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Mirrors > Home > MPE Home > Th. List > lspss | Structured version Visualization version Unicode version |
Description: Span preserves subset ordering. (spanss 27001 analog.) (Contributed by NM, 11-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
lspss.v |
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lspss.n |
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Ref | Expression |
---|---|
lspss |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl3 1013 |
. . . . 5
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2 | sstr2 3439 |
. . . . 5
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3 | 1, 2 | syl 17 |
. . . 4
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4 | 3 | ss2rabdv 3510 |
. . 3
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5 | intss 4255 |
. . 3
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6 | 4, 5 | syl 17 |
. 2
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7 | simp1 1008 |
. . 3
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8 | simp3 1010 |
. . . 4
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9 | simp2 1009 |
. . . 4
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10 | 8, 9 | sstrd 3442 |
. . 3
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11 | lspss.v |
. . . 4
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12 | eqid 2451 |
. . . 4
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13 | lspss.n |
. . . 4
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14 | 11, 12, 13 | lspval 18198 |
. . 3
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15 | 7, 10, 14 | syl2anc 667 |
. 2
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16 | 11, 12, 13 | lspval 18198 |
. . 3
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17 | 16 | 3adant3 1028 |
. 2
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18 | 6, 15, 17 | 3sstr4d 3475 |
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 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 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-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-nul 3732 df-if 3882 df-pw 3953 df-sn 3969 df-pr 3971 df-op 3975 df-uni 4199 df-int 4235 df-iun 4280 df-br 4403 df-opab 4462 df-mpt 4463 df-id 4749 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-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-0g 15340 df-mgm 16488 df-sgrp 16527 df-mnd 16537 df-grp 16673 df-lmod 18093 df-lss 18156 df-lsp 18195 |
This theorem is referenced by: lspun 18210 lspssp 18211 lspprid1 18220 lbspss 18305 lspsolvlem 18365 lspsolv 18366 lsppratlem3 18372 lbsextlem2 18382 lbsextlem3 18383 lbsextlem4 18384 lindfrn 19379 f1lindf 19380 lssats 32578 lpssat 32579 lssatle 32581 lssat 32582 dvhdimlem 35012 dvh3dim3N 35017 mapdindp2 35289 lspindp5 35338 |
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