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Mirrors > Home > MPE Home > Th. List > lsppratlem2 | Structured version Visualization version GIF version |
Description: Lemma for lspprat 20162. Show that if 𝑋 and 𝑌 are both in (𝑁‘{𝑥, 𝑦}) (which will be our goal for each of the two cases above), then (𝑁‘{𝑋, 𝑌}) ⊆ 𝑈, contradicting the hypothesis for 𝑈. (Contributed by NM, 29-Aug-2014.) (Revised by Mario Carneiro, 5-Sep-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}))) |
lsppratlem2.x1 | ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑥, 𝑦})) |
lsppratlem2.y1 | ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑥, 𝑦})) |
Ref | Expression |
---|---|
lsppratlem2 | ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspprat.s | . . 3 ⊢ 𝑆 = (LSubSp‘𝑊) | |
2 | lspprat.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑊) | |
3 | lspprat.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
4 | lveclmod 20115 | . . . 4 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) |
6 | lsppratlem1.x2 | . . . . . . 7 ⊢ (𝜑 → 𝑥 ∈ (𝑈 ∖ { 0 })) | |
7 | 6 | eldifad 3869 | . . . . . 6 ⊢ (𝜑 → 𝑥 ∈ 𝑈) |
8 | lsppratlem1.y2 | . . . . . . 7 ⊢ (𝜑 → 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥}))) | |
9 | 8 | eldifad 3869 | . . . . . 6 ⊢ (𝜑 → 𝑦 ∈ 𝑈) |
10 | 7, 9 | prssd 4725 | . . . . 5 ⊢ (𝜑 → {𝑥, 𝑦} ⊆ 𝑈) |
11 | lspprat.u | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
12 | lspprat.v | . . . . . . 7 ⊢ 𝑉 = (Base‘𝑊) | |
13 | 12, 1 | lssss 19945 | . . . . . 6 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ⊆ 𝑉) |
14 | 11, 13 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑈 ⊆ 𝑉) |
15 | 10, 14 | sstrd 3901 | . . . 4 ⊢ (𝜑 → {𝑥, 𝑦} ⊆ 𝑉) |
16 | 12, 1, 2 | lspcl 19985 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ {𝑥, 𝑦} ⊆ 𝑉) → (𝑁‘{𝑥, 𝑦}) ∈ 𝑆) |
17 | 5, 15, 16 | syl2anc 587 | . . 3 ⊢ (𝜑 → (𝑁‘{𝑥, 𝑦}) ∈ 𝑆) |
18 | lsppratlem2.x1 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑥, 𝑦})) | |
19 | lsppratlem2.y1 | . . 3 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑥, 𝑦})) | |
20 | 1, 2, 5, 17, 18, 19 | lspprss 20001 | . 2 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ (𝑁‘{𝑥, 𝑦})) |
21 | 1, 2, 5, 11, 7, 9 | lspprss 20001 | . 2 ⊢ (𝜑 → (𝑁‘{𝑥, 𝑦}) ⊆ 𝑈) |
22 | 20, 21 | sstrd 3901 | 1 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 ∖ cdif 3854 ⊆ wss 3857 ⊊ wpss 3858 {csn 4531 {cpr 4533 ‘cfv 6369 Basecbs 16684 0gc0g 16916 LModclmod 19871 LSubSpclss 19940 LSpanclspn 19980 LVecclvec 20111 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-2 11876 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-plusg 16780 df-0g 16918 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-grp 18340 df-minusg 18341 df-sbg 18342 df-mgp 19477 df-ur 19489 df-ring 19536 df-lmod 19873 df-lss 19941 df-lsp 19981 df-lvec 20112 |
This theorem is referenced by: lsppratlem5 20160 |
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