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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 41725, 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 41734) and a multiplicative semigroup (see 2zrngmsgrp 41737). (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 41734 | . 2 ⊢ 𝑅 ∈ Abel |
4 | 2zrngmmgm.1 | . . 3 ⊢ 𝑀 = (mulGrp‘𝑅) | |
5 | 1, 2, 4 | 2zrngmsgrp 41737 | . 2 ⊢ 𝑀 ∈ SGrp |
6 | elrabi 3328 | . . . . . 6 ⊢ (𝑎 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} → 𝑎 ∈ ℤ) | |
7 | 6 | zcnd 11359 | . . . . 5 ⊢ (𝑎 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} → 𝑎 ∈ ℂ) |
8 | 7, 1 | eleq2s 2706 | . . . 4 ⊢ (𝑎 ∈ 𝐸 → 𝑎 ∈ ℂ) |
9 | elrabi 3328 | . . . . . 6 ⊢ (𝑏 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} → 𝑏 ∈ ℤ) | |
10 | 9 | zcnd 11359 | . . . . 5 ⊢ (𝑏 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} → 𝑏 ∈ ℂ) |
11 | 10, 1 | eleq2s 2706 | . . . 4 ⊢ (𝑏 ∈ 𝐸 → 𝑏 ∈ ℂ) |
12 | elrabi 3328 | . . . . . 6 ⊢ (𝑦 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} → 𝑦 ∈ ℤ) | |
13 | 12 | zcnd 11359 | . . . . 5 ⊢ (𝑦 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} → 𝑦 ∈ ℂ) |
14 | 13, 1 | eleq2s 2706 | . . . 4 ⊢ (𝑦 ∈ 𝐸 → 𝑦 ∈ ℂ) |
15 | adddi 9904 | . . . . 5 ⊢ ((𝑎 ∈ ℂ ∧ 𝑏 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑎 · (𝑏 + 𝑦)) = ((𝑎 · 𝑏) + (𝑎 · 𝑦))) | |
16 | adddir 9910 | . . . . 5 ⊢ ((𝑎 ∈ ℂ ∧ 𝑏 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑎 + 𝑏) · 𝑦) = ((𝑎 · 𝑦) + (𝑏 · 𝑦))) | |
17 | 15, 16 | jca 553 | . . . 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 1360 | . . 3 ⊢ ((𝑎 ∈ 𝐸 ∧ 𝑏 ∈ 𝐸 ∧ 𝑦 ∈ 𝐸) → ((𝑎 · (𝑏 + 𝑦)) = ((𝑎 · 𝑏) + (𝑎 · 𝑦)) ∧ ((𝑎 + 𝑏) · 𝑦) = ((𝑎 · 𝑦) + (𝑏 · 𝑦)))) |
19 | 18 | rgen3 2959 | . 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 41726 | . . 3 ⊢ 𝐸 = (Base‘𝑅) |
21 | 1, 2 | 2zrngadd 41727 | . . 3 ⊢ + = (+g‘𝑅) |
22 | 1, 2 | 2zrngmul 41735 | . . 3 ⊢ · = (.r‘𝑅) |
23 | 20, 4, 21, 22 | isrng 41666 | . 2 ⊢ (𝑅 ∈ Rng ↔ (𝑅 ∈ Abel ∧ 𝑀 ∈ SGrp ∧ ∀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 1237 | 1 ⊢ 𝑅 ∈ Rng |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 {crab 2900 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 + caddc 9818 · cmul 9820 2c2 10947 ℤcz 11254 ↾s cress 15696 SGrpcsgrp 17106 Abelcabl 18017 mulGrpcmgp 18312 ℂfldccnfld 19567 Rngcrng 41664 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ring 18372 df-cring 18373 df-cnfld 19568 df-rng0 41665 |
This theorem is referenced by: (None) |
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