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Mirrors > Home > MPE Home > Th. List > lspexchn2 | Structured version Visualization version GIF version |
Description: Exchange property for span of a pair with negated membership. TODO: look at uses of lspexch 19895 to see if this will shorten proofs. (Contributed by NM, 24-May-2015.) |
Ref | Expression |
---|---|
lspexchn2.v | ⊢ 𝑉 = (Base‘𝑊) |
lspexchn2.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lspexchn2.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lspexchn2.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
lspexchn2.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
lspexchn2.z | ⊢ (𝜑 → 𝑍 ∈ 𝑉) |
lspexchn2.q | ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘{𝑍})) |
lspexchn2.e | ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑍, 𝑌})) |
Ref | Expression |
---|---|
lspexchn2 | ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑋})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspexchn2.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
2 | lspexchn2.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑊) | |
3 | lspexchn2.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
4 | lspexchn2.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
5 | lspexchn2.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
6 | lspexchn2.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑉) | |
7 | lspexchn2.q | . . 3 ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘{𝑍})) | |
8 | lspexchn2.e | . . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑍, 𝑌})) | |
9 | prcom 4662 | . . . . . 6 ⊢ {𝑍, 𝑌} = {𝑌, 𝑍} | |
10 | 9 | fveq2i 6668 | . . . . 5 ⊢ (𝑁‘{𝑍, 𝑌}) = (𝑁‘{𝑌, 𝑍}) |
11 | 10 | eleq2i 2904 | . . . 4 ⊢ (𝑋 ∈ (𝑁‘{𝑍, 𝑌}) ↔ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) |
12 | 8, 11 | sylnib 330 | . . 3 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) |
13 | 1, 2, 3, 4, 5, 6, 7, 12 | lspexchn1 19896 | . 2 ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘{𝑋, 𝑍})) |
14 | prcom 4662 | . . . 4 ⊢ {𝑋, 𝑍} = {𝑍, 𝑋} | |
15 | 14 | fveq2i 6668 | . . 3 ⊢ (𝑁‘{𝑋, 𝑍}) = (𝑁‘{𝑍, 𝑋}) |
16 | 15 | eleq2i 2904 | . 2 ⊢ (𝑌 ∈ (𝑁‘{𝑋, 𝑍}) ↔ 𝑌 ∈ (𝑁‘{𝑍, 𝑋})) |
17 | 13, 16 | sylnib 330 | 1 ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑋})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1533 ∈ wcel 2110 {csn 4561 {cpr 4563 ‘cfv 6350 Basecbs 16477 LSpanclspn 19737 LVecclvec 19868 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-tpos 7886 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-grp 18100 df-minusg 18101 df-sbg 18102 df-subg 18270 df-cntz 18441 df-lsm 18755 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-oppr 19367 df-dvdsr 19385 df-unit 19386 df-invr 19416 df-drng 19498 df-lmod 19630 df-lss 19698 df-lsp 19738 df-lvec 19869 |
This theorem is referenced by: baerlem5amN 38846 baerlem5bmN 38847 baerlem5abmN 38848 |
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