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Mathbox for Norm Megill |
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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 34587 | . 2 ⊢ (𝜑 → 𝑈 ≠ 𝑉) |
6 | 3 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑊 ∈ LMod) |
7 | eqid 2651 | . . . . . . . . 9 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
8 | 7 | lsssssubg 19006 | . . . . . . . 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 34586 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ (LSubSp‘𝑊)) |
11 | 10 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑈 ∈ (LSubSp‘𝑊)) |
12 | 9, 11 | sseldd 3637 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑈 ∈ (SubGrp‘𝑊)) |
13 | lshpnel.x | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
14 | 13 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ 𝑉) |
15 | lshpnel.n | . . . . . . . . 9 ⊢ 𝑁 = (LSpan‘𝑊) | |
16 | 1, 7, 15 | lspsncl 19025 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑊)) |
17 | 6, 14, 16 | syl2anc 694 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑊)) |
18 | 9, 17 | sseldd 3637 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑊)) |
19 | simpr 476 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ 𝑈) | |
20 | 7, 15, 6, 11, 19 | lspsnel5a 19044 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝑁‘{𝑋}) ⊆ 𝑈) |
21 | lshpnel.p | . . . . . . 7 ⊢ ⊕ = (LSSum‘𝑊) | |
22 | 21 | lsmss2 18127 | . . . . . 6 ⊢ ((𝑈 ∈ (SubGrp‘𝑊) ∧ (𝑁‘{𝑋}) ∈ (SubGrp‘𝑊) ∧ (𝑁‘{𝑋}) ⊆ 𝑈) → (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑈) |
23 | 12, 18, 20, 22 | syl3anc 1366 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑈) |
24 | lshpnel.e | . . . . . 6 ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) | |
25 | 24 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) |
26 | 23, 25 | eqtr3d 2687 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑈 = 𝑉) |
27 | 26 | ex 449 | . . 3 ⊢ (𝜑 → (𝑋 ∈ 𝑈 → 𝑈 = 𝑉)) |
28 | 27 | necon3ad 2836 | . 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 383 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 ⊆ wss 3607 {csn 4210 ‘cfv 5926 (class class class)co 6690 Basecbs 15904 SubGrpcsubg 17635 LSSumclsm 18095 LModclmod 18911 LSubSpclss 18980 LSpanclspn 19019 LSHypclsh 34580 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-0g 16149 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-submnd 17383 df-grp 17472 df-minusg 17473 df-sbg 17474 df-subg 17638 df-lsm 18097 df-mgp 18536 df-ur 18548 df-ring 18595 df-lmod 18913 df-lss 18981 df-lsp 19020 df-lshyp 34582 |
This theorem is referenced by: lshpnelb 34589 lshpne0 34591 lshpdisj 34592 |
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