 ~QG 𝐼))
1411, 13eqtri 2762 . . . . . . . . . 10 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))
15 eqid 2734 . . . . . . . . . 10 (2Ideal‘𝑅) = (2Ideal‘𝑅)
1614, 15qus2idrng 21300 . . . . . . . . 9 ((𝑅 ∈ Rng ∧ 𝐼 ∈ (2Ideal‘𝑅) ∧ 𝐼 ∈ (SubGrp‘𝑅)) → 𝑄 ∈ Rng)
171, 2, 10, 16syl3anc 1370 . . . . . . . 8 (𝜑𝑄 ∈ Rng)
18 rnggrp 20175 . . . . . . . . 9 (𝑄 ∈ Rng → 𝑄 ∈ Grp)
1918grpmndd 18976 . . . . . . . 8 (𝑄 ∈ Rng → 𝑄 ∈ Mnd)
2017, 19syl 17 . . . . . . 7 (𝜑𝑄 ∈ Mnd)
21 ringmnd 20260 . . . . . . . 8 (𝐽 ∈ Ring → 𝐽 ∈ Mnd)
224, 21syl 17 . . . . . . 7 (𝜑𝐽 ∈ Mnd)
23 rngqiprngim.p . . . . . . . 8 𝑃 = (𝑄 ×s 𝐽)
2423xpsmnd0 18803 . . . . . . 7 ((𝑄 ∈ Mnd ∧ 𝐽 ∈ Mnd) → (0g𝑃) = ⟨(0g𝑄), (0g𝐽)⟩)
2520, 22, 24syl2anc 584 . . . . . 6 (𝜑 → (0g𝑃) = ⟨(0g𝑄), (0g𝐽)⟩)
2625sneqd 4642 . . . . 5 (𝜑 → {(0g𝑃)} = {⟨(0g𝑄), (0g𝐽)⟩})
2726imaeq2d 6079 . . . 4 (𝜑 → (𝐹 “ {(0g𝑃)}) = (𝐹 “ {⟨(0g𝑄), (0g𝐽)⟩}))
28 nfv 1911 . . . . . 6 𝑥𝜑
29 opex 5474 . . . . . . 7 ⟨[𝑥] , ( 1 · 𝑥)⟩ ∈ V
3029a1i 11 . . . . . 6 ((𝜑𝑥𝐵) → ⟨[𝑥] , ( 1 · 𝑥)⟩ ∈ V)
31 rngqiprngim.f . . . . . 6 𝐹 = (𝑥𝐵 ↦ ⟨[𝑥] , ( 1 · 𝑥)⟩)
3228, 30, 31fnmptd 6709 . . . . 5 (𝜑𝐹 Fn 𝐵)
33 fncnvima2 7080 . . . . 5 (𝐹 Fn 𝐵 → (𝐹 “ {⟨(0g𝑄), (0g𝐽)⟩}) = {𝑎𝐵 ∣ (𝐹𝑎) ∈ {⟨(0g𝑄), (0g𝐽)⟩}})
3432, 33syl 17 . . . 4 (𝜑 → (𝐹 “ {⟨(0g𝑄), (0g𝐽)⟩}) = {𝑎𝐵 ∣ (𝐹𝑎) ∈ {⟨(0g𝑄), (0g𝐽)⟩}})
35 rng2idlring.b . . . . . . . 8 𝐵 = (Base‘𝑅)
36 rng2idlring.t . . . . . . . 8 · = (.r𝑅)
37 rng2idlring.1 . . . . . . . 8 1 = (1r𝐽)
38 rngqiprngim.c . . . . . . . 8 𝐶 = (Base‘𝑄)
391, 2, 3, 4, 35, 36, 37, 12, 11, 38, 23, 31rngqiprngimfv 21325 . . . . . . 7 ((𝜑𝑎𝐵) → (𝐹𝑎) = ⟨[𝑎] , ( 1 · 𝑎)⟩)
4039eleq1d 2823 . . . . . 6 ((𝜑𝑎𝐵) → ((𝐹𝑎) ∈ {⟨(0g𝑄), (0g𝐽)⟩} ↔ ⟨[𝑎] , ( 1 · 𝑎)⟩ ∈ {⟨(0g𝑄), (0g𝐽)⟩}))
4140rabbidva 3439 . . . . 5 (𝜑 → {𝑎𝐵 ∣ (𝐹𝑎) ∈ {⟨(0g𝑄), (0g𝐽)⟩}} = {𝑎𝐵 ∣ ⟨[𝑎] , ( 1 · 𝑎)⟩ ∈ {⟨(0g𝑄), (0g𝐽)⟩}})
42 eceq1 8782 . . . . . . . 8 (𝑎 = (0g𝑅) → [𝑎] = [(0g𝑅)] )
43 oveq2 7438 . . . . . . . 8 (𝑎 = (0g𝑅) → ( 1 · 𝑎) = ( 1 · (0g𝑅)))
4442, 43opeq12d 4885 . . . . . . 7 (𝑎 = (0g𝑅) → ⟨[𝑎] , ( 1 · 𝑎)⟩ = ⟨[(0g𝑅)] , ( 1 · (0g𝑅))⟩)
4544eleq1d 2823 . . . . . 6 (𝑎 = (0g𝑅) → (⟨[𝑎] , ( 1 · 𝑎)⟩ ∈ {⟨(0g𝑄), (0g𝐽)⟩} ↔ ⟨[(0g𝑅)] , ( 1 · (0g𝑅))⟩ ∈ {⟨(0g𝑄), (0g𝐽)⟩}))
46 rnggrp 20175 . . . . . . . . 9 (𝑅 ∈ Rng → 𝑅 ∈ Grp)
471, 46syl 17 . . . . . . . 8 (𝜑𝑅 ∈ Grp)
4847grpmndd 18976 . . . . . . 7 (𝜑𝑅 ∈ Mnd)
49 eqid 2734 . . . . . . . 8 (0g𝑅) = (0g𝑅)
5035, 49mndidcl 18774 . . . . . . 7 (𝑅 ∈ Mnd → (0g𝑅) ∈ 𝐵)
5148, 50syl 17 . . . . . 6 (𝜑 → (0g𝑅) ∈ 𝐵)
5212eceq2i 8785 . . . . . . . . 9 [(0g𝑅)] = [(0g𝑅)](𝑅 ~QG 𝐼)
5314, 49qus0 19219 . . . . . . . . . 10 (𝐼 ∈ (NrmSGrp‘𝑅) → [(0g𝑅)](𝑅 ~QG 𝐼) = (0g𝑄))
548, 53syl 17 . . . . . . . . 9 (𝜑 → [(0g𝑅)](𝑅 ~QG 𝐼) = (0g𝑄))
5552, 54eqtrid 2786 . . . . . . . 8 (𝜑 → [(0g𝑅)] = (0g𝑄))
561, 2, 7rng2idl0 21294 . . . . . . . . . . 11 (𝜑 → (0g𝑅) ∈ 𝐼)
5735, 152idlss 21289 . . . . . . . . . . . 12 (𝐼 ∈ (2Ideal‘𝑅) → 𝐼𝐵)
582, 57syl 17 . . . . . . . . . . 11 (𝜑𝐼𝐵)
593, 35, 49ress0g 18787 . . . . . . . . . . 11 ((𝑅 ∈ Mnd ∧ (0g𝑅) ∈ 𝐼𝐼𝐵) → (0g𝑅) = (0g𝐽))
6048, 56, 58, 59syl3anc 1370 . . . . . . . . . 10 (𝜑 → (0g𝑅) = (0g𝐽))
6160oveq2d 7446 . . . . . . . . 9 (𝜑 → ( 1 · (0g𝑅)) = ( 1 · (0g𝐽)))
623, 36ressmulr 17352 . . . . . . . . . . 11 (𝐼 ∈ (2Ideal‘𝑅) → · = (.r𝐽))
632, 62syl 17 . . . . . . . . . 10 (𝜑· = (.r𝐽))
6463oveqd 7447 . . . . . . . . 9 (𝜑 → ( 1 · (0g𝐽)) = ( 1 (.r𝐽)(0g𝐽)))
65 eqid 2734 . . . . . . . . . . 11 (Base‘𝐽) = (Base‘𝐽)
6665, 37ringidcl 20279 . . . . . . . . . 10 (𝐽 ∈ Ring → 1 ∈ (Base‘𝐽))
67 eqid 2734 . . . . . . . . . . 11 (.r𝐽) = (.r𝐽)
68 eqid 2734 . . . . . . . . . . 11 (0g𝐽) = (0g𝐽)
6965, 67, 68ringrz 20307 . . . . . . . . . 10 ((𝐽 ∈ Ring ∧ 1 ∈ (Base‘𝐽)) → ( 1 (.r𝐽)(0g𝐽)) = (0g𝐽))
704, 66, 69syl2anc2 585 . . . . . . . . 9 (𝜑 → ( 1 (.r𝐽)(0g𝐽)) = (0g𝐽))
7161, 64, 703eqtrd 2778 . . . . . . . 8 (𝜑 → ( 1 · (0g𝑅)) = (0g𝐽))
7255, 71opeq12d 4885 . . . . . . 7 (𝜑 → ⟨[(0g𝑅)] , ( 1 · (0g𝑅))⟩ = ⟨(0g𝑄), (0g𝐽)⟩)
73 opex 5474 . . . . . . . 8 ⟨[(0g𝑅)] , ( 1 · (0g𝑅))⟩ ∈ V
7473elsn 4645 . . . . . . 7 (⟨[(0g𝑅)] , ( 1 · (0g𝑅))⟩ ∈ {⟨(0g𝑄), (0g𝐽)⟩} ↔ ⟨[(0g𝑅)] , ( 1 · (0g𝑅))⟩ = ⟨(0g𝑄), (0g𝐽)⟩)
7572, 74sylibr 234 . . . . . 6 (𝜑 → ⟨[(0g𝑅)] , ( 1 · (0g𝑅))⟩ ∈ {⟨(0g𝑄), (0g𝐽)⟩})
76 opex 5474 . . . . . . . . . 10 ⟨[𝑎] , ( 1 · 𝑎)⟩ ∈ V
7776elsn 4645 . . . . . . . . 9 (⟨[𝑎] , ( 1 · 𝑎)⟩ ∈ {⟨(0g𝑄), (0g𝐽)⟩} ↔ ⟨[𝑎] , ( 1 · 𝑎)⟩ = ⟨(0g𝑄), (0g𝐽)⟩)
7812ovexi 7464 . . . . . . . . . . 11 ∈ V
79 ecexg 8747 . . . . . . . . . . 11 ( ∈ V → [𝑎] ∈ V)
8078, 79ax-mp 5 . . . . . . . . . 10 [𝑎] ∈ V
81 ovex 7463 . . . . . . . . . 10 ( 1 · 𝑎) ∈ V
8280, 81opth 5486 . . . . . . . . 9 (⟨[𝑎] , ( 1 · 𝑎)⟩ = ⟨(0g𝑄), (0g𝐽)⟩ ↔ ([𝑎] = (0g𝑄) ∧ ( 1 · 𝑎) = (0g𝐽)))
8377, 82bitri 275 . . . . . . . 8 (⟨[𝑎] , ( 1 · 𝑎)⟩ ∈ {⟨(0g𝑄), (0g𝐽)⟩} ↔ ([𝑎] = (0g𝑄) ∧ ( 1 · 𝑎) = (0g𝐽)))
841, 2, 3, 4, 35, 36, 37, 12, 11rngqiprngimf1lem 21321 . . . . . . . 8 ((𝜑𝑎𝐵) → (([𝑎] = (0g𝑄) ∧ ( 1 · 𝑎) = (0g𝐽)) → 𝑎 = (0g𝑅)))
8583, 84biimtrid 242 . . . . . . 7 ((𝜑𝑎𝐵) → (⟨[𝑎] , ( 1 · 𝑎)⟩ ∈ {⟨(0g𝑄), (0g𝐽)⟩} → 𝑎 = (0g𝑅)))
8685imp 406 . . . . . 6 (((𝜑𝑎𝐵) ∧ ⟨[𝑎] , ( 1 · 𝑎)⟩ ∈ {⟨(0g𝑄), (0g𝐽)⟩}) → 𝑎 = (0g𝑅))
8745, 51, 75, 86rabeqsnd 4673 . . . . 5 (𝜑 → {𝑎𝐵 ∣ ⟨[𝑎] , ( 1 · 𝑎)⟩ ∈ {⟨(0g𝑄), (0g𝐽)⟩}} = {(0g𝑅)})
8841, 87eqtrd 2774 . . . 4 (𝜑 → {𝑎𝐵 ∣ (𝐹𝑎) ∈ {⟨(0g𝑄), (0g𝐽)⟩}} = {(0g𝑅)})
8927, 34, 883eqtrd 2778 . . 3 (𝜑 → (𝐹 “ {(0g𝑃)}) = {(0g𝑅)})
901, 2, 3, 4, 35, 36, 37, 12, 11, 38, 23, 31rngqiprngghm 21326 . . . 4 (𝜑𝐹 ∈ (𝑅 GrpHom 𝑃))
91 eqid 2734 . . . . 5 (Base‘𝑃) = (Base‘𝑃)
92 eqid 2734 . . . . 5 (0g𝑃) = (0g𝑃)
9335, 91, 49, 92kerf1ghm 19277 . . . 4 (𝐹 ∈ (𝑅 GrpHom 𝑃) → (𝐹:𝐵1-1→(Base‘𝑃) ↔ (𝐹 “ {(0g𝑃)}) = {(0g𝑅)}))
9490, 93syl 17 . . 3 (𝜑 → (𝐹:𝐵1-1→(Base‘𝑃) ↔ (𝐹 “ {(0g𝑃)}) = {(0g𝑅)}))
9589, 94mpbird 257 . 2 (𝜑𝐹:𝐵1-1→(Base‘𝑃))
96 eqidd 2735 . . 3 (𝜑𝐹 = 𝐹)
97 eqidd 2735 . . 3 (𝜑𝐵 = 𝐵)
981, 2, 3, 4, 35, 36, 37, 12, 11, 38, 23rngqipbas 21322 . . 3 (𝜑 → (Base‘𝑃) = (𝐶 × 𝐼))
9996, 97, 98f1eq123d 6840 . 2 (𝜑 → (𝐹:𝐵1-1→(Base‘𝑃) ↔ 𝐹:𝐵1-1→(𝐶 × 𝐼)))
10095, 99mpbid 232 1 (𝜑𝐹:𝐵1-1→(𝐶 × 𝐼))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  {crab 3432  Vcvv 3477  wss 3962  {csn 4630  cop 4636  cmpt 5230   × cxp 5686  ccnv 5687  cima 5691   Fn wfn 6557  1-1wf1 6559  cfv 6562  (class class class)co 7430  [cec 8741  Basecbs 17244  s cress 17273  .rcmulr 17298  0gc0g 17485   /s cqus 17551   ×s cxps 17552  Mndcmnd 18759  Grpcgrp 18963  SubGrpcsubg 19150  NrmSGrpcnsg 19151   ~QG cqg 19152   GrpHom cghm 19242  Rngcrng 20169  1rcur 20198  Ringcrg 20250  2Idealc2idl 21276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-tpos 8249  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-ec 8745  df-qs 8749  df-map 8866  df-ixp 8936  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-sup 9479  df-inf 9480  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-dec 12731  df-uz 12876  df-fz 13544  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-hom 17321  df-cco 17322  df-0g 17487  df-prds 17493  df-imas 17554  df-qus 17555  df-xps 17556  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-grp 18966  df-minusg 18967  df-sbg 18968  df-subg 19153  df-nsg 19154  df-eqg 19155  df-ghm 19243  df-cmn 19814  df-abl 19815  df-mgp 20152  df-rng 20170  df-ur 20199  df-ring 20252  df-oppr 20350  df-dvdsr 20373  df-unit 20374  df-invr 20404  df-subrng 20562  df-lss 20947  df-sra 21189  df-rgmod 21190  df-lidl 21235  df-2idl 21277
This theorem is referenced by:  rngqiprngim  21331
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