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| Mirrors > Home > MPE Home > Th. List > pzriprnglem13 | Structured version Visualization version GIF version | ||
| Description: Lemma 13 for pzriprng 21508: 𝑄 is a unital ring. (Contributed by AV, 23-Mar-2025.) |
| Ref | Expression |
|---|---|
| pzriprng.r | ⊢ 𝑅 = (ℤring ×s ℤring) |
| pzriprng.i | ⊢ 𝐼 = (ℤ × {0}) |
| pzriprng.j | ⊢ 𝐽 = (𝑅 ↾s 𝐼) |
| pzriprng.1 | ⊢ 1 = (1r‘𝐽) |
| pzriprng.g | ⊢ ∼ = (𝑅 ~QG 𝐼) |
| pzriprng.q | ⊢ 𝑄 = (𝑅 /s ∼ ) |
| Ref | Expression |
|---|---|
| pzriprnglem13 | ⊢ 𝑄 ∈ Ring |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | pzriprng.r | . . . 4 ⊢ 𝑅 = (ℤring ×s ℤring) | |
| 2 | 1 | pzriprnglem1 21492 | . . 3 ⊢ 𝑅 ∈ Rng |
| 3 | pzriprng.i | . . . 4 ⊢ 𝐼 = (ℤ × {0}) | |
| 4 | pzriprng.j | . . . 4 ⊢ 𝐽 = (𝑅 ↾s 𝐼) | |
| 5 | 1, 3, 4 | pzriprnglem8 21499 | . . 3 ⊢ 𝐼 ∈ (2Ideal‘𝑅) |
| 6 | 1, 3 | pzriprnglem4 21495 | . . 3 ⊢ 𝐼 ∈ (SubGrp‘𝑅) |
| 7 | pzriprng.q | . . . . 5 ⊢ 𝑄 = (𝑅 /s ∼ ) | |
| 8 | pzriprng.g | . . . . . 6 ⊢ ∼ = (𝑅 ~QG 𝐼) | |
| 9 | 8 | oveq2i 7442 | . . . . 5 ⊢ (𝑅 /s ∼ ) = (𝑅 /s (𝑅 ~QG 𝐼)) |
| 10 | 7, 9 | eqtri 2765 | . . . 4 ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼)) |
| 11 | eqid 2737 | . . . 4 ⊢ (2Ideal‘𝑅) = (2Ideal‘𝑅) | |
| 12 | 10, 11 | qus2idrng 21283 | . . 3 ⊢ ((𝑅 ∈ Rng ∧ 𝐼 ∈ (2Ideal‘𝑅) ∧ 𝐼 ∈ (SubGrp‘𝑅)) → 𝑄 ∈ Rng) |
| 13 | 2, 5, 6, 12 | mp3an 1463 | . 2 ⊢ 𝑄 ∈ Rng |
| 14 | 1z 12647 | . . . . . 6 ⊢ 1 ∈ ℤ | |
| 15 | zex 12622 | . . . . . . . 8 ⊢ ℤ ∈ V | |
| 16 | snex 5436 | . . . . . . . 8 ⊢ {1} ∈ V | |
| 17 | 15, 16 | xpex 7773 | . . . . . . 7 ⊢ (ℤ × {1}) ∈ V |
| 18 | 17 | snid 4662 | . . . . . 6 ⊢ (ℤ × {1}) ∈ {(ℤ × {1})} |
| 19 | sneq 4636 | . . . . . . . . . 10 ⊢ (𝑦 = 1 → {𝑦} = {1}) | |
| 20 | 19 | xpeq2d 5715 | . . . . . . . . 9 ⊢ (𝑦 = 1 → (ℤ × {𝑦}) = (ℤ × {1})) |
| 21 | 20 | sneqd 4638 | . . . . . . . 8 ⊢ (𝑦 = 1 → {(ℤ × {𝑦})} = {(ℤ × {1})}) |
| 22 | 21 | eleq2d 2827 | . . . . . . 7 ⊢ (𝑦 = 1 → ((ℤ × {1}) ∈ {(ℤ × {𝑦})} ↔ (ℤ × {1}) ∈ {(ℤ × {1})})) |
| 23 | 22 | rspcev 3622 | . . . . . 6 ⊢ ((1 ∈ ℤ ∧ (ℤ × {1}) ∈ {(ℤ × {1})}) → ∃𝑦 ∈ ℤ (ℤ × {1}) ∈ {(ℤ × {𝑦})}) |
| 24 | 14, 18, 23 | mp2an 692 | . . . . 5 ⊢ ∃𝑦 ∈ ℤ (ℤ × {1}) ∈ {(ℤ × {𝑦})} |
| 25 | eliun 4995 | . . . . 5 ⊢ ((ℤ × {1}) ∈ ∪ 𝑦 ∈ ℤ {(ℤ × {𝑦})} ↔ ∃𝑦 ∈ ℤ (ℤ × {1}) ∈ {(ℤ × {𝑦})}) | |
| 26 | 24, 25 | mpbir 231 | . . . 4 ⊢ (ℤ × {1}) ∈ ∪ 𝑦 ∈ ℤ {(ℤ × {𝑦})} |
| 27 | pzriprng.1 | . . . . 5 ⊢ 1 = (1r‘𝐽) | |
| 28 | 1, 3, 4, 27, 8, 7 | pzriprnglem11 21502 | . . . 4 ⊢ (Base‘𝑄) = ∪ 𝑦 ∈ ℤ {(ℤ × {𝑦})} |
| 29 | 26, 28 | eleqtrri 2840 | . . 3 ⊢ (ℤ × {1}) ∈ (Base‘𝑄) |
| 30 | oveq1 7438 | . . . . . 6 ⊢ (𝑖 = (ℤ × {1}) → (𝑖(.r‘𝑄)𝑥) = ((ℤ × {1})(.r‘𝑄)𝑥)) | |
| 31 | 30 | eqeq1d 2739 | . . . . 5 ⊢ (𝑖 = (ℤ × {1}) → ((𝑖(.r‘𝑄)𝑥) = 𝑥 ↔ ((ℤ × {1})(.r‘𝑄)𝑥) = 𝑥)) |
| 32 | 31 | ovanraleqv 7455 | . . . 4 ⊢ (𝑖 = (ℤ × {1}) → (∀𝑥 ∈ (Base‘𝑄)((𝑖(.r‘𝑄)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑄)𝑖) = 𝑥) ↔ ∀𝑥 ∈ (Base‘𝑄)(((ℤ × {1})(.r‘𝑄)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑄)(ℤ × {1})) = 𝑥))) |
| 33 | id 22 | . . . 4 ⊢ ((ℤ × {1}) ∈ (Base‘𝑄) → (ℤ × {1}) ∈ (Base‘𝑄)) | |
| 34 | 1, 3, 4, 27, 8, 7 | pzriprnglem12 21503 | . . . . . 6 ⊢ (𝑥 ∈ (Base‘𝑄) → (((ℤ × {1})(.r‘𝑄)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑄)(ℤ × {1})) = 𝑥)) |
| 35 | 34 | a1i 11 | . . . . 5 ⊢ ((ℤ × {1}) ∈ (Base‘𝑄) → (𝑥 ∈ (Base‘𝑄) → (((ℤ × {1})(.r‘𝑄)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑄)(ℤ × {1})) = 𝑥))) |
| 36 | 35 | ralrimiv 3145 | . . . 4 ⊢ ((ℤ × {1}) ∈ (Base‘𝑄) → ∀𝑥 ∈ (Base‘𝑄)(((ℤ × {1})(.r‘𝑄)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑄)(ℤ × {1})) = 𝑥)) |
| 37 | 32, 33, 36 | rspcedvdw 3625 | . . 3 ⊢ ((ℤ × {1}) ∈ (Base‘𝑄) → ∃𝑖 ∈ (Base‘𝑄)∀𝑥 ∈ (Base‘𝑄)((𝑖(.r‘𝑄)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑄)𝑖) = 𝑥)) |
| 38 | 29, 37 | ax-mp 5 | . 2 ⊢ ∃𝑖 ∈ (Base‘𝑄)∀𝑥 ∈ (Base‘𝑄)((𝑖(.r‘𝑄)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑄)𝑖) = 𝑥) |
| 39 | eqid 2737 | . . 3 ⊢ (Base‘𝑄) = (Base‘𝑄) | |
| 40 | eqid 2737 | . . 3 ⊢ (.r‘𝑄) = (.r‘𝑄) | |
| 41 | 39, 40 | isringrng 20284 | . 2 ⊢ (𝑄 ∈ Ring ↔ (𝑄 ∈ Rng ∧ ∃𝑖 ∈ (Base‘𝑄)∀𝑥 ∈ (Base‘𝑄)((𝑖(.r‘𝑄)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑄)𝑖) = 𝑥))) |
| 42 | 13, 38, 41 | mpbir2an 711 | 1 ⊢ 𝑄 ∈ Ring |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 {csn 4626 ∪ ciun 4991 × cxp 5683 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 ℤcz 12613 Basecbs 17247 ↾s cress 17274 .rcmulr 17298 /s cqus 17550 ×s cxps 17551 SubGrpcsubg 19138 ~QG cqg 19140 Rngcrng 20149 1rcur 20178 Ringcrg 20230 2Idealc2idl 21259 ℤringczring 21457 |
| 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 ax-addf 11234 ax-mulf 11235 |
| 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-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 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-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-nsg 19142 df-eqg 19143 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-oppr 20334 df-subrng 20546 df-subrg 20570 df-lss 20930 df-sra 21172 df-rgmod 21173 df-lidl 21218 df-2idl 21260 df-cnfld 21365 df-zring 21458 |
| This theorem is referenced by: pzriprnglem14 21505 pzriprngALT 21506 pzriprng1ALT 21507 |
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