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| Mirrors > Home > MPE Home > Th. List > pzriprnglem7 | Structured version Visualization version GIF version | ||
| Description: Lemma 7 for pzriprng 21508: 𝐽 is a unital ring. (Contributed by AV, 19-Mar-2025.) |
| Ref | Expression |
|---|---|
| pzriprng.r | ⊢ 𝑅 = (ℤring ×s ℤring) |
| pzriprng.i | ⊢ 𝐼 = (ℤ × {0}) |
| pzriprng.j | ⊢ 𝐽 = (𝑅 ↾s 𝐼) |
| Ref | Expression |
|---|---|
| pzriprnglem7 | ⊢ 𝐽 ∈ Ring |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | pzriprng.r | . . . 4 ⊢ 𝑅 = (ℤring ×s ℤring) | |
| 2 | pzriprng.i | . . . 4 ⊢ 𝐼 = (ℤ × {0}) | |
| 3 | 1, 2 | pzriprnglem5 21496 | . . 3 ⊢ 𝐼 ∈ (SubRng‘𝑅) |
| 4 | pzriprng.j | . . . 4 ⊢ 𝐽 = (𝑅 ↾s 𝐼) | |
| 5 | 4 | subrngrng 20550 | . . 3 ⊢ (𝐼 ∈ (SubRng‘𝑅) → 𝐽 ∈ Rng) |
| 6 | 3, 5 | ax-mp 5 | . 2 ⊢ 𝐽 ∈ Rng |
| 7 | 1z 12647 | . . . . 5 ⊢ 1 ∈ ℤ | |
| 8 | c0ex 11255 | . . . . . 6 ⊢ 0 ∈ V | |
| 9 | 8 | snid 4662 | . . . . 5 ⊢ 0 ∈ {0} |
| 10 | 7, 9 | opelxpii 5723 | . . . 4 ⊢ 〈1, 0〉 ∈ (ℤ × {0}) |
| 11 | 4 | subrngbas 20554 | . . . . . 6 ⊢ (𝐼 ∈ (SubRng‘𝑅) → 𝐼 = (Base‘𝐽)) |
| 12 | 3, 11 | ax-mp 5 | . . . . 5 ⊢ 𝐼 = (Base‘𝐽) |
| 13 | 12, 2 | eqtr3i 2767 | . . . 4 ⊢ (Base‘𝐽) = (ℤ × {0}) |
| 14 | 10, 13 | eleqtrri 2840 | . . 3 ⊢ 〈1, 0〉 ∈ (Base‘𝐽) |
| 15 | oveq1 7438 | . . . . . 6 ⊢ (𝑖 = 〈1, 0〉 → (𝑖(.r‘𝐽)𝑥) = (〈1, 0〉(.r‘𝐽)𝑥)) | |
| 16 | 15 | eqeq1d 2739 | . . . . 5 ⊢ (𝑖 = 〈1, 0〉 → ((𝑖(.r‘𝐽)𝑥) = 𝑥 ↔ (〈1, 0〉(.r‘𝐽)𝑥) = 𝑥)) |
| 17 | 16 | ovanraleqv 7455 | . . . 4 ⊢ (𝑖 = 〈1, 0〉 → (∀𝑥 ∈ (Base‘𝐽)((𝑖(.r‘𝐽)𝑥) = 𝑥 ∧ (𝑥(.r‘𝐽)𝑖) = 𝑥) ↔ ∀𝑥 ∈ (Base‘𝐽)((〈1, 0〉(.r‘𝐽)𝑥) = 𝑥 ∧ (𝑥(.r‘𝐽)〈1, 0〉) = 𝑥))) |
| 18 | id 22 | . . . 4 ⊢ (〈1, 0〉 ∈ (Base‘𝐽) → 〈1, 0〉 ∈ (Base‘𝐽)) | |
| 19 | 12 | eleq2i 2833 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐼 ↔ 𝑥 ∈ (Base‘𝐽)) |
| 20 | 1, 2, 4 | pzriprnglem6 21497 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐼 → ((〈1, 0〉(.r‘𝐽)𝑥) = 𝑥 ∧ (𝑥(.r‘𝐽)〈1, 0〉) = 𝑥)) |
| 21 | 19, 20 | sylbir 235 | . . . . . 6 ⊢ (𝑥 ∈ (Base‘𝐽) → ((〈1, 0〉(.r‘𝐽)𝑥) = 𝑥 ∧ (𝑥(.r‘𝐽)〈1, 0〉) = 𝑥)) |
| 22 | 21 | a1i 11 | . . . . 5 ⊢ (〈1, 0〉 ∈ (Base‘𝐽) → (𝑥 ∈ (Base‘𝐽) → ((〈1, 0〉(.r‘𝐽)𝑥) = 𝑥 ∧ (𝑥(.r‘𝐽)〈1, 0〉) = 𝑥))) |
| 23 | 22 | ralrimiv 3145 | . . . 4 ⊢ (〈1, 0〉 ∈ (Base‘𝐽) → ∀𝑥 ∈ (Base‘𝐽)((〈1, 0〉(.r‘𝐽)𝑥) = 𝑥 ∧ (𝑥(.r‘𝐽)〈1, 0〉) = 𝑥)) |
| 24 | 17, 18, 23 | rspcedvdw 3625 | . . 3 ⊢ (〈1, 0〉 ∈ (Base‘𝐽) → ∃𝑖 ∈ (Base‘𝐽)∀𝑥 ∈ (Base‘𝐽)((𝑖(.r‘𝐽)𝑥) = 𝑥 ∧ (𝑥(.r‘𝐽)𝑖) = 𝑥)) |
| 25 | 14, 24 | ax-mp 5 | . 2 ⊢ ∃𝑖 ∈ (Base‘𝐽)∀𝑥 ∈ (Base‘𝐽)((𝑖(.r‘𝐽)𝑥) = 𝑥 ∧ (𝑥(.r‘𝐽)𝑖) = 𝑥) |
| 26 | eqid 2737 | . . 3 ⊢ (Base‘𝐽) = (Base‘𝐽) | |
| 27 | eqid 2737 | . . 3 ⊢ (.r‘𝐽) = (.r‘𝐽) | |
| 28 | 26, 27 | isringrng 20284 | . 2 ⊢ (𝐽 ∈ Ring ↔ (𝐽 ∈ Rng ∧ ∃𝑖 ∈ (Base‘𝐽)∀𝑥 ∈ (Base‘𝐽)((𝑖(.r‘𝐽)𝑥) = 𝑥 ∧ (𝑥(.r‘𝐽)𝑖) = 𝑥))) |
| 29 | 6, 25, 28 | mpbir2an 711 | 1 ⊢ 𝐽 ∈ Ring |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 {csn 4626 〈cop 4632 × cxp 5683 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 ℤcz 12613 Basecbs 17247 ↾s cress 17274 .rcmulr 17298 ×s cxps 17551 Rngcrng 20149 Ringcrg 20230 SubRngcsubrng 20545 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-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 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-xps 17555 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-cring 20233 df-subrng 20546 df-subrg 20570 df-cnfld 21365 df-zring 21458 |
| This theorem is referenced by: pzriprnglem9 21500 pzriprngALT 21506 pzriprng1ALT 21507 |
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