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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 10623 | . . . . . 6 ⊢ 0 ∈ V | |
2 | 1ex 10625 | . . . . . 6 ⊢ 1 ∈ V | |
3 | 1, 2 | pm3.2i 471 | . . . . 5 ⊢ (0 ∈ V ∧ 1 ∈ V) |
4 | 0ne1 11696 | . . . . 5 ⊢ 0 ≠ 1 | |
5 | fprg 6909 | . . . . 5 ⊢ (((0 ∈ V ∧ 1 ∈ V) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 0 ≠ 1) → {〈0, 𝐴〉, 〈1, 𝐵〉}:{0, 1}⟶{𝐴, 𝐵}) | |
6 | 3, 4, 5 | mp3an13 1443 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {〈0, 𝐴〉, 〈1, 𝐵〉}:{0, 1}⟶{𝐴, 𝐵}) |
7 | prssi 4746 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {𝐴, 𝐵} ⊆ ℤ) | |
8 | zringbas 20551 | . . . . 5 ⊢ ℤ = (Base‘ℤring) | |
9 | 7, 8 | sseqtrdi 4014 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {𝐴, 𝐵} ⊆ (Base‘ℤring)) |
10 | 6, 9 | fssd 6521 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {〈0, 𝐴〉, 〈1, 𝐵〉}:{0, 1}⟶(Base‘ℤring)) |
11 | fvex 6676 | . . . . 5 ⊢ (Base‘ℤring) ∈ V | |
12 | prex 5323 | . . . . 5 ⊢ {0, 1} ∈ V | |
13 | 11, 12 | pm3.2i 471 | . . . 4 ⊢ ((Base‘ℤring) ∈ V ∧ {0, 1} ∈ V) |
14 | elmapg 8408 | . . . 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 258 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {〈0, 𝐴〉, 〈1, 𝐵〉} ∈ ((Base‘ℤring) ↑m {0, 1})) |
17 | zringring 20548 | . . . 4 ⊢ ℤring ∈ Ring | |
18 | prfi 8781 | . . . 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 2818 | . . . 4 ⊢ (Base‘ℤring) = (Base‘ℤring) | |
22 | 20, 21 | frlmfibas 20834 | . . 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 2912 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {〈0, 𝐴〉, 〈1, 𝐵〉} ∈ (Base‘𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 Vcvv 3492 {cpr 4559 〈cop 4563 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ↑m cmap 8395 Fincfn 8497 0cc0 10525 1c1 10526 ℤcz 11969 Basecbs 16471 Ringcrg 19226 ℤringzring 20545 freeLMod cfrlm 20818 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-sup 8894 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-hom 16577 df-cco 16578 df-0g 16703 df-prds 16709 df-pws 16711 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-subg 18214 df-cmn 18837 df-mgp 19169 df-ur 19181 df-ring 19228 df-cring 19229 df-subrg 19462 df-sra 19873 df-rgmod 19874 df-cnfld 20474 df-zring 20546 df-dsmm 20804 df-frlm 20819 |
This theorem is referenced by: zlmodzxzscm 44333 zlmodzxzadd 44334 zlmodzxzsubm 44335 zlmodzxzsub 44336 zlmodzxzldeplem3 44485 zlmodzxzldep 44487 ldepsnlinclem1 44488 ldepsnlinclem2 44489 ldepsnlinc 44491 |
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