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| Mirrors > Home > MPE Home > Th. List > lsppratlem1 | Structured version Visualization version GIF version | ||
| Description: Lemma for lspprat 21155. 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 21145 (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 480 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ (𝑁‘{𝑌})) → 𝑊 ∈ LVec) |
| 3 | lspprat.y | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
| 4 | 3 | snssd 4809 | . . . . . 6 ⊢ (𝜑 → {𝑌} ⊆ 𝑉) |
| 5 | 4 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ (𝑁‘{𝑌})) → {𝑌} ⊆ 𝑉) |
| 6 | lspprat.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
| 7 | 6 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ (𝑁‘{𝑌})) → 𝑋 ∈ 𝑉) |
| 8 | lspprat.p | . . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ⊊ (𝑁‘{𝑋, 𝑌})) | |
| 9 | 8 | pssssd 4100 | . . . . . . . . 9 ⊢ (𝜑 → 𝑈 ⊆ (𝑁‘{𝑋, 𝑌})) |
| 10 | lsppratlem1.x2 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑥 ∈ (𝑈 ∖ { 0 })) | |
| 11 | 10 | eldifad 3963 | . . . . . . . . 9 ⊢ (𝜑 → 𝑥 ∈ 𝑈) |
| 12 | 9, 11 | sseldd 3984 | . . . . . . . 8 ⊢ (𝜑 → 𝑥 ∈ (𝑁‘{𝑋, 𝑌})) |
| 13 | prcom 4732 | . . . . . . . . . 10 ⊢ {𝑋, 𝑌} = {𝑌, 𝑋} | |
| 14 | df-pr 4629 | . . . . . . . . . 10 ⊢ {𝑌, 𝑋} = ({𝑌} ∪ {𝑋}) | |
| 15 | 13, 14 | eqtri 2765 | . . . . . . . . 9 ⊢ {𝑋, 𝑌} = ({𝑌} ∪ {𝑋}) |
| 16 | 15 | fveq2i 6909 | . . . . . . . 8 ⊢ (𝑁‘{𝑋, 𝑌}) = (𝑁‘({𝑌} ∪ {𝑋})) |
| 17 | 12, 16 | eleqtrdi 2851 | . . . . . . 7 ⊢ (𝜑 → 𝑥 ∈ (𝑁‘({𝑌} ∪ {𝑋}))) |
| 18 | 17 | anim1i 615 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ (𝑁‘{𝑌})) → (𝑥 ∈ (𝑁‘({𝑌} ∪ {𝑋})) ∧ ¬ 𝑥 ∈ (𝑁‘{𝑌}))) |
| 19 | eldif 3961 | . . . . . 6 ⊢ (𝑥 ∈ ((𝑁‘({𝑌} ∪ {𝑋})) ∖ (𝑁‘{𝑌})) ↔ (𝑥 ∈ (𝑁‘({𝑌} ∪ {𝑋})) ∧ ¬ 𝑥 ∈ (𝑁‘{𝑌}))) | |
| 20 | 18, 19 | sylibr 234 | . . . . 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 21145 | . . . . 5 ⊢ ((𝑊 ∈ LVec ∧ ({𝑌} ⊆ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ 𝑥 ∈ ((𝑁‘({𝑌} ∪ {𝑋})) ∖ (𝑁‘{𝑌})))) → 𝑋 ∈ (𝑁‘({𝑌} ∪ {𝑥}))) |
| 25 | 2, 5, 7, 20, 24 | syl13anc 1374 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ (𝑁‘{𝑌})) → 𝑋 ∈ (𝑁‘({𝑌} ∪ {𝑥}))) |
| 26 | df-pr 4629 | . . . . . 6 ⊢ {𝑌, 𝑥} = ({𝑌} ∪ {𝑥}) | |
| 27 | prcom 4732 | . . . . . 6 ⊢ {𝑌, 𝑥} = {𝑥, 𝑌} | |
| 28 | 26, 27 | eqtr3i 2767 | . . . . 5 ⊢ ({𝑌} ∪ {𝑥}) = {𝑥, 𝑌} |
| 29 | 28 | fveq2i 6909 | . . . 4 ⊢ (𝑁‘({𝑌} ∪ {𝑥})) = (𝑁‘{𝑥, 𝑌}) |
| 30 | 25, 29 | eleqtrdi 2851 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ (𝑁‘{𝑌})) → 𝑋 ∈ (𝑁‘{𝑥, 𝑌})) |
| 31 | 30 | ex 412 | . 2 ⊢ (𝜑 → (¬ 𝑥 ∈ (𝑁‘{𝑌}) → 𝑋 ∈ (𝑁‘{𝑥, 𝑌}))) |
| 32 | 31 | orrd 864 | 1 ⊢ (𝜑 → (𝑥 ∈ (𝑁‘{𝑌}) ∨ 𝑋 ∈ (𝑁‘{𝑥, 𝑌}))) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ∖ cdif 3948 ∪ cun 3949 ⊆ wss 3951 ⊊ wpss 3952 {csn 4626 {cpr 4628 ‘cfv 6561 Basecbs 17247 0gc0g 17484 LSubSpclss 20929 LSpanclspn 20969 LVecclvec 21101 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-sbg 18956 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-drng 20731 df-lmod 20860 df-lss 20930 df-lsp 20970 df-lvec 21102 |
| This theorem is referenced by: lsppratlem5 21153 |
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