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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > islininds2 | Structured version Visualization version Unicode version |
Description: Implication of being a linearly independent subset of a (left) module over a nonzero ring. (Contributed by AV, 29-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.) |
Ref | Expression |
---|---|
islindeps2.b |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
islindeps2.z |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
islindeps2.r |
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
islindeps2.e |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
islindeps2.0 |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Ref | Expression |
---|---|
islininds2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lindepsnlininds 40349 |
. . . . 5
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2 | 1 | ancoms 455 |
. . . 4
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3 | 2 | 3adant3 1029 |
. . 3
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4 | 3 | con2bid 331 |
. 2
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5 | notnot 293 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | nne 2630 |
. . . . . . . . . . 11
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7 | 6 | bicomi 206 |
. . . . . . . . . 10
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8 | 5, 7 | anbi12i 704 |
. . . . . . . . 9
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9 | pm4.56 498 |
. . . . . . . . 9
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10 | 8, 9 | bitri 253 |
. . . . . . . 8
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11 | 10 | rexbii 2891 |
. . . . . . 7
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12 | rexnal 2838 |
. . . . . . 7
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13 | 11, 12 | bitri 253 |
. . . . . 6
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14 | 13 | rexbii 2891 |
. . . . 5
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15 | rexnal 2838 |
. . . . 5
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16 | 14, 15 | bitri 253 |
. . . 4
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17 | islindeps2.b |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
18 | islindeps2.z |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
19 | islindeps2.r |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
20 | islindeps2.e |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
21 | islindeps2.0 |
. . . . 5
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22 | 17, 18, 19, 20, 21 | islindeps2 40380 |
. . . 4
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23 | 16, 22 | syl5bir 222 |
. . 3
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24 | 23 | con1d 128 |
. 2
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25 | 4, 24 | sylbid 219 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1671 ax-4 1684 ax-5 1760 ax-6 1807 ax-7 1853 ax-8 1891 ax-9 1898 ax-10 1917 ax-11 1922 ax-12 1935 ax-13 2093 ax-ext 2433 ax-rep 4518 ax-sep 4528 ax-nul 4537 ax-pow 4584 ax-pr 4642 ax-un 6588 ax-inf2 8151 ax-cnex 9600 ax-resscn 9601 ax-1cn 9602 ax-icn 9603 ax-addcl 9604 ax-addrcl 9605 ax-mulcl 9606 ax-mulrcl 9607 ax-mulcom 9608 ax-addass 9609 ax-mulass 9610 ax-distr 9611 ax-i2m1 9612 ax-1ne0 9613 ax-1rid 9614 ax-rnegex 9615 ax-rrecex 9616 ax-cnre 9617 ax-pre-lttri 9618 ax-pre-lttrn 9619 ax-pre-ltadd 9620 ax-pre-mulgt0 9621 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 987 df-3an 988 df-tru 1449 df-ex 1666 df-nf 1670 df-sb 1800 df-eu 2305 df-mo 2306 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2583 df-ne 2626 df-nel 2627 df-ral 2744 df-rex 2745 df-reu 2746 df-rmo 2747 df-rab 2748 df-v 3049 df-sbc 3270 df-csb 3366 df-dif 3409 df-un 3411 df-in 3413 df-ss 3420 df-pss 3422 df-nul 3734 df-if 3884 df-pw 3955 df-sn 3971 df-pr 3973 df-tp 3975 df-op 3977 df-uni 4202 df-int 4238 df-iun 4283 df-iin 4284 df-br 4406 df-opab 4465 df-mpt 4466 df-tr 4501 df-eprel 4748 df-id 4752 df-po 4758 df-so 4759 df-fr 4796 df-se 4797 df-we 4798 df-xp 4843 df-rel 4844 df-cnv 4845 df-co 4846 df-dm 4847 df-rn 4848 df-res 4849 df-ima 4850 df-pred 5383 df-ord 5429 df-on 5430 df-lim 5431 df-suc 5432 df-iota 5549 df-fun 5587 df-fn 5588 df-f 5589 df-f1 5590 df-fo 5591 df-f1o 5592 df-fv 5593 df-isom 5594 df-riota 6257 df-ov 6298 df-oprab 6299 df-mpt2 6300 df-of 6536 df-om 6698 df-1st 6798 df-2nd 6799 df-supp 6920 df-tpos 6978 df-wrecs 7033 df-recs 7095 df-rdg 7133 df-1o 7187 df-oadd 7191 df-er 7368 df-map 7479 df-en 7575 df-dom 7576 df-sdom 7577 df-fin 7578 df-fsupp 7889 df-oi 8030 df-card 8378 df-pnf 9682 df-mnf 9683 df-xr 9684 df-ltxr 9685 df-le 9686 df-sub 9867 df-neg 9868 df-nn 10617 df-2 10675 df-3 10676 df-n0 10877 df-z 10945 df-uz 11167 df-fz 11792 df-fzo 11923 df-seq 12221 df-hash 12523 df-ndx 15136 df-slot 15137 df-base 15138 df-sets 15139 df-ress 15140 df-plusg 15215 df-mulr 15216 df-0g 15352 df-gsum 15353 df-mre 15504 df-mrc 15505 df-acs 15507 df-mgm 16500 df-sgrp 16539 df-mnd 16549 df-submnd 16595 df-grp 16685 df-minusg 16686 df-mulg 16688 df-cntz 16983 df-cmn 17444 df-abl 17445 df-mgp 17736 df-ur 17748 df-ring 17794 df-oppr 17863 df-dvdsr 17881 df-unit 17882 df-invr 17912 df-lmod 18105 df-nzr 18494 df-linc 40303 df-lininds 40339 df-lindeps 40341 |
This theorem is referenced by: (None) |
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