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| Mirrors > Home > MPE Home > Th. List > rngqipring1 | Structured version Visualization version GIF version | ||
| 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.) | 
| 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) | 
| Ref | Expression | 
|---|---|
| rngqipring1 | ⊢ (𝜑 → (1r‘𝑃) = 〈[𝐸] ∼ , 1 〉) | 
| 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 20331 | . 2 ⊢ (𝜑 → (1r‘𝑃) = 〈(1r‘𝑄), (1r‘𝐽)〉) | 
| 5 | rngqiprngfu.e | . . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ (1r‘𝑄)) | |
| 6 | 5 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐸 ∈ (1r‘𝑄)) | 
| 7 | eleq2 2829 | . . . . . . . . . . 11 ⊢ ((1r‘𝑄) = [𝑥] ∼ → (𝐸 ∈ (1r‘𝑄) ↔ 𝐸 ∈ [𝑥] ∼ )) | |
| 8 | 7 | adantl 481 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (1r‘𝑄) = [𝑥] ∼ ) → (𝐸 ∈ (1r‘𝑄) ↔ 𝐸 ∈ [𝑥] ∼ )) | 
| 9 | elecg 8790 | . . . . . . . . . . . . 13 ⊢ ((𝐸 ∈ (1r‘𝑄) ∧ 𝑥 ∈ 𝐵) → (𝐸 ∈ [𝑥] ∼ ↔ 𝑥 ∼ 𝐸)) | |
| 10 | 5, 9 | sylan 580 | . . . . . . . . . . . 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 20283 | . . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐽 ∈ Ring → 𝐽 ∈ Rng) | |
| 15 | 3, 14 | syl 17 | . . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐽 ∈ Rng) | 
| 16 | 13, 15 | eqeltrrid 2845 | . . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑅 ↾s 𝐼) ∈ Rng) | 
| 17 | 11, 12, 16 | rng2idlnsg 21277 | . . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅)) | 
| 18 | nsgsubg 19177 | . . . . . . . . . . . . . . . . . . 19 ⊢ (𝐼 ∈ (NrmSGrp‘𝑅) → 𝐼 ∈ (SubGrp‘𝑅)) | |
| 19 | 17, 18 | syl 17 | . . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐼 ∈ (SubGrp‘𝑅)) | 
| 20 | 19 | adantr 480 | . . . . . . . . . . . . . . . . 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 19197 | . . . . . . . . . . . . . . . . 17 ⊢ (𝐼 ∈ (SubGrp‘𝑅) → ∼ Er 𝐵) | 
| 24 | 20, 23 | syl 17 | . . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∼ Er 𝐵) | 
| 25 | simpr 484 | . . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵) | |
| 26 | 24, 25 | erth 8797 | . . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 ∼ 𝐸 ↔ [𝑥] ∼ = [𝐸] ∼ )) | 
| 27 | 26 | biimpa 476 | . . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑥 ∼ 𝐸) → [𝑥] ∼ = [𝐸] ∼ ) | 
| 28 | 27 | eqcomd 2742 | . . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑥 ∼ 𝐸) → [𝐸] ∼ = [𝑥] ∼ ) | 
| 29 | 28 | ex 412 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 ∼ 𝐸 → [𝐸] ∼ = [𝑥] ∼ )) | 
| 30 | 10, 29 | sylbid 240 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐸 ∈ [𝑥] ∼ → [𝐸] ∼ = [𝑥] ∼ )) | 
| 31 | 30 | adantr 480 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (1r‘𝑄) = [𝑥] ∼ ) → (𝐸 ∈ [𝑥] ∼ → [𝐸] ∼ = [𝑥] ∼ )) | 
| 32 | 8, 31 | sylbid 240 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (1r‘𝑄) = [𝑥] ∼ ) → (𝐸 ∈ (1r‘𝑄) → [𝐸] ∼ = [𝑥] ∼ )) | 
| 33 | 32 | ex 412 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((1r‘𝑄) = [𝑥] ∼ → (𝐸 ∈ (1r‘𝑄) → [𝐸] ∼ = [𝑥] ∼ ))) | 
| 34 | 6, 33 | mpid 44 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((1r‘𝑄) = [𝑥] ∼ → [𝐸] ∼ = [𝑥] ∼ )) | 
| 35 | 34 | imp 406 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (1r‘𝑄) = [𝑥] ∼ ) → [𝐸] ∼ = [𝑥] ∼ ) | 
| 36 | simpr 484 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (1r‘𝑄) = [𝑥] ∼ ) → (1r‘𝑄) = [𝑥] ∼ ) | |
| 37 | 35, 36 | eqtr4d 2779 | . . . . 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 21322 | . . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 (1r‘𝑄) = [𝑥] ∼ ) | 
| 42 | 37, 41 | r19.29a 3161 | . . . 4 ⊢ (𝜑 → [𝐸] ∼ = (1r‘𝑄)) | 
| 43 | 42 | eqcomd 2742 | . . 3 ⊢ (𝜑 → (1r‘𝑄) = [𝐸] ∼ ) | 
| 44 | 39 | eqcomi 2745 | . . . 4 ⊢ (1r‘𝐽) = 1 | 
| 45 | 44 | a1i 11 | . . 3 ⊢ (𝜑 → (1r‘𝐽) = 1 ) | 
| 46 | 43, 45 | opeq12d 4880 | . 2 ⊢ (𝜑 → 〈(1r‘𝑄), (1r‘𝐽)〉 = 〈[𝐸] ∼ , 1 〉) | 
| 47 | 4, 46 | eqtrd 2776 | 1 ⊢ (𝜑 → (1r‘𝑃) = 〈[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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