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Mirrors > Home > MPE Home > Th. List > Mathboxes > zlmodzxz0 | Structured version Visualization version GIF version |
Description: The 0 of the ℤ-module ℤ × ℤ. (Contributed by AV, 20-May-2019.) (Revised by AV, 10-Jun-2019.) |
Ref | Expression |
---|---|
zlmodzxz.z | ⊢ 𝑍 = (ℤring freeLMod {0, 1}) |
zlmodzxz.o | ⊢ 0 = {〈0, 0〉, 〈1, 0〉} |
Ref | Expression |
---|---|
zlmodzxz0 | ⊢ 0 = (0g‘𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zlmodzxz.o | . 2 ⊢ 0 = {〈0, 0〉, 〈1, 0〉} | |
2 | c0ex 10810 | . . 3 ⊢ 0 ∈ V | |
3 | 1ex 10812 | . . 3 ⊢ 1 ∈ V | |
4 | xpprsng 6944 | . . 3 ⊢ ((0 ∈ V ∧ 1 ∈ V ∧ 0 ∈ V) → ({0, 1} × {0}) = {〈0, 0〉, 〈1, 0〉}) | |
5 | 2, 3, 2, 4 | mp3an 1463 | . 2 ⊢ ({0, 1} × {0}) = {〈0, 0〉, 〈1, 0〉} |
6 | zringring 20410 | . . 3 ⊢ ℤring ∈ Ring | |
7 | prex 5314 | . . 3 ⊢ {0, 1} ∈ V | |
8 | zlmodzxz.z | . . . 4 ⊢ 𝑍 = (ℤring freeLMod {0, 1}) | |
9 | zring0 20417 | . . . 4 ⊢ 0 = (0g‘ℤring) | |
10 | 8, 9 | frlm0 20688 | . . 3 ⊢ ((ℤring ∈ Ring ∧ {0, 1} ∈ V) → ({0, 1} × {0}) = (0g‘𝑍)) |
11 | 6, 7, 10 | mp2an 692 | . 2 ⊢ ({0, 1} × {0}) = (0g‘𝑍) |
12 | 1, 5, 11 | 3eqtr2i 2768 | 1 ⊢ 0 = (0g‘𝑍) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2110 Vcvv 3401 {csn 4531 {cpr 4533 〈cop 4537 × cxp 5538 ‘cfv 6369 (class class class)co 7202 0cc0 10712 1c1 10713 0gc0g 16916 Ringcrg 19534 ℤringzring 20407 freeLMod cfrlm 20680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-addf 10791 ax-mulf 10792 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-map 8499 df-ixp 8568 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-sup 9047 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-7 11881 df-8 11882 df-9 11883 df-n0 12074 df-z 12160 df-dec 12277 df-uz 12422 df-fz 13079 df-struct 16686 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-mulr 16781 df-starv 16782 df-sca 16783 df-vsca 16784 df-ip 16785 df-tset 16786 df-ple 16787 df-ds 16789 df-unif 16790 df-hom 16791 df-cco 16792 df-0g 16918 df-prds 16924 df-pws 16926 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-grp 18340 df-minusg 18341 df-sbg 18342 df-subg 18512 df-cmn 19144 df-mgp 19477 df-ur 19489 df-ring 19536 df-cring 19537 df-subrg 19770 df-lmod 19873 df-lss 19941 df-sra 20181 df-rgmod 20182 df-cnfld 20336 df-zring 20408 df-dsmm 20666 df-frlm 20681 |
This theorem is referenced by: zlmodzxzldeplem3 45470 |
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