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Mirrors > Home > MPE Home > Th. List > pzriprnglem13 | Structured version Visualization version GIF version |
Description: Lemma 13 for pzriprng 21525: 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 21509 | . . 3 ⊢ 𝑅 ∈ Rng |
3 | pzriprng.i | . . . 4 ⊢ 𝐼 = (ℤ × {0}) | |
4 | pzriprng.j | . . . 4 ⊢ 𝐽 = (𝑅 ↾s 𝐼) | |
5 | 1, 3, 4 | pzriprnglem8 21516 | . . 3 ⊢ 𝐼 ∈ (2Ideal‘𝑅) |
6 | 1, 3 | pzriprnglem4 21512 | . . 3 ⊢ 𝐼 ∈ (SubGrp‘𝑅) |
7 | pzriprng.q | . . . . 5 ⊢ 𝑄 = (𝑅 /s ∼ ) | |
8 | pzriprng.g | . . . . . 6 ⊢ ∼ = (𝑅 ~QG 𝐼) | |
9 | 8 | oveq2i 7441 | . . . . 5 ⊢ (𝑅 /s ∼ ) = (𝑅 /s (𝑅 ~QG 𝐼)) |
10 | 7, 9 | eqtri 2762 | . . . 4 ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼)) |
11 | eqid 2734 | . . . 4 ⊢ (2Ideal‘𝑅) = (2Ideal‘𝑅) | |
12 | 10, 11 | qus2idrng 21300 | . . 3 ⊢ ((𝑅 ∈ Rng ∧ 𝐼 ∈ (2Ideal‘𝑅) ∧ 𝐼 ∈ (SubGrp‘𝑅)) → 𝑄 ∈ Rng) |
13 | 2, 5, 6, 12 | mp3an 1460 | . 2 ⊢ 𝑄 ∈ Rng |
14 | 1z 12644 | . . . . . 6 ⊢ 1 ∈ ℤ | |
15 | zex 12619 | . . . . . . . 8 ⊢ ℤ ∈ V | |
16 | snex 5441 | . . . . . . . 8 ⊢ {1} ∈ V | |
17 | 15, 16 | xpex 7771 | . . . . . . 7 ⊢ (ℤ × {1}) ∈ V |
18 | 17 | snid 4666 | . . . . . 6 ⊢ (ℤ × {1}) ∈ {(ℤ × {1})} |
19 | sneq 4640 | . . . . . . . . . 10 ⊢ (𝑦 = 1 → {𝑦} = {1}) | |
20 | 19 | xpeq2d 5718 | . . . . . . . . 9 ⊢ (𝑦 = 1 → (ℤ × {𝑦}) = (ℤ × {1})) |
21 | 20 | sneqd 4642 | . . . . . . . 8 ⊢ (𝑦 = 1 → {(ℤ × {𝑦})} = {(ℤ × {1})}) |
22 | 21 | eleq2d 2824 | . . . . . . 7 ⊢ (𝑦 = 1 → ((ℤ × {1}) ∈ {(ℤ × {𝑦})} ↔ (ℤ × {1}) ∈ {(ℤ × {1})})) |
23 | 22 | rspcev 3621 | . . . . . 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 4999 | . . . . 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 21519 | . . . 4 ⊢ (Base‘𝑄) = ∪ 𝑦 ∈ ℤ {(ℤ × {𝑦})} |
29 | 26, 28 | eleqtrri 2837 | . . 3 ⊢ (ℤ × {1}) ∈ (Base‘𝑄) |
30 | oveq1 7437 | . . . . . 6 ⊢ (𝑖 = (ℤ × {1}) → (𝑖(.r‘𝑄)𝑥) = ((ℤ × {1})(.r‘𝑄)𝑥)) | |
31 | 30 | eqeq1d 2736 | . . . . 5 ⊢ (𝑖 = (ℤ × {1}) → ((𝑖(.r‘𝑄)𝑥) = 𝑥 ↔ ((ℤ × {1})(.r‘𝑄)𝑥) = 𝑥)) |
32 | 31 | ovanraleqv 7454 | . . . 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 21520 | . . . . . 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 3142 | . . . 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 3624 | . . 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 2734 | . . 3 ⊢ (Base‘𝑄) = (Base‘𝑄) | |
40 | eqid 2734 | . . 3 ⊢ (.r‘𝑄) = (.r‘𝑄) | |
41 | 39, 40 | isringrng 20300 | . 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 1536 ∈ wcel 2105 ∀wral 3058 ∃wrex 3067 {csn 4630 ∪ ciun 4995 × cxp 5686 ‘cfv 6562 (class class class)co 7430 0cc0 11152 1c1 11153 ℤcz 12610 Basecbs 17244 ↾s cress 17273 .rcmulr 17298 /s cqus 17551 ×s cxps 17552 SubGrpcsubg 19150 ~QG cqg 19152 Rngcrng 20169 1rcur 20198 Ringcrg 20250 2Idealc2idl 21276 ℤringczring 21474 |
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 ax-addf 11231 ax-mulf 11232 |
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-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 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-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-cring 20253 df-oppr 20350 df-subrng 20562 df-subrg 20586 df-lss 20947 df-sra 21189 df-rgmod 21190 df-lidl 21235 df-2idl 21277 df-cnfld 21382 df-zring 21475 |
This theorem is referenced by: pzriprnglem14 21522 pzriprngALT 21523 pzriprng1ALT 21524 |
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