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| Mirrors > Home > MPE Home > Th. List > rngqiprngu | Structured version Visualization version GIF version | ||
| Description: If a non-unital ring has a (two-sided) ideal which is unital, and the quotient of the ring and the ideal is also unital, then the ring is also unital with a ring unity which can be constructed from the ring unity of the ideal and a representative of the ring unity of the quotient. (Contributed by AV, 17-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 ) | 
| Ref | Expression | 
|---|---|
| rngqiprngu | ⊢ (𝜑 → (1r‘𝑅) = 𝑈) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqid 2737 | . . . 4 ⊢ (𝑄 ×s 𝐽) = (𝑄 ×s 𝐽) | |
| 2 | rngqiprngfu.v | . . . 4 ⊢ (𝜑 → 𝑄 ∈ Ring) | |
| 3 | rngqiprngfu.u | . . . 4 ⊢ (𝜑 → 𝐽 ∈ Ring) | |
| 4 | 1, 2, 3 | xpsringd 20329 | . . 3 ⊢ (𝜑 → (𝑄 ×s 𝐽) ∈ Ring) | 
| 5 | rngqiprngfu.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ Rng) | |
| 6 | rngqiprngfu.i | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅)) | |
| 7 | rngqiprngfu.j | . . . . 5 ⊢ 𝐽 = (𝑅 ↾s 𝐼) | |
| 8 | rngqiprngfu.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
| 9 | rngqiprngfu.t | . . . . 5 ⊢ · = (.r‘𝑅) | |
| 10 | rngqiprngfu.1 | . . . . 5 ⊢ 1 = (1r‘𝐽) | |
| 11 | rngqiprngfu.g | . . . . 5 ⊢ ∼ = (𝑅 ~QG 𝐼) | |
| 12 | rngqiprngfu.q | . . . . 5 ⊢ 𝑄 = (𝑅 /s ∼ ) | |
| 13 | eqid 2737 | . . . . 5 ⊢ (Base‘𝑄) = (Base‘𝑄) | |
| 14 | eqid 2737 | . . . . 5 ⊢ (𝑥 ∈ 𝐵 ↦ 〈[𝑥] ∼ , ( 1 · 𝑥)〉) = (𝑥 ∈ 𝐵 ↦ 〈[𝑥] ∼ , ( 1 · 𝑥)〉) | |
| 15 | 5, 6, 7, 3, 8, 9, 10, 11, 12, 13, 1, 14 | rngqiprngim 21314 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 〈[𝑥] ∼ , ( 1 · 𝑥)〉) ∈ (𝑅 RngIso (𝑄 ×s 𝐽))) | 
| 16 | rngimcnv 20456 | . . . 4 ⊢ ((𝑥 ∈ 𝐵 ↦ 〈[𝑥] ∼ , ( 1 · 𝑥)〉) ∈ (𝑅 RngIso (𝑄 ×s 𝐽)) → ◡(𝑥 ∈ 𝐵 ↦ 〈[𝑥] ∼ , ( 1 · 𝑥)〉) ∈ ((𝑄 ×s 𝐽) RngIso 𝑅)) | |
| 17 | 15, 16 | syl 17 | . . 3 ⊢ (𝜑 → ◡(𝑥 ∈ 𝐵 ↦ 〈[𝑥] ∼ , ( 1 · 𝑥)〉) ∈ ((𝑄 ×s 𝐽) RngIso 𝑅)) | 
| 18 | rngisomring1 20468 | . . 3 ⊢ (((𝑄 ×s 𝐽) ∈ Ring ∧ 𝑅 ∈ Rng ∧ ◡(𝑥 ∈ 𝐵 ↦ 〈[𝑥] ∼ , ( 1 · 𝑥)〉) ∈ ((𝑄 ×s 𝐽) RngIso 𝑅)) → (1r‘𝑅) = (◡(𝑥 ∈ 𝐵 ↦ 〈[𝑥] ∼ , ( 1 · 𝑥)〉)‘(1r‘(𝑄 ×s 𝐽)))) | |
| 19 | 4, 5, 17, 18 | syl3anc 1373 | . 2 ⊢ (𝜑 → (1r‘𝑅) = (◡(𝑥 ∈ 𝐵 ↦ 〈[𝑥] ∼ , ( 1 · 𝑥)〉)‘(1r‘(𝑄 ×s 𝐽)))) | 
| 20 | rngqiprngfu.e | . . . . 5 ⊢ (𝜑 → 𝐸 ∈ (1r‘𝑄)) | |
| 21 | rngqiprngfu.m | . . . . 5 ⊢ − = (-g‘𝑅) | |
| 22 | rngqiprngfu.a | . . . . 5 ⊢ + = (+g‘𝑅) | |
| 23 | rngqiprngfu.n | . . . . 5 ⊢ 𝑈 = ((𝐸 − ( 1 · 𝐸)) + 1 ) | |
| 24 | 5, 6, 7, 3, 8, 9, 10, 11, 12, 2, 20, 21, 22, 23, 14 | rngqiprngfu 21327 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ 〈[𝑥] ∼ , ( 1 · 𝑥)〉)‘𝑈) = 〈[𝐸] ∼ , 1 〉) | 
| 25 | 5, 6, 7, 3, 8, 9, 10, 11, 12, 2, 20, 21, 22, 23, 1 | rngqipring1 21326 | . . . 4 ⊢ (𝜑 → (1r‘(𝑄 ×s 𝐽)) = 〈[𝐸] ∼ , 1 〉) | 
| 26 | 24, 25 | eqtr4d 2780 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ 〈[𝑥] ∼ , ( 1 · 𝑥)〉)‘𝑈) = (1r‘(𝑄 ×s 𝐽))) | 
| 27 | eqid 2737 | . . . . . 6 ⊢ (Base‘(𝑄 ×s 𝐽)) = (Base‘(𝑄 ×s 𝐽)) | |
| 28 | 8, 27 | rngimf1o 20454 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ↦ 〈[𝑥] ∼ , ( 1 · 𝑥)〉) ∈ (𝑅 RngIso (𝑄 ×s 𝐽)) → (𝑥 ∈ 𝐵 ↦ 〈[𝑥] ∼ , ( 1 · 𝑥)〉):𝐵–1-1-onto→(Base‘(𝑄 ×s 𝐽))) | 
| 29 | 15, 28 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 〈[𝑥] ∼ , ( 1 · 𝑥)〉):𝐵–1-1-onto→(Base‘(𝑄 ×s 𝐽))) | 
| 30 | 5, 6, 7, 3, 8, 9, 10, 11, 12, 2, 20, 21, 22, 23 | rngqiprngfulem3 21323 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝐵) | 
| 31 | eqid 2737 | . . . . . 6 ⊢ (1r‘(𝑄 ×s 𝐽)) = (1r‘(𝑄 ×s 𝐽)) | |
| 32 | 27, 31 | ringidcl 20262 | . . . . 5 ⊢ ((𝑄 ×s 𝐽) ∈ Ring → (1r‘(𝑄 ×s 𝐽)) ∈ (Base‘(𝑄 ×s 𝐽))) | 
| 33 | 4, 32 | syl 17 | . . . 4 ⊢ (𝜑 → (1r‘(𝑄 ×s 𝐽)) ∈ (Base‘(𝑄 ×s 𝐽))) | 
| 34 | f1ocnvfvb 7299 | . . . 4 ⊢ (((𝑥 ∈ 𝐵 ↦ 〈[𝑥] ∼ , ( 1 · 𝑥)〉):𝐵–1-1-onto→(Base‘(𝑄 ×s 𝐽)) ∧ 𝑈 ∈ 𝐵 ∧ (1r‘(𝑄 ×s 𝐽)) ∈ (Base‘(𝑄 ×s 𝐽))) → (((𝑥 ∈ 𝐵 ↦ 〈[𝑥] ∼ , ( 1 · 𝑥)〉)‘𝑈) = (1r‘(𝑄 ×s 𝐽)) ↔ (◡(𝑥 ∈ 𝐵 ↦ 〈[𝑥] ∼ , ( 1 · 𝑥)〉)‘(1r‘(𝑄 ×s 𝐽))) = 𝑈)) | |
| 35 | 29, 30, 33, 34 | syl3anc 1373 | . . 3 ⊢ (𝜑 → (((𝑥 ∈ 𝐵 ↦ 〈[𝑥] ∼ , ( 1 · 𝑥)〉)‘𝑈) = (1r‘(𝑄 ×s 𝐽)) ↔ (◡(𝑥 ∈ 𝐵 ↦ 〈[𝑥] ∼ , ( 1 · 𝑥)〉)‘(1r‘(𝑄 ×s 𝐽))) = 𝑈)) | 
| 36 | 26, 35 | mpbid 232 | . 2 ⊢ (𝜑 → (◡(𝑥 ∈ 𝐵 ↦ 〈[𝑥] ∼ , ( 1 · 𝑥)〉)‘(1r‘(𝑄 ×s 𝐽))) = 𝑈) | 
| 37 | 19, 36 | eqtrd 2777 | 1 ⊢ (𝜑 → (1r‘𝑅) = 𝑈) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 〈cop 4632 ↦ cmpt 5225 ◡ccnv 5684 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 [cec 8743 Basecbs 17247 ↾s cress 17274 +gcplusg 17297 .rcmulr 17298 /s cqus 17550 ×s cxps 17551 -gcsg 18953 ~QG cqg 19140 Rngcrng 20149 1rcur 20178 Ringcrg 20230 RngIso crngim 20435 2Idealc2idl 21259 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-ec 8747 df-qs 8751 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 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 17486 df-prds 17492 df-imas 17553 df-qus 17554 df-xps 17555 df-mgm 18653 df-mgmhm 18705 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-nsg 19142 df-eqg 19143 df-ghm 19231 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-rnghm 20436 df-rngim 20437 df-subrng 20546 df-lss 20930 df-sra 21172 df-rgmod 21173 df-lidl 21218 df-2idl 21260 | 
| This theorem is referenced by: ring2idlqus1 21329 | 
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