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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 20691 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 2731 | . . 3 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
4 | lspexchn1.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑊) | |
5 | lspexchn1.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
6 | 5 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑋, 𝑍})) → 𝑊 ∈ LVec) |
7 | eqid 2731 | . . . . 5 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
8 | lveclmod 20666 | . . . . . 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 20537 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑍 ∈ 𝑉) → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊)) |
12 | 9, 10, 11 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊)) |
13 | lspexchn1.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
14 | lspexchn1.q | . . . . 5 ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘{𝑍})) | |
15 | 3, 7, 9, 12, 13, 14 | lssneln0 20512 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ {(0g‘𝑊)})) |
16 | 15 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑋, 𝑍})) → 𝑌 ∈ (𝑉 ∖ {(0g‘𝑊)})) |
17 | lspexchn1.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
18 | 17 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑋, 𝑍})) → 𝑋 ∈ 𝑉) |
19 | 10 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑋, 𝑍})) → 𝑍 ∈ 𝑉) |
20 | 2, 4, 9, 13, 10, 14 | lspsnne2 20680 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) |
21 | 20 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑋, 𝑍})) → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) |
22 | simpr 485 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑋, 𝑍})) → 𝑌 ∈ (𝑁‘{𝑋, 𝑍})) | |
23 | 2, 3, 4, 6, 16, 18, 19, 21, 22 | lspexch 20691 | . 2 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑋, 𝑍})) → 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) |
24 | 1, 23 | mtand 814 | 1 ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘{𝑋, 𝑍})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2939 ∖ cdif 3941 {csn 4622 {cpr 4624 ‘cfv 6532 Basecbs 17126 0gc0g 17367 LModclmod 20420 LSubSpclss 20491 LSpanclspn 20531 LVecclvec 20662 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 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 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-tpos 8193 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-nn 12195 df-2 12257 df-3 12258 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17127 df-ress 17156 df-plusg 17192 df-mulr 17193 df-0g 17369 df-mgm 18543 df-sgrp 18592 df-mnd 18603 df-submnd 18648 df-grp 18797 df-minusg 18798 df-sbg 18799 df-subg 18975 df-cntz 19147 df-lsm 19468 df-cmn 19614 df-abl 19615 df-mgp 19947 df-ur 19964 df-ring 20016 df-oppr 20102 df-dvdsr 20123 df-unit 20124 df-invr 20154 df-drng 20267 df-lmod 20422 df-lss 20492 df-lsp 20532 df-lvec 20663 |
This theorem is referenced by: lspexchn2 20693 |
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