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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 11212 | . . . . . 6 ⊢ 0 ∈ V | |
2 | 1ex 11214 | . . . . . 6 ⊢ 1 ∈ V | |
3 | 1, 2 | pm3.2i 470 | . . . . 5 ⊢ (0 ∈ V ∧ 1 ∈ V) |
4 | 0ne1 12287 | . . . . 5 ⊢ 0 ≠ 1 | |
5 | fprg 7149 | . . . . 5 ⊢ (((0 ∈ V ∧ 1 ∈ V) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 0 ≠ 1) → {⟨0, 𝐴⟩, ⟨1, 𝐵⟩}:{0, 1}⟶{𝐴, 𝐵}) | |
6 | 3, 4, 5 | mp3an13 1448 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {⟨0, 𝐴⟩, ⟨1, 𝐵⟩}:{0, 1}⟶{𝐴, 𝐵}) |
7 | prssi 4819 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {𝐴, 𝐵} ⊆ ℤ) | |
8 | zringbas 21340 | . . . . 5 ⊢ ℤ = (Base‘ℤring) | |
9 | 7, 8 | sseqtrdi 4027 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {𝐴, 𝐵} ⊆ (Base‘ℤring)) |
10 | 6, 9 | fssd 6729 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {⟨0, 𝐴⟩, ⟨1, 𝐵⟩}:{0, 1}⟶(Base‘ℤring)) |
11 | fvex 6898 | . . . . 5 ⊢ (Base‘ℤring) ∈ V | |
12 | prex 5425 | . . . . 5 ⊢ {0, 1} ∈ V | |
13 | 11, 12 | pm3.2i 470 | . . . 4 ⊢ ((Base‘ℤring) ∈ V ∧ {0, 1} ∈ V) |
14 | elmapg 8835 | . . . 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 257 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∈ ((Base‘ℤring) ↑m {0, 1})) |
17 | zringring 21336 | . . . 4 ⊢ ℤring ∈ Ring | |
18 | prfi 9324 | . . . 4 ⊢ {0, 1} ∈ Fin | |
19 | 17, 18 | pm3.2i 470 | . . 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 21657 | . . 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 2829 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∈ (Base‘𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 Vcvv 3468 {cpr 4625 ⟨cop 4629 ⟶wf 6533 ‘cfv 6537 (class class class)co 7405 ↑m cmap 8822 Fincfn 8941 0cc0 11112 1c1 11113 ℤcz 12562 Basecbs 17153 Ringcrg 20138 ℤringczring 21333 freeLMod cfrlm 21641 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-addf 11191 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13491 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-starv 17221 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-unif 17229 df-hom 17230 df-cco 17231 df-0g 17396 df-prds 17402 df-pws 17404 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-grp 18866 df-minusg 18867 df-subg 19050 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-ring 20140 df-cring 20141 df-subrng 20446 df-subrg 20471 df-sra 21021 df-rgmod 21022 df-cnfld 21241 df-zring 21334 df-dsmm 21627 df-frlm 21642 |
This theorem is referenced by: zlmodzxzscm 47309 zlmodzxzadd 47310 zlmodzxzsubm 47311 zlmodzxzsub 47312 zlmodzxzldeplem3 47458 zlmodzxzldep 47460 ldepsnlinclem1 47461 ldepsnlinclem2 47462 ldepsnlinc 47464 |
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