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Mirrors > Home > MPE Home > Th. List > Mathboxes > 2zrngacmnd | Structured version Visualization version GIF version |
Description: R is a commutative (additive) monoid. (Contributed by AV, 11-Feb-2020.) |
Ref | Expression |
---|---|
2zrng.e | ⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} |
2zrngbas.r | ⊢ 𝑅 = (ℂfld ↾s 𝐸) |
Ref | Expression |
---|---|
2zrngacmnd | ⊢ 𝑅 ∈ CMnd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2zrng.e | . . 3 ⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} | |
2 | 1 | 0even 42928 | . 2 ⊢ 0 ∈ 𝐸 |
3 | 2zrngbas.r | . . . . 5 ⊢ 𝑅 = (ℂfld ↾s 𝐸) | |
4 | 1, 3 | 2zrngbas 42933 | . . . 4 ⊢ 𝐸 = (Base‘𝑅) |
5 | 4 | a1i 11 | . . 3 ⊢ (0 ∈ 𝐸 → 𝐸 = (Base‘𝑅)) |
6 | 1, 3 | 2zrngadd 42934 | . . . 4 ⊢ + = (+g‘𝑅) |
7 | 6 | a1i 11 | . . 3 ⊢ (0 ∈ 𝐸 → + = (+g‘𝑅)) |
8 | 1, 3 | 2zrngamnd 42938 | . . . 4 ⊢ 𝑅 ∈ Mnd |
9 | 8 | a1i 11 | . . 3 ⊢ (0 ∈ 𝐸 → 𝑅 ∈ Mnd) |
10 | elrabi 3566 | . . . . . . . 8 ⊢ (𝑥 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} → 𝑥 ∈ ℤ) | |
11 | 10 | zcnd 11835 | . . . . . . 7 ⊢ (𝑥 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} → 𝑥 ∈ ℂ) |
12 | 11, 1 | eleq2s 2876 | . . . . . 6 ⊢ (𝑥 ∈ 𝐸 → 𝑥 ∈ ℂ) |
13 | 12 | adantr 474 | . . . . 5 ⊢ ((𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐸) → 𝑥 ∈ ℂ) |
14 | elrabi 3566 | . . . . . . . 8 ⊢ (𝑦 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} → 𝑦 ∈ ℤ) | |
15 | 14 | zcnd 11835 | . . . . . . 7 ⊢ (𝑦 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} → 𝑦 ∈ ℂ) |
16 | 15, 1 | eleq2s 2876 | . . . . . 6 ⊢ (𝑦 ∈ 𝐸 → 𝑦 ∈ ℂ) |
17 | 16 | adantl 475 | . . . . 5 ⊢ ((𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐸) → 𝑦 ∈ ℂ) |
18 | 13, 17 | addcomd 10578 | . . . 4 ⊢ ((𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐸) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) |
19 | 18 | 3adant1 1121 | . . 3 ⊢ ((0 ∈ 𝐸 ∧ 𝑥 ∈ 𝐸 ∧ 𝑦 ∈ 𝐸) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) |
20 | 5, 7, 9, 19 | iscmnd 18591 | . 2 ⊢ (0 ∈ 𝐸 → 𝑅 ∈ CMnd) |
21 | 2, 20 | ax-mp 5 | 1 ⊢ 𝑅 ∈ CMnd |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 386 = wceq 1601 ∈ wcel 2106 ∃wrex 3090 {crab 3093 ‘cfv 6135 (class class class)co 6922 ℂcc 10270 0cc0 10272 + caddc 10275 · cmul 10277 2c2 11430 ℤcz 11728 Basecbs 16255 ↾s cress 16256 +gcplusg 16338 Mndcmnd 17680 CMndccmn 18579 ℂfldccnfld 20142 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-addf 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-fz 12644 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-starv 16353 df-tset 16357 df-ple 16358 df-ds 16360 df-unif 16361 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-cmn 18581 df-cnfld 20143 |
This theorem is referenced by: 2zrngaabl 42941 |
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