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Mirrors > Home > MPE Home > Th. List > lsppratlem1 | Structured version Visualization version GIF version |
Description: Lemma for lspprat 20717. Let 𝑥 ∈ (𝑈 ∖ {0}) (if there is no such 𝑥 then 𝑈 is the zero subspace), and let 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥})) (assuming the conclusion is false). The goal is to write 𝑋, 𝑌 in terms of 𝑥, 𝑦, which would normally be done by solving the system of linear equations. The span equivalent of this process is lspsolv 20707 (hence the name), which we use extensively below. In this lemma, we show that since 𝑥 ∈ (𝑁‘{𝑋, 𝑌}), either 𝑥 ∈ (𝑁‘{𝑌}) or 𝑋 ∈ (𝑁‘{𝑥, 𝑌}). (Contributed by NM, 29-Aug-2014.) |
Ref | Expression |
---|---|
lspprat.v | ⊢ 𝑉 = (Base‘𝑊) |
lspprat.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lspprat.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lspprat.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lspprat.u | ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
lspprat.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
lspprat.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
lspprat.p | ⊢ (𝜑 → 𝑈 ⊊ (𝑁‘{𝑋, 𝑌})) |
lsppratlem1.o | ⊢ 0 = (0g‘𝑊) |
lsppratlem1.x2 | ⊢ (𝜑 → 𝑥 ∈ (𝑈 ∖ { 0 })) |
lsppratlem1.y2 | ⊢ (𝜑 → 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥}))) |
Ref | Expression |
---|---|
lsppratlem1 | ⊢ (𝜑 → (𝑥 ∈ (𝑁‘{𝑌}) ∨ 𝑋 ∈ (𝑁‘{𝑥, 𝑌}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspprat.w | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
2 | 1 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ (𝑁‘{𝑌})) → 𝑊 ∈ LVec) |
3 | lspprat.y | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
4 | 3 | snssd 4806 | . . . . . 6 ⊢ (𝜑 → {𝑌} ⊆ 𝑉) |
5 | 4 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ (𝑁‘{𝑌})) → {𝑌} ⊆ 𝑉) |
6 | lspprat.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
7 | 6 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ (𝑁‘{𝑌})) → 𝑋 ∈ 𝑉) |
8 | lspprat.p | . . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ⊊ (𝑁‘{𝑋, 𝑌})) | |
9 | 8 | pssssd 4094 | . . . . . . . . 9 ⊢ (𝜑 → 𝑈 ⊆ (𝑁‘{𝑋, 𝑌})) |
10 | lsppratlem1.x2 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑥 ∈ (𝑈 ∖ { 0 })) | |
11 | 10 | eldifad 3957 | . . . . . . . . 9 ⊢ (𝜑 → 𝑥 ∈ 𝑈) |
12 | 9, 11 | sseldd 3980 | . . . . . . . 8 ⊢ (𝜑 → 𝑥 ∈ (𝑁‘{𝑋, 𝑌})) |
13 | prcom 4730 | . . . . . . . . . 10 ⊢ {𝑋, 𝑌} = {𝑌, 𝑋} | |
14 | df-pr 4626 | . . . . . . . . . 10 ⊢ {𝑌, 𝑋} = ({𝑌} ∪ {𝑋}) | |
15 | 13, 14 | eqtri 2760 | . . . . . . . . 9 ⊢ {𝑋, 𝑌} = ({𝑌} ∪ {𝑋}) |
16 | 15 | fveq2i 6882 | . . . . . . . 8 ⊢ (𝑁‘{𝑋, 𝑌}) = (𝑁‘({𝑌} ∪ {𝑋})) |
17 | 12, 16 | eleqtrdi 2843 | . . . . . . 7 ⊢ (𝜑 → 𝑥 ∈ (𝑁‘({𝑌} ∪ {𝑋}))) |
18 | 17 | anim1i 615 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ (𝑁‘{𝑌})) → (𝑥 ∈ (𝑁‘({𝑌} ∪ {𝑋})) ∧ ¬ 𝑥 ∈ (𝑁‘{𝑌}))) |
19 | eldif 3955 | . . . . . 6 ⊢ (𝑥 ∈ ((𝑁‘({𝑌} ∪ {𝑋})) ∖ (𝑁‘{𝑌})) ↔ (𝑥 ∈ (𝑁‘({𝑌} ∪ {𝑋})) ∧ ¬ 𝑥 ∈ (𝑁‘{𝑌}))) | |
20 | 18, 19 | sylibr 233 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ (𝑁‘{𝑌})) → 𝑥 ∈ ((𝑁‘({𝑌} ∪ {𝑋})) ∖ (𝑁‘{𝑌}))) |
21 | lspprat.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑊) | |
22 | lspprat.s | . . . . . 6 ⊢ 𝑆 = (LSubSp‘𝑊) | |
23 | lspprat.n | . . . . . 6 ⊢ 𝑁 = (LSpan‘𝑊) | |
24 | 21, 22, 23 | lspsolv 20707 | . . . . 5 ⊢ ((𝑊 ∈ LVec ∧ ({𝑌} ⊆ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ 𝑥 ∈ ((𝑁‘({𝑌} ∪ {𝑋})) ∖ (𝑁‘{𝑌})))) → 𝑋 ∈ (𝑁‘({𝑌} ∪ {𝑥}))) |
25 | 2, 5, 7, 20, 24 | syl13anc 1372 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ (𝑁‘{𝑌})) → 𝑋 ∈ (𝑁‘({𝑌} ∪ {𝑥}))) |
26 | df-pr 4626 | . . . . . 6 ⊢ {𝑌, 𝑥} = ({𝑌} ∪ {𝑥}) | |
27 | prcom 4730 | . . . . . 6 ⊢ {𝑌, 𝑥} = {𝑥, 𝑌} | |
28 | 26, 27 | eqtr3i 2762 | . . . . 5 ⊢ ({𝑌} ∪ {𝑥}) = {𝑥, 𝑌} |
29 | 28 | fveq2i 6882 | . . . 4 ⊢ (𝑁‘({𝑌} ∪ {𝑥})) = (𝑁‘{𝑥, 𝑌}) |
30 | 25, 29 | eleqtrdi 2843 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ (𝑁‘{𝑌})) → 𝑋 ∈ (𝑁‘{𝑥, 𝑌})) |
31 | 30 | ex 413 | . 2 ⊢ (𝜑 → (¬ 𝑥 ∈ (𝑁‘{𝑌}) → 𝑋 ∈ (𝑁‘{𝑥, 𝑌}))) |
32 | 31 | orrd 861 | 1 ⊢ (𝜑 → (𝑥 ∈ (𝑁‘{𝑌}) ∨ 𝑋 ∈ (𝑁‘{𝑥, 𝑌}))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ∖ cdif 3942 ∪ cun 3943 ⊆ wss 3945 ⊊ wpss 3946 {csn 4623 {cpr 4625 ‘cfv 6533 Basecbs 17128 0gc0g 17369 LSubSpclss 20493 LSpanclspn 20533 LVecclvec 20664 |
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 2703 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5357 ax-pr 5421 ax-un 7709 ax-cnex 11150 ax-resscn 11151 ax-1cn 11152 ax-icn 11153 ax-addcl 11154 ax-addrcl 11155 ax-mulcl 11156 ax-mulrcl 11157 ax-mulcom 11158 ax-addass 11159 ax-mulass 11160 ax-distr 11161 ax-i2m1 11162 ax-1ne0 11163 ax-1rid 11164 ax-rnegex 11165 ax-rrecex 11166 ax-cnre 11167 ax-pre-lttri 11168 ax-pre-lttrn 11169 ax-pre-ltadd 11170 ax-pre-mulgt0 11171 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7350 df-ov 7397 df-oprab 7398 df-mpo 7399 df-om 7840 df-1st 7959 df-2nd 7960 df-tpos 8195 df-frecs 8250 df-wrecs 8281 df-recs 8355 df-rdg 8394 df-er 8688 df-en 8925 df-dom 8926 df-sdom 8927 df-pnf 11234 df-mnf 11235 df-xr 11236 df-ltxr 11237 df-le 11238 df-sub 11430 df-neg 11431 df-nn 12197 df-2 12259 df-3 12260 df-sets 17081 df-slot 17099 df-ndx 17111 df-base 17129 df-ress 17158 df-plusg 17194 df-mulr 17195 df-0g 17371 df-mgm 18545 df-sgrp 18594 df-mnd 18605 df-grp 18799 df-minusg 18800 df-sbg 18801 df-cmn 19616 df-abl 19617 df-mgp 19949 df-ur 19966 df-ring 20018 df-oppr 20104 df-dvdsr 20125 df-unit 20126 df-invr 20156 df-drng 20269 df-lmod 20424 df-lss 20494 df-lsp 20534 df-lvec 20665 |
This theorem is referenced by: lsppratlem5 20715 |
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