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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > zlmodzxzel | Structured version Visualization version GIF version |
Description: An element of the (base set of the) ℤ-module ℤ × ℤ. (Contributed by AV, 21-May-2019.) (Revised by AV, 10-Jun-2019.) |
Ref | Expression |
---|---|
zlmodzxz.z | ⊢ 𝑍 = (ℤring freeLMod {0, 1}) |
Ref | Expression |
---|---|
zlmodzxzel | ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {〈0, 𝐴〉, 〈1, 𝐵〉} ∈ (Base‘𝑍)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | c0ex 10435 | . . . . . 6 ⊢ 0 ∈ V | |
2 | 1ex 10437 | . . . . . 6 ⊢ 1 ∈ V | |
3 | 1, 2 | pm3.2i 463 | . . . . 5 ⊢ (0 ∈ V ∧ 1 ∈ V) |
4 | 0ne1 11514 | . . . . 5 ⊢ 0 ≠ 1 | |
5 | fprg 6742 | . . . . 5 ⊢ (((0 ∈ V ∧ 1 ∈ V) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 0 ≠ 1) → {〈0, 𝐴〉, 〈1, 𝐵〉}:{0, 1}⟶{𝐴, 𝐵}) | |
6 | 3, 4, 5 | mp3an13 1431 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {〈0, 𝐴〉, 〈1, 𝐵〉}:{0, 1}⟶{𝐴, 𝐵}) |
7 | prssi 4629 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {𝐴, 𝐵} ⊆ ℤ) | |
8 | zringbas 20328 | . . . . 5 ⊢ ℤ = (Base‘ℤring) | |
9 | 7, 8 | syl6sseq 3909 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {𝐴, 𝐵} ⊆ (Base‘ℤring)) |
10 | 6, 9 | fssd 6360 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {〈0, 𝐴〉, 〈1, 𝐵〉}:{0, 1}⟶(Base‘ℤring)) |
11 | fvex 6514 | . . . . 5 ⊢ (Base‘ℤring) ∈ V | |
12 | prex 5190 | . . . . 5 ⊢ {0, 1} ∈ V | |
13 | 11, 12 | pm3.2i 463 | . . . 4 ⊢ ((Base‘ℤring) ∈ V ∧ {0, 1} ∈ V) |
14 | elmapg 8221 | . . . 4 ⊢ (((Base‘ℤring) ∈ V ∧ {0, 1} ∈ V) → ({〈0, 𝐴〉, 〈1, 𝐵〉} ∈ ((Base‘ℤring) ↑𝑚 {0, 1}) ↔ {〈0, 𝐴〉, 〈1, 𝐵〉}:{0, 1}⟶(Base‘ℤring))) | |
15 | 13, 14 | mp1i 13 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ({〈0, 𝐴〉, 〈1, 𝐵〉} ∈ ((Base‘ℤring) ↑𝑚 {0, 1}) ↔ {〈0, 𝐴〉, 〈1, 𝐵〉}:{0, 1}⟶(Base‘ℤring))) |
16 | 10, 15 | mpbird 249 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {〈0, 𝐴〉, 〈1, 𝐵〉} ∈ ((Base‘ℤring) ↑𝑚 {0, 1})) |
17 | zringring 20325 | . . . 4 ⊢ ℤring ∈ Ring | |
18 | prfi 8590 | . . . 4 ⊢ {0, 1} ∈ Fin | |
19 | 17, 18 | pm3.2i 463 | . . 3 ⊢ (ℤring ∈ Ring ∧ {0, 1} ∈ Fin) |
20 | zlmodzxz.z | . . . 4 ⊢ 𝑍 = (ℤring freeLMod {0, 1}) | |
21 | eqid 2778 | . . . 4 ⊢ (Base‘ℤring) = (Base‘ℤring) | |
22 | 20, 21 | frlmfibas 20611 | . . 3 ⊢ ((ℤring ∈ Ring ∧ {0, 1} ∈ Fin) → ((Base‘ℤring) ↑𝑚 {0, 1}) = (Base‘𝑍)) |
23 | 19, 22 | mp1i 13 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((Base‘ℤring) ↑𝑚 {0, 1}) = (Base‘𝑍)) |
24 | 16, 23 | eleqtrd 2868 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {〈0, 𝐴〉, 〈1, 𝐵〉} ∈ (Base‘𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ≠ wne 2967 Vcvv 3415 {cpr 4444 〈cop 4448 ⟶wf 6186 ‘cfv 6190 (class class class)co 6978 ↑𝑚 cmap 8208 Fincfn 8308 0cc0 10337 1c1 10338 ℤcz 11796 Basecbs 16342 Ringcrg 19023 ℤringzring 20322 freeLMod cfrlm 20595 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5050 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-cnex 10393 ax-resscn 10394 ax-1cn 10395 ax-icn 10396 ax-addcl 10397 ax-addrcl 10398 ax-mulcl 10399 ax-mulrcl 10400 ax-mulcom 10401 ax-addass 10402 ax-mulass 10403 ax-distr 10404 ax-i2m1 10405 ax-1ne0 10406 ax-1rid 10407 ax-rnegex 10408 ax-rrecex 10409 ax-cnre 10410 ax-pre-lttri 10411 ax-pre-lttrn 10412 ax-pre-ltadd 10413 ax-pre-mulgt0 10414 ax-addf 10416 ax-mulf 10417 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-int 4751 df-iun 4795 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-riota 6939 df-ov 6981 df-oprab 6982 df-mpo 6983 df-om 7399 df-1st 7503 df-2nd 7504 df-supp 7636 df-wrecs 7752 df-recs 7814 df-rdg 7852 df-1o 7907 df-oadd 7911 df-er 8091 df-map 8210 df-ixp 8262 df-en 8309 df-dom 8310 df-sdom 8311 df-fin 8312 df-fsupp 8631 df-sup 8703 df-pnf 10478 df-mnf 10479 df-xr 10480 df-ltxr 10481 df-le 10482 df-sub 10674 df-neg 10675 df-nn 11442 df-2 11506 df-3 11507 df-4 11508 df-5 11509 df-6 11510 df-7 11511 df-8 11512 df-9 11513 df-n0 11711 df-z 11797 df-dec 11915 df-uz 12062 df-fz 12712 df-struct 16344 df-ndx 16345 df-slot 16346 df-base 16348 df-sets 16349 df-ress 16350 df-plusg 16437 df-mulr 16438 df-starv 16439 df-sca 16440 df-vsca 16441 df-ip 16442 df-tset 16443 df-ple 16444 df-ds 16446 df-unif 16447 df-hom 16448 df-cco 16449 df-0g 16574 df-prds 16580 df-pws 16582 df-mgm 17713 df-sgrp 17755 df-mnd 17766 df-grp 17897 df-minusg 17898 df-subg 18063 df-cmn 18671 df-mgp 18966 df-ur 18978 df-ring 19025 df-cring 19026 df-subrg 19259 df-sra 19669 df-rgmod 19670 df-cnfld 20251 df-zring 20323 df-dsmm 20581 df-frlm 20596 |
This theorem is referenced by: zlmodzxzscm 43770 zlmodzxzadd 43771 zlmodzxzsubm 43772 zlmodzxzsub 43773 zlmodzxzldeplem3 43925 zlmodzxzldep 43927 ldepsnlinclem1 43928 ldepsnlinclem2 43929 ldepsnlinc 43931 |
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