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Mirrors > Home > MPE Home > Th. List > lsppratlem2 | Structured version Visualization version GIF version |
Description: Lemma for lspprat 19153. 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 19106 | . . . 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 3586 | . . . . . 6 ⊢ (𝜑 → 𝑥 ∈ 𝑈) |
8 | lsppratlem1.y2 | . . . . . . 7 ⊢ (𝜑 → 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥}))) | |
9 | 8 | eldifad 3586 | . . . . . 6 ⊢ (𝜑 → 𝑦 ∈ 𝑈) |
10 | prssi 4353 | . . . . . 6 ⊢ ((𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈) → {𝑥, 𝑦} ⊆ 𝑈) | |
11 | 7, 9, 10 | syl2anc 693 | . . . . 5 ⊢ (𝜑 → {𝑥, 𝑦} ⊆ 𝑈) |
12 | lspprat.u | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
13 | lspprat.v | . . . . . . 7 ⊢ 𝑉 = (Base‘𝑊) | |
14 | 13, 1 | lssss 18937 | . . . . . 6 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ⊆ 𝑉) |
15 | 12, 14 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑈 ⊆ 𝑉) |
16 | 11, 15 | sstrd 3613 | . . . 4 ⊢ (𝜑 → {𝑥, 𝑦} ⊆ 𝑉) |
17 | 13, 1, 2 | lspcl 18976 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ {𝑥, 𝑦} ⊆ 𝑉) → (𝑁‘{𝑥, 𝑦}) ∈ 𝑆) |
18 | 5, 16, 17 | syl2anc 693 | . . 3 ⊢ (𝜑 → (𝑁‘{𝑥, 𝑦}) ∈ 𝑆) |
19 | lsppratlem2.x1 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑥, 𝑦})) | |
20 | lsppratlem2.y1 | . . 3 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑥, 𝑦})) | |
21 | 1, 2, 5, 18, 19, 20 | lspprss 18992 | . 2 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ (𝑁‘{𝑥, 𝑦})) |
22 | 1, 2, 5, 12, 7, 9 | lspprss 18992 | . 2 ⊢ (𝜑 → (𝑁‘{𝑥, 𝑦}) ⊆ 𝑈) |
23 | 21, 22 | sstrd 3613 | 1 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ∖ cdif 3571 ⊆ wss 3574 ⊊ wpss 3575 {csn 4177 {cpr 4179 ‘cfv 5888 Basecbs 15857 0gc0g 16100 LModclmod 18863 LSubSpclss 18932 LSpanclspn 18971 LVecclvec 19102 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-plusg 15954 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mgp 18490 df-ur 18502 df-ring 18549 df-lmod 18865 df-lss 18933 df-lsp 18972 df-lvec 19103 |
This theorem is referenced by: lsppratlem5 19151 |
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