Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > lshpnel | Structured version Visualization version GIF version |
Description: A hyperplane's generating vector does not belong to the hyperplane. (Contributed by NM, 3-Jul-2014.) |
Ref | Expression |
---|---|
lshpnel.v | ⊢ 𝑉 = (Base‘𝑊) |
lshpnel.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lshpnel.p | ⊢ ⊕ = (LSSum‘𝑊) |
lshpnel.h | ⊢ 𝐻 = (LSHyp‘𝑊) |
lshpnel.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lshpnel.u | ⊢ (𝜑 → 𝑈 ∈ 𝐻) |
lshpnel.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
lshpnel.e | ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) |
Ref | Expression |
---|---|
lshpnel | ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lshpnel.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
2 | lshpnel.h | . . 3 ⊢ 𝐻 = (LSHyp‘𝑊) | |
3 | lshpnel.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
4 | lshpnel.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝐻) | |
5 | 1, 2, 3, 4 | lshpne 36992 | . 2 ⊢ (𝜑 → 𝑈 ≠ 𝑉) |
6 | 3 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑊 ∈ LMod) |
7 | eqid 2740 | . . . . . . . . 9 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
8 | 7 | lsssssubg 20218 | . . . . . . . 8 ⊢ (𝑊 ∈ LMod → (LSubSp‘𝑊) ⊆ (SubGrp‘𝑊)) |
9 | 6, 8 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (LSubSp‘𝑊) ⊆ (SubGrp‘𝑊)) |
10 | 7, 2, 3, 4 | lshplss 36991 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ (LSubSp‘𝑊)) |
11 | 10 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑈 ∈ (LSubSp‘𝑊)) |
12 | 9, 11 | sseldd 3927 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑈 ∈ (SubGrp‘𝑊)) |
13 | lshpnel.x | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
14 | 13 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ 𝑉) |
15 | lshpnel.n | . . . . . . . . 9 ⊢ 𝑁 = (LSpan‘𝑊) | |
16 | 1, 7, 15 | lspsncl 20237 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑊)) |
17 | 6, 14, 16 | syl2anc 584 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑊)) |
18 | 9, 17 | sseldd 3927 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑊)) |
19 | simpr 485 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ 𝑈) | |
20 | 7, 15, 6, 11, 19 | lspsnel5a 20256 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝑁‘{𝑋}) ⊆ 𝑈) |
21 | lshpnel.p | . . . . . . 7 ⊢ ⊕ = (LSSum‘𝑊) | |
22 | 21 | lsmss2 19271 | . . . . . 6 ⊢ ((𝑈 ∈ (SubGrp‘𝑊) ∧ (𝑁‘{𝑋}) ∈ (SubGrp‘𝑊) ∧ (𝑁‘{𝑋}) ⊆ 𝑈) → (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑈) |
23 | 12, 18, 20, 22 | syl3anc 1370 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑈) |
24 | lshpnel.e | . . . . . 6 ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) | |
25 | 24 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) |
26 | 23, 25 | eqtr3d 2782 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑈 = 𝑉) |
27 | 26 | ex 413 | . . 3 ⊢ (𝜑 → (𝑋 ∈ 𝑈 → 𝑈 = 𝑉)) |
28 | 27 | necon3ad 2958 | . 2 ⊢ (𝜑 → (𝑈 ≠ 𝑉 → ¬ 𝑋 ∈ 𝑈)) |
29 | 5, 28 | mpd 15 | 1 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ≠ wne 2945 ⊆ wss 3892 {csn 4567 ‘cfv 6432 (class class class)co 7271 Basecbs 16910 SubGrpcsubg 18747 LSSumclsm 19237 LModclmod 20121 LSubSpclss 20191 LSpanclspn 20231 LSHypclsh 36985 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-er 8481 df-en 8717 df-dom 8718 df-sdom 8719 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-0g 17150 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-submnd 18429 df-grp 18578 df-minusg 18579 df-sbg 18580 df-subg 18750 df-lsm 19239 df-mgp 19719 df-ur 19736 df-ring 19783 df-lmod 20123 df-lss 20192 df-lsp 20232 df-lshyp 36987 |
This theorem is referenced by: lshpnelb 36994 lshpne0 36996 lshpdisj 36997 |
Copyright terms: Public domain | W3C validator |