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Mirrors > Home > MPE Home > Th. List > lspexchn1 | Structured version Visualization version GIF version |
Description: Exchange property for span of a pair with negated membership. TODO: look at uses of lspexch 20888 to see if this will shorten proofs. (Contributed by NM, 20-May-2015.) |
Ref | Expression |
---|---|
lspexchn1.v | β’ π = (Baseβπ) |
lspexchn1.n | β’ π = (LSpanβπ) |
lspexchn1.w | β’ (π β π β LVec) |
lspexchn1.x | β’ (π β π β π) |
lspexchn1.y | β’ (π β π β π) |
lspexchn1.z | β’ (π β π β π) |
lspexchn1.q | β’ (π β Β¬ π β (πβ{π})) |
lspexchn1.e | β’ (π β Β¬ π β (πβ{π, π})) |
Ref | Expression |
---|---|
lspexchn1 | β’ (π β Β¬ π β (πβ{π, π})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspexchn1.e | . 2 β’ (π β Β¬ π β (πβ{π, π})) | |
2 | lspexchn1.v | . . 3 β’ π = (Baseβπ) | |
3 | eqid 2731 | . . 3 β’ (0gβπ) = (0gβπ) | |
4 | lspexchn1.n | . . 3 β’ π = (LSpanβπ) | |
5 | lspexchn1.w | . . . 4 β’ (π β π β LVec) | |
6 | 5 | adantr 480 | . . 3 β’ ((π β§ π β (πβ{π, π})) β π β LVec) |
7 | eqid 2731 | . . . . 5 β’ (LSubSpβπ) = (LSubSpβπ) | |
8 | lveclmod 20862 | . . . . . 6 β’ (π β LVec β π β LMod) | |
9 | 5, 8 | syl 17 | . . . . 5 β’ (π β π β LMod) |
10 | lspexchn1.z | . . . . . 6 β’ (π β π β π) | |
11 | 2, 7, 4 | lspsncl 20733 | . . . . . 6 β’ ((π β LMod β§ π β π) β (πβ{π}) β (LSubSpβπ)) |
12 | 9, 10, 11 | syl2anc 583 | . . . . 5 β’ (π β (πβ{π}) β (LSubSpβπ)) |
13 | lspexchn1.y | . . . . 5 β’ (π β π β π) | |
14 | lspexchn1.q | . . . . 5 β’ (π β Β¬ π β (πβ{π})) | |
15 | 3, 7, 9, 12, 13, 14 | lssneln0 20708 | . . . 4 β’ (π β π β (π β {(0gβπ)})) |
16 | 15 | adantr 480 | . . 3 β’ ((π β§ π β (πβ{π, π})) β π β (π β {(0gβπ)})) |
17 | lspexchn1.x | . . . 4 β’ (π β π β π) | |
18 | 17 | adantr 480 | . . 3 β’ ((π β§ π β (πβ{π, π})) β π β π) |
19 | 10 | adantr 480 | . . 3 β’ ((π β§ π β (πβ{π, π})) β π β π) |
20 | 2, 4, 9, 13, 10, 14 | lspsnne2 20877 | . . . 4 β’ (π β (πβ{π}) β (πβ{π})) |
21 | 20 | adantr 480 | . . 3 β’ ((π β§ π β (πβ{π, π})) β (πβ{π}) β (πβ{π})) |
22 | simpr 484 | . . 3 β’ ((π β§ π β (πβ{π, π})) β π β (πβ{π, π})) | |
23 | 2, 3, 4, 6, 16, 18, 19, 21, 22 | lspexch 20888 | . 2 β’ ((π β§ π β (πβ{π, π})) β π β (πβ{π, π})) |
24 | 1, 23 | mtand 813 | 1 β’ (π β Β¬ π β (πβ{π, π})) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 395 = wceq 1540 β wcel 2105 β wne 2939 β cdif 3945 {csn 4628 {cpr 4630 βcfv 6543 Basecbs 17149 0gc0g 17390 LModclmod 20615 LSubSpclss 20687 LSpanclspn 20727 LVecclvec 20858 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-tpos 8215 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-3 12281 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-0g 17392 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-submnd 18707 df-grp 18859 df-minusg 18860 df-sbg 18861 df-subg 19040 df-cntz 19223 df-lsm 19546 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-oppr 20226 df-dvdsr 20249 df-unit 20250 df-invr 20280 df-drng 20503 df-lmod 20617 df-lss 20688 df-lsp 20728 df-lvec 20859 |
This theorem is referenced by: lspexchn2 20890 |
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