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Mirrors > Home > MPE Home > Th. List > mreclatdemoBAD | Structured version Visualization version Unicode version |
Description: The closed subspaces of a topology-bearing module form a complete lattice. Demonstration for mreclatBAD 16433. (Contributed by Stefan O'Rear, 31-Jan-2015.) TODO (df-riota 6252 update): This proof uses the old df-clat 16354 and references the required instance of mreclatBAD 16433 as a hypothesis. When mreclatBAD 16433 is corrected to become mreclat, delete this theorem and uncomment the mreclatdemo below. |
Ref | Expression |
---|---|
mreclatBAD. |
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Ref | Expression |
---|---|
mreclatdemoBAD |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 5875 |
. . . . 5
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2 | 1 | uniex 6587 |
. . . 4
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3 | mremre 15510 |
. . . 4
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4 | 2, 3 | mp1i 13 |
. . 3
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5 | inss2 3653 |
. . . . . 6
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6 | 5 | sseli 3428 |
. . . . 5
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7 | eqid 2451 |
. . . . . 6
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8 | eqid 2451 |
. . . . . 6
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9 | 7, 8 | lssmre 18189 |
. . . . 5
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10 | 6, 9 | syl 17 |
. . . 4
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11 | inss1 3652 |
. . . . . 6
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12 | 11 | sseli 3428 |
. . . . 5
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13 | eqid 2451 |
. . . . . . 7
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14 | 7, 13 | tpsuni 19953 |
. . . . . 6
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15 | 14 | fveq2d 5869 |
. . . . 5
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16 | 12, 15 | syl 17 |
. . . 4
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17 | 10, 16 | eleqtrd 2531 |
. . 3
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18 | 13 | tpstop 19954 |
. . . 4
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19 | eqid 2451 |
. . . . 5
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20 | 19 | cldmre 20094 |
. . . 4
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21 | 12, 18, 20 | 3syl 18 |
. . 3
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22 | mreincl 15505 |
. . 3
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23 | 4, 17, 21, 22 | syl3anc 1268 |
. 2
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24 | mreclatBAD. |
. 2
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25 | 23, 24 | syl 17 |
1
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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 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-8 1889 ax-9 1896 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 ax-rep 4515 ax-sep 4525 ax-nul 4534 ax-pow 4581 ax-pr 4639 ax-un 6583 ax-cnex 9595 ax-resscn 9596 ax-1cn 9597 ax-icn 9598 ax-addcl 9599 ax-addrcl 9600 ax-mulcl 9601 ax-mulrcl 9602 ax-mulcom 9603 ax-addass 9604 ax-mulass 9605 ax-distr 9606 ax-i2m1 9607 ax-1ne0 9608 ax-1rid 9609 ax-rnegex 9610 ax-rrecex 9611 ax-cnre 9612 ax-pre-lttri 9613 ax-pre-lttrn 9614 ax-pre-ltadd 9615 ax-pre-mulgt0 9616 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 986 df-3an 987 df-tru 1447 df-ex 1664 df-nf 1668 df-sb 1798 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2624 df-nel 2625 df-ral 2742 df-rex 2743 df-reu 2744 df-rmo 2745 df-rab 2746 df-v 3047 df-sbc 3268 df-csb 3364 df-dif 3407 df-un 3409 df-in 3411 df-ss 3418 df-pss 3420 df-nul 3732 df-if 3882 df-pw 3953 df-sn 3969 df-pr 3971 df-tp 3973 df-op 3975 df-uni 4199 df-int 4235 df-iun 4280 df-iin 4281 df-br 4403 df-opab 4462 df-mpt 4463 df-tr 4498 df-eprel 4745 df-id 4749 df-po 4755 df-so 4756 df-fr 4793 df-we 4795 df-xp 4840 df-rel 4841 df-cnv 4842 df-co 4843 df-dm 4844 df-rn 4845 df-res 4846 df-ima 4847 df-pred 5380 df-ord 5426 df-on 5427 df-lim 5428 df-suc 5429 df-iota 5546 df-fun 5584 df-fn 5585 df-f 5586 df-f1 5587 df-fo 5588 df-f1o 5589 df-fv 5590 df-riota 6252 df-ov 6293 df-oprab 6294 df-mpt2 6295 df-om 6693 df-1st 6793 df-2nd 6794 df-wrecs 7028 df-recs 7090 df-rdg 7128 df-er 7363 df-en 7570 df-dom 7571 df-sdom 7572 df-pnf 9677 df-mnf 9678 df-xr 9679 df-ltxr 9680 df-le 9681 df-sub 9862 df-neg 9863 df-nn 10610 df-2 10668 df-ndx 15124 df-slot 15125 df-base 15126 df-sets 15127 df-plusg 15203 df-0g 15340 df-mre 15492 df-mgm 16488 df-sgrp 16527 df-mnd 16537 df-grp 16673 df-minusg 16674 df-sbg 16675 df-mgp 17724 df-ur 17736 df-ring 17782 df-lmod 18093 df-lss 18156 df-top 19921 df-topon 19923 df-topsp 19924 df-cld 20034 |
This theorem is referenced by: (None) |
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