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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 10970 | . . . . . 6 ⊢ 0 ∈ V | |
2 | 1ex 10972 | . . . . . 6 ⊢ 1 ∈ V | |
3 | 1, 2 | pm3.2i 471 | . . . . 5 ⊢ (0 ∈ V ∧ 1 ∈ V) |
4 | 0ne1 12044 | . . . . 5 ⊢ 0 ≠ 1 | |
5 | fprg 7024 | . . . . 5 ⊢ (((0 ∈ V ∧ 1 ∈ V) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 0 ≠ 1) → {〈0, 𝐴〉, 〈1, 𝐵〉}:{0, 1}⟶{𝐴, 𝐵}) | |
6 | 3, 4, 5 | mp3an13 1451 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {〈0, 𝐴〉, 〈1, 𝐵〉}:{0, 1}⟶{𝐴, 𝐵}) |
7 | prssi 4760 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {𝐴, 𝐵} ⊆ ℤ) | |
8 | zringbas 20674 | . . . . 5 ⊢ ℤ = (Base‘ℤring) | |
9 | 7, 8 | sseqtrdi 3976 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {𝐴, 𝐵} ⊆ (Base‘ℤring)) |
10 | 6, 9 | fssd 6616 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {〈0, 𝐴〉, 〈1, 𝐵〉}:{0, 1}⟶(Base‘ℤring)) |
11 | fvex 6784 | . . . . 5 ⊢ (Base‘ℤring) ∈ V | |
12 | prex 5359 | . . . . 5 ⊢ {0, 1} ∈ V | |
13 | 11, 12 | pm3.2i 471 | . . . 4 ⊢ ((Base‘ℤring) ∈ V ∧ {0, 1} ∈ V) |
14 | elmapg 8611 | . . . 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 20671 | . . . 4 ⊢ ℤring ∈ Ring | |
18 | prfi 9067 | . . . 4 ⊢ {0, 1} ∈ Fin | |
19 | 17, 18 | pm3.2i 471 | . . 3 ⊢ (ℤring ∈ Ring ∧ {0, 1} ∈ Fin) |
20 | zlmodzxz.z | . . . 4 ⊢ 𝑍 = (ℤring freeLMod {0, 1}) | |
21 | eqid 2740 | . . . 4 ⊢ (Base‘ℤring) = (Base‘ℤring) | |
22 | 20, 21 | frlmfibas 20967 | . . 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 2843 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {〈0, 𝐴〉, 〈1, 𝐵〉} ∈ (Base‘𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ≠ wne 2945 Vcvv 3431 {cpr 4569 〈cop 4573 ⟶wf 6428 ‘cfv 6432 (class class class)co 7271 ↑m cmap 8598 Fincfn 8716 0cc0 10872 1c1 10873 ℤcz 12319 Basecbs 16910 Ringcrg 19781 ℤringczring 20668 freeLMod cfrlm 20951 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-addf 10951 ax-mulf 10952 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-supp 7969 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-er 8481 df-map 8600 df-ixp 8669 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-fsupp 9107 df-sup 9179 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12437 df-uz 12582 df-fz 13239 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-starv 16975 df-sca 16976 df-vsca 16977 df-ip 16978 df-tset 16979 df-ple 16980 df-ds 16982 df-unif 16983 df-hom 16984 df-cco 16985 df-0g 17150 df-prds 17156 df-pws 17158 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-grp 18578 df-minusg 18579 df-subg 18750 df-cmn 19386 df-mgp 19719 df-ur 19736 df-ring 19783 df-cring 19784 df-subrg 20020 df-sra 20432 df-rgmod 20433 df-cnfld 20596 df-zring 20669 df-dsmm 20937 df-frlm 20952 |
This theorem is referenced by: zlmodzxzscm 45662 zlmodzxzadd 45663 zlmodzxzsubm 45664 zlmodzxzsub 45665 zlmodzxzldeplem3 45812 zlmodzxzldep 45814 ldepsnlinclem1 45815 ldepsnlinclem2 45816 ldepsnlinc 45818 |
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