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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 18950 to see if this will shorten proofs. (Contributed by NM, 20-May-2015.) |
Ref | Expression |
---|---|
lspexchn1.v | ⊢ 𝑉 = (Base‘𝑊) |
lspexchn1.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lspexchn1.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lspexchn1.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
lspexchn1.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
lspexchn1.z | ⊢ (𝜑 → 𝑍 ∈ 𝑉) |
lspexchn1.q | ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘{𝑍})) |
lspexchn1.e | ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) |
Ref | Expression |
---|---|
lspexchn1 | ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘{𝑋, 𝑍})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspexchn1.e | . 2 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) | |
2 | lspexchn1.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
3 | eqid 2610 | . . 3 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
4 | lspexchn1.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑊) | |
5 | lspexchn1.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
6 | 5 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑋, 𝑍})) → 𝑊 ∈ LVec) |
7 | eqid 2610 | . . . . 5 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
8 | lveclmod 18927 | . . . . . 6 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
9 | 5, 8 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) |
10 | lspexchn1.z | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑉) | |
11 | 2, 7, 4 | lspsncl 18798 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑍 ∈ 𝑉) → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊)) |
12 | 9, 10, 11 | syl2anc 691 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊)) |
13 | lspexchn1.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
14 | lspexchn1.q | . . . . 5 ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘{𝑍})) | |
15 | 2, 3, 7, 9, 12, 13, 14 | lssneln0 18773 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ {(0g‘𝑊)})) |
16 | 15 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑋, 𝑍})) → 𝑌 ∈ (𝑉 ∖ {(0g‘𝑊)})) |
17 | lspexchn1.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
18 | 17 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑋, 𝑍})) → 𝑋 ∈ 𝑉) |
19 | 10 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑋, 𝑍})) → 𝑍 ∈ 𝑉) |
20 | 2, 4, 9, 13, 10, 14 | lspsnne2 18939 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) |
21 | 20 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑋, 𝑍})) → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) |
22 | simpr 476 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑋, 𝑍})) → 𝑌 ∈ (𝑁‘{𝑋, 𝑍})) | |
23 | 2, 3, 4, 6, 16, 18, 19, 21, 22 | lspexch 18950 | . 2 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑋, 𝑍})) → 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) |
24 | 1, 23 | mtand 689 | 1 ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘{𝑋, 𝑍})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∖ cdif 3537 {csn 4125 {cpr 4127 ‘cfv 5804 Basecbs 15695 0gc0g 15923 LModclmod 18686 LSubSpclss 18753 LSpanclspn 18792 LVecclvec 18923 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-subg 17414 df-cntz 17573 df-lsm 17874 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-drng 18572 df-lmod 18688 df-lss 18754 df-lsp 18793 df-lvec 18924 |
This theorem is referenced by: lspexchn2 18952 |
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