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| Mirrors > Home > MPE Home > Th. List > Mathboxes > 2zrngALT | Structured version Visualization version GIF version | ||
| Description: The ring of integers restricted to the even integers is a non-unital ring, the "ring of even integers". Alternate version of 2zrng 48162, based on a restriction of the field of the complex numbers. The proof is based on the facts that the ring of even integers is an additive abelian group (see 2zrngaabl 48171) and a multiplicative semigroup (see 2zrngmsgrp 48174). (Contributed by AV, 11-Feb-2020.) (New usage is discouraged.) (Proof modification is discouraged.) | 
| Ref | Expression | 
|---|---|
| 2zrng.e | ⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} | 
| 2zrngbas.r | ⊢ 𝑅 = (ℂfld ↾s 𝐸) | 
| 2zrngmmgm.1 | ⊢ 𝑀 = (mulGrp‘𝑅) | 
| Ref | Expression | 
|---|---|
| 2zrngALT | ⊢ 𝑅 ∈ Rng | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 2zrng.e | . . 3 ⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} | |
| 2 | 2zrngbas.r | . . 3 ⊢ 𝑅 = (ℂfld ↾s 𝐸) | |
| 3 | 1, 2 | 2zrngaabl 48171 | . 2 ⊢ 𝑅 ∈ Abel | 
| 4 | 2zrngmmgm.1 | . . 3 ⊢ 𝑀 = (mulGrp‘𝑅) | |
| 5 | 1, 2, 4 | 2zrngmsgrp 48174 | . 2 ⊢ 𝑀 ∈ Smgrp | 
| 6 | elrabi 3686 | . . . . . 6 ⊢ (𝑎 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} → 𝑎 ∈ ℤ) | |
| 7 | 6 | zcnd 12725 | . . . . 5 ⊢ (𝑎 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} → 𝑎 ∈ ℂ) | 
| 8 | 7, 1 | eleq2s 2858 | . . . 4 ⊢ (𝑎 ∈ 𝐸 → 𝑎 ∈ ℂ) | 
| 9 | elrabi 3686 | . . . . . 6 ⊢ (𝑏 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} → 𝑏 ∈ ℤ) | |
| 10 | 9 | zcnd 12725 | . . . . 5 ⊢ (𝑏 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} → 𝑏 ∈ ℂ) | 
| 11 | 10, 1 | eleq2s 2858 | . . . 4 ⊢ (𝑏 ∈ 𝐸 → 𝑏 ∈ ℂ) | 
| 12 | elrabi 3686 | . . . . . 6 ⊢ (𝑦 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} → 𝑦 ∈ ℤ) | |
| 13 | 12 | zcnd 12725 | . . . . 5 ⊢ (𝑦 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} → 𝑦 ∈ ℂ) | 
| 14 | 13, 1 | eleq2s 2858 | . . . 4 ⊢ (𝑦 ∈ 𝐸 → 𝑦 ∈ ℂ) | 
| 15 | adddi 11245 | . . . . 5 ⊢ ((𝑎 ∈ ℂ ∧ 𝑏 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑎 · (𝑏 + 𝑦)) = ((𝑎 · 𝑏) + (𝑎 · 𝑦))) | |
| 16 | adddir 11253 | . . . . 5 ⊢ ((𝑎 ∈ ℂ ∧ 𝑏 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑎 + 𝑏) · 𝑦) = ((𝑎 · 𝑦) + (𝑏 · 𝑦))) | |
| 17 | 15, 16 | jca 511 | . . . 4 ⊢ ((𝑎 ∈ ℂ ∧ 𝑏 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑎 · (𝑏 + 𝑦)) = ((𝑎 · 𝑏) + (𝑎 · 𝑦)) ∧ ((𝑎 + 𝑏) · 𝑦) = ((𝑎 · 𝑦) + (𝑏 · 𝑦)))) | 
| 18 | 8, 11, 14, 17 | syl3an 1160 | . . 3 ⊢ ((𝑎 ∈ 𝐸 ∧ 𝑏 ∈ 𝐸 ∧ 𝑦 ∈ 𝐸) → ((𝑎 · (𝑏 + 𝑦)) = ((𝑎 · 𝑏) + (𝑎 · 𝑦)) ∧ ((𝑎 + 𝑏) · 𝑦) = ((𝑎 · 𝑦) + (𝑏 · 𝑦)))) | 
| 19 | 18 | rgen3 3203 | . 2 ⊢ ∀𝑎 ∈ 𝐸 ∀𝑏 ∈ 𝐸 ∀𝑦 ∈ 𝐸 ((𝑎 · (𝑏 + 𝑦)) = ((𝑎 · 𝑏) + (𝑎 · 𝑦)) ∧ ((𝑎 + 𝑏) · 𝑦) = ((𝑎 · 𝑦) + (𝑏 · 𝑦))) | 
| 20 | 1, 2 | 2zrngbas 48163 | . . 3 ⊢ 𝐸 = (Base‘𝑅) | 
| 21 | 1, 2 | 2zrngadd 48164 | . . 3 ⊢ + = (+g‘𝑅) | 
| 22 | 1, 2 | 2zrngmul 48172 | . . 3 ⊢ · = (.r‘𝑅) | 
| 23 | 20, 4, 21, 22 | isrng 20152 | . 2 ⊢ (𝑅 ∈ Rng ↔ (𝑅 ∈ Abel ∧ 𝑀 ∈ Smgrp ∧ ∀𝑎 ∈ 𝐸 ∀𝑏 ∈ 𝐸 ∀𝑦 ∈ 𝐸 ((𝑎 · (𝑏 + 𝑦)) = ((𝑎 · 𝑏) + (𝑎 · 𝑦)) ∧ ((𝑎 + 𝑏) · 𝑦) = ((𝑎 · 𝑦) + (𝑏 · 𝑦))))) | 
| 24 | 3, 5, 19, 23 | mpbir3an 1341 | 1 ⊢ 𝑅 ∈ Rng | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ∀wral 3060 ∃wrex 3069 {crab 3435 ‘cfv 6560 (class class class)co 7432 ℂcc 11154 + caddc 11159 · cmul 11161 2c2 12322 ℤcz 12615 ↾s cress 17275 Smgrpcsgrp 18732 Abelcabl 19800 mulGrpcmgp 20138 Rngcrng 20150 ℂfldccnfld 21365 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-addf 11235 ax-mulf 11236 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-fz 13549 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-starv 17313 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-0g 17487 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-grp 18955 df-cmn 19801 df-abl 19802 df-mgp 20139 df-rng 20151 df-ring 20233 df-cring 20234 df-cnfld 21366 | 
| This theorem is referenced by: (None) | 
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