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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 11249 | . . . . . 6 ⊢ 0 ∈ V | |
2 | 1ex 11251 | . . . . . 6 ⊢ 1 ∈ V | |
3 | 1, 2 | pm3.2i 469 | . . . . 5 ⊢ (0 ∈ V ∧ 1 ∈ V) |
4 | 0ne1 12329 | . . . . 5 ⊢ 0 ≠ 1 | |
5 | fprg 7161 | . . . . 5 ⊢ (((0 ∈ V ∧ 1 ∈ V) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 0 ≠ 1) → {〈0, 𝐴〉, 〈1, 𝐵〉}:{0, 1}⟶{𝐴, 𝐵}) | |
6 | 3, 4, 5 | mp3an13 1449 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {〈0, 𝐴〉, 〈1, 𝐵〉}:{0, 1}⟶{𝐴, 𝐵}) |
7 | prssi 4820 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {𝐴, 𝐵} ⊆ ℤ) | |
8 | zringbas 21439 | . . . . 5 ⊢ ℤ = (Base‘ℤring) | |
9 | 7, 8 | sseqtrdi 4029 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {𝐴, 𝐵} ⊆ (Base‘ℤring)) |
10 | 6, 9 | fssd 6737 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {〈0, 𝐴〉, 〈1, 𝐵〉}:{0, 1}⟶(Base‘ℤring)) |
11 | fvex 6906 | . . . . 5 ⊢ (Base‘ℤring) ∈ V | |
12 | prex 5430 | . . . . 5 ⊢ {0, 1} ∈ V | |
13 | 11, 12 | pm3.2i 469 | . . . 4 ⊢ ((Base‘ℤring) ∈ V ∧ {0, 1} ∈ V) |
14 | elmapg 8860 | . . . 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 256 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {〈0, 𝐴〉, 〈1, 𝐵〉} ∈ ((Base‘ℤring) ↑m {0, 1})) |
17 | zringring 21435 | . . . 4 ⊢ ℤring ∈ Ring | |
18 | prfi 9358 | . . . 4 ⊢ {0, 1} ∈ Fin | |
19 | 17, 18 | pm3.2i 469 | . . 3 ⊢ (ℤring ∈ Ring ∧ {0, 1} ∈ Fin) |
20 | zlmodzxz.z | . . . 4 ⊢ 𝑍 = (ℤring freeLMod {0, 1}) | |
21 | eqid 2726 | . . . 4 ⊢ (Base‘ℤring) = (Base‘ℤring) | |
22 | 20, 21 | frlmfibas 21756 | . . 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 2828 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {〈0, 𝐴〉, 〈1, 𝐵〉} ∈ (Base‘𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 Vcvv 3462 {cpr 4625 〈cop 4629 ⟶wf 6542 ‘cfv 6546 (class class class)co 7416 ↑m cmap 8847 Fincfn 8966 0cc0 11149 1c1 11150 ℤcz 12604 Basecbs 17208 Ringcrg 20212 ℤringczring 21432 freeLMod cfrlm 21740 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-addf 11228 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7995 df-2nd 7996 df-supp 8167 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-2o 8489 df-er 8726 df-map 8849 df-ixp 8919 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-fsupp 9399 df-sup 9478 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-nn 12259 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12519 df-z 12605 df-dec 12724 df-uz 12869 df-fz 13533 df-struct 17144 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-mulr 17275 df-starv 17276 df-sca 17277 df-vsca 17278 df-ip 17279 df-tset 17280 df-ple 17281 df-ds 17283 df-unif 17284 df-hom 17285 df-cco 17286 df-0g 17451 df-prds 17457 df-pws 17459 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-grp 18926 df-minusg 18927 df-subg 19113 df-cmn 19776 df-abl 19777 df-mgp 20114 df-rng 20132 df-ur 20161 df-ring 20214 df-cring 20215 df-subrng 20524 df-subrg 20549 df-sra 21147 df-rgmod 21148 df-cnfld 21340 df-zring 21433 df-dsmm 21726 df-frlm 21741 |
This theorem is referenced by: zlmodzxzscm 47772 zlmodzxzadd 47773 zlmodzxzsubm 47774 zlmodzxzsub 47775 zlmodzxzldeplem3 47921 zlmodzxzldep 47923 ldepsnlinclem1 47924 ldepsnlinclem2 47925 ldepsnlinc 47927 |
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