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| Mirrors > Home > MPE Home > Th. List > rngqiprngimf | Structured version Visualization version GIF version | ||
| Description: 𝐹 is a function from (the base set of) a non-unital ring to the product of the (base set 𝐶 of the) quotient with a two-sided ideal and the (base set 𝐼 of the) two-sided ideal (because of 2idlbas 21273, (Base‘𝐽) = 𝐼!) (Contributed by AV, 21-Feb-2025.) | 
| Ref | Expression | 
|---|---|
| rng2idlring.r | ⊢ (𝜑 → 𝑅 ∈ Rng) | 
| rng2idlring.i | ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅)) | 
| rng2idlring.j | ⊢ 𝐽 = (𝑅 ↾s 𝐼) | 
| rng2idlring.u | ⊢ (𝜑 → 𝐽 ∈ Ring) | 
| rng2idlring.b | ⊢ 𝐵 = (Base‘𝑅) | 
| rng2idlring.t | ⊢ · = (.r‘𝑅) | 
| rng2idlring.1 | ⊢ 1 = (1r‘𝐽) | 
| rngqiprngim.g | ⊢ ∼ = (𝑅 ~QG 𝐼) | 
| rngqiprngim.q | ⊢ 𝑄 = (𝑅 /s ∼ ) | 
| rngqiprngim.c | ⊢ 𝐶 = (Base‘𝑄) | 
| rngqiprngim.p | ⊢ 𝑃 = (𝑄 ×s 𝐽) | 
| rngqiprngim.f | ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ 〈[𝑥] ∼ , ( 1 · 𝑥)〉) | 
| Ref | Expression | 
|---|---|
| rngqiprngimf | ⊢ (𝜑 → 𝐹:𝐵⟶(𝐶 × 𝐼)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | rngqiprngim.g | . . . . . . 7 ⊢ ∼ = (𝑅 ~QG 𝐼) | |
| 2 | 1 | ovexi 7465 | . . . . . 6 ⊢ ∼ ∈ V | 
| 3 | 2 | ecelqsi 8813 | . . . . 5 ⊢ (𝑥 ∈ 𝐵 → [𝑥] ∼ ∈ (𝐵 / ∼ )) | 
| 4 | 3 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → [𝑥] ∼ ∈ (𝐵 / ∼ )) | 
| 5 | rngqiprngim.q | . . . . . . 7 ⊢ 𝑄 = (𝑅 /s ∼ ) | |
| 6 | 5 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑄 = (𝑅 /s ∼ )) | 
| 7 | rng2idlring.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑅) | |
| 8 | 7 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐵 = (Base‘𝑅)) | 
| 9 | 2 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∼ ∈ V) | 
| 10 | rng2idlring.r | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Rng) | |
| 11 | 10 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑅 ∈ Rng) | 
| 12 | 6, 8, 9, 11 | qusbas 17590 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐵 / ∼ ) = (Base‘𝑄)) | 
| 13 | rngqiprngim.c | . . . . 5 ⊢ 𝐶 = (Base‘𝑄) | |
| 14 | 12, 13 | eqtr4di 2795 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐵 / ∼ ) = 𝐶) | 
| 15 | 4, 14 | eleqtrd 2843 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → [𝑥] ∼ ∈ 𝐶) | 
| 16 | rng2idlring.i | . . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅)) | |
| 17 | rng2idlring.j | . . . . . . 7 ⊢ 𝐽 = (𝑅 ↾s 𝐼) | |
| 18 | eqid 2737 | . . . . . . 7 ⊢ (Base‘𝐽) = (Base‘𝐽) | |
| 19 | 16, 17, 18 | 2idlbas 21273 | . . . . . 6 ⊢ (𝜑 → (Base‘𝐽) = 𝐼) | 
| 20 | 16, 17, 18 | 2idlelbas 21274 | . . . . . . 7 ⊢ (𝜑 → ((Base‘𝐽) ∈ (LIdeal‘𝑅) ∧ (Base‘𝐽) ∈ (LIdeal‘(oppr‘𝑅)))) | 
| 21 | 20 | simprd 495 | . . . . . 6 ⊢ (𝜑 → (Base‘𝐽) ∈ (LIdeal‘(oppr‘𝑅))) | 
| 22 | 19, 21 | eqeltrrd 2842 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘(oppr‘𝑅))) | 
| 23 | rng2idlring.u | . . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ Ring) | |
| 24 | ringrng 20282 | . . . . . . . 8 ⊢ (𝐽 ∈ Ring → 𝐽 ∈ Rng) | |
| 25 | 23, 24 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ Rng) | 
| 26 | 17, 25 | eqeltrrid 2846 | . . . . . 6 ⊢ (𝜑 → (𝑅 ↾s 𝐼) ∈ Rng) | 
| 27 | 10, 16, 26 | rng2idl0 21277 | . . . . 5 ⊢ (𝜑 → (0g‘𝑅) ∈ 𝐼) | 
| 28 | 10, 22, 27 | 3jca 1129 | . . . 4 ⊢ (𝜑 → (𝑅 ∈ Rng ∧ 𝐼 ∈ (LIdeal‘(oppr‘𝑅)) ∧ (0g‘𝑅) ∈ 𝐼)) | 
| 29 | rng2idlring.1 | . . . . . . . 8 ⊢ 1 = (1r‘𝐽) | |
| 30 | 18, 29 | ringidcl 20262 | . . . . . . 7 ⊢ (𝐽 ∈ Ring → 1 ∈ (Base‘𝐽)) | 
| 31 | 23, 30 | syl 17 | . . . . . 6 ⊢ (𝜑 → 1 ∈ (Base‘𝐽)) | 
| 32 | 31, 19 | eleqtrd 2843 | . . . . 5 ⊢ (𝜑 → 1 ∈ 𝐼) | 
| 33 | 32 | anim1ci 616 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ 𝐵 ∧ 1 ∈ 𝐼)) | 
| 34 | eqid 2737 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
| 35 | rng2idlring.t | . . . . 5 ⊢ · = (.r‘𝑅) | |
| 36 | eqid 2737 | . . . . 5 ⊢ (LIdeal‘(oppr‘𝑅)) = (LIdeal‘(oppr‘𝑅)) | |
| 37 | 34, 7, 35, 36 | rngridlmcl 21227 | . . . 4 ⊢ (((𝑅 ∈ Rng ∧ 𝐼 ∈ (LIdeal‘(oppr‘𝑅)) ∧ (0g‘𝑅) ∈ 𝐼) ∧ (𝑥 ∈ 𝐵 ∧ 1 ∈ 𝐼)) → ( 1 · 𝑥) ∈ 𝐼) | 
| 38 | 28, 33, 37 | syl2an2r 685 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 1 · 𝑥) ∈ 𝐼) | 
| 39 | 15, 38 | opelxpd 5724 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 〈[𝑥] ∼ , ( 1 · 𝑥)〉 ∈ (𝐶 × 𝐼)) | 
| 40 | rngqiprngim.f | . 2 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ 〈[𝑥] ∼ , ( 1 · 𝑥)〉) | |
| 41 | 39, 40 | fmptd 7134 | 1 ⊢ (𝜑 → 𝐹:𝐵⟶(𝐶 × 𝐼)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 Vcvv 3480 〈cop 4632 ↦ cmpt 5225 × cxp 5683 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 [cec 8743 / cqs 8744 Basecbs 17247 ↾s cress 17274 .rcmulr 17298 0gc0g 17484 /s cqus 17550 ×s cxps 17551 ~QG cqg 19140 Rngcrng 20149 1rcur 20178 Ringcrg 20230 opprcoppr 20333 LIdealclidl 21216 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-er 8745 df-ec 8747 df-qs 8751 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-0g 17486 df-imas 17553 df-qus 17554 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-subg 19141 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-oppr 20334 df-subrng 20546 df-lss 20930 df-sra 21172 df-rgmod 21173 df-lidl 21218 df-2idl 21260 | 
| This theorem is referenced by: rngqiprngghm 21309 rngqiprngimfo 21311 | 
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