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Mirrors > Home > MPE Home > Th. List > Mathboxes > lshpnel2N | Structured version Visualization version GIF version |
Description: Condition that determines a hyperplane. (Contributed by NM, 3-Oct-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
lshpnel2.v | ⊢ 𝑉 = (Base‘𝑊) |
lshpnel2.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lshpnel2.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lshpnel2.p | ⊢ ⊕ = (LSSum‘𝑊) |
lshpnel2.h | ⊢ 𝐻 = (LSHyp‘𝑊) |
lshpnel2.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lshpnel2.u | ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
lshpnel2.t | ⊢ (𝜑 → 𝑈 ≠ 𝑉) |
lshpnel2.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
lshpnel2.e | ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈) |
Ref | Expression |
---|---|
lshpnel2N | ⊢ (𝜑 → (𝑈 ∈ 𝐻 ↔ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lshpnel2.e | . . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈) | |
2 | 1 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐻) → ¬ 𝑋 ∈ 𝑈) |
3 | lshpnel2.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
4 | lshpnel2.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑊) | |
5 | lshpnel2.p | . . . 4 ⊢ ⊕ = (LSSum‘𝑊) | |
6 | lshpnel2.h | . . . 4 ⊢ 𝐻 = (LSHyp‘𝑊) | |
7 | lshpnel2.w | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
8 | 7 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐻) → 𝑊 ∈ LVec) |
9 | simpr 476 | . . . 4 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐻) → 𝑈 ∈ 𝐻) | |
10 | lshpnel2.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
11 | 10 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐻) → 𝑋 ∈ 𝑉) |
12 | 3, 4, 5, 6, 8, 9, 11 | lshpnelb 34589 | . . 3 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐻) → (¬ 𝑋 ∈ 𝑈 ↔ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉)) |
13 | 2, 12 | mpbid 222 | . 2 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐻) → (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) |
14 | lshpnel2.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
15 | 14 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) → 𝑈 ∈ 𝑆) |
16 | lshpnel2.t | . . . 4 ⊢ (𝜑 → 𝑈 ≠ 𝑉) | |
17 | 16 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) → 𝑈 ≠ 𝑉) |
18 | 10 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) → 𝑋 ∈ 𝑉) |
19 | lveclmod 19154 | . . . . . . . . . . 11 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
20 | 7, 19 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ LMod) |
21 | lshpnel2.s | . . . . . . . . . . 11 ⊢ 𝑆 = (LSubSp‘𝑊) | |
22 | 21, 4 | lspid 19030 | . . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (𝑁‘𝑈) = 𝑈) |
23 | 20, 14, 22 | syl2anc 694 | . . . . . . . . 9 ⊢ (𝜑 → (𝑁‘𝑈) = 𝑈) |
24 | 23 | uneq1d 3799 | . . . . . . . 8 ⊢ (𝜑 → ((𝑁‘𝑈) ∪ (𝑁‘{𝑋})) = (𝑈 ∪ (𝑁‘{𝑋}))) |
25 | 24 | fveq2d 6233 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘((𝑁‘𝑈) ∪ (𝑁‘{𝑋}))) = (𝑁‘(𝑈 ∪ (𝑁‘{𝑋})))) |
26 | 3, 21 | lssss 18985 | . . . . . . . . 9 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ⊆ 𝑉) |
27 | 14, 26 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ⊆ 𝑉) |
28 | 10 | snssd 4372 | . . . . . . . 8 ⊢ (𝜑 → {𝑋} ⊆ 𝑉) |
29 | 3, 4 | lspun 19035 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉 ∧ {𝑋} ⊆ 𝑉) → (𝑁‘(𝑈 ∪ {𝑋})) = (𝑁‘((𝑁‘𝑈) ∪ (𝑁‘{𝑋})))) |
30 | 20, 27, 28, 29 | syl3anc 1366 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘(𝑈 ∪ {𝑋})) = (𝑁‘((𝑁‘𝑈) ∪ (𝑁‘{𝑋})))) |
31 | 3, 21, 4 | lspsncl 19025 | . . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ 𝑆) |
32 | 20, 10, 31 | syl2anc 694 | . . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ 𝑆) |
33 | 21, 4, 5 | lsmsp 19134 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ (𝑁‘{𝑋}) ∈ 𝑆) → (𝑈 ⊕ (𝑁‘{𝑋})) = (𝑁‘(𝑈 ∪ (𝑁‘{𝑋})))) |
34 | 20, 14, 32, 33 | syl3anc 1366 | . . . . . . 7 ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑋})) = (𝑁‘(𝑈 ∪ (𝑁‘{𝑋})))) |
35 | 25, 30, 34 | 3eqtr4rd 2696 | . . . . . 6 ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑋})) = (𝑁‘(𝑈 ∪ {𝑋}))) |
36 | 35 | eqeq1d 2653 | . . . . 5 ⊢ (𝜑 → ((𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉 ↔ (𝑁‘(𝑈 ∪ {𝑋})) = 𝑉)) |
37 | 36 | biimpa 500 | . . . 4 ⊢ ((𝜑 ∧ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) → (𝑁‘(𝑈 ∪ {𝑋})) = 𝑉) |
38 | sneq 4220 | . . . . . . . 8 ⊢ (𝑣 = 𝑋 → {𝑣} = {𝑋}) | |
39 | 38 | uneq2d 3800 | . . . . . . 7 ⊢ (𝑣 = 𝑋 → (𝑈 ∪ {𝑣}) = (𝑈 ∪ {𝑋})) |
40 | 39 | fveq2d 6233 | . . . . . 6 ⊢ (𝑣 = 𝑋 → (𝑁‘(𝑈 ∪ {𝑣})) = (𝑁‘(𝑈 ∪ {𝑋}))) |
41 | 40 | eqeq1d 2653 | . . . . 5 ⊢ (𝑣 = 𝑋 → ((𝑁‘(𝑈 ∪ {𝑣})) = 𝑉 ↔ (𝑁‘(𝑈 ∪ {𝑋})) = 𝑉)) |
42 | 41 | rspcev 3340 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑁‘(𝑈 ∪ {𝑋})) = 𝑉) → ∃𝑣 ∈ 𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉) |
43 | 18, 37, 42 | syl2anc 694 | . . 3 ⊢ ((𝜑 ∧ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) → ∃𝑣 ∈ 𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉) |
44 | 7 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) → 𝑊 ∈ LVec) |
45 | 3, 4, 21, 6 | islshp 34584 | . . . 4 ⊢ (𝑊 ∈ LVec → (𝑈 ∈ 𝐻 ↔ (𝑈 ∈ 𝑆 ∧ 𝑈 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉))) |
46 | 44, 45 | syl 17 | . . 3 ⊢ ((𝜑 ∧ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) → (𝑈 ∈ 𝐻 ↔ (𝑈 ∈ 𝑆 ∧ 𝑈 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉))) |
47 | 15, 17, 43, 46 | mpbir3and 1264 | . 2 ⊢ ((𝜑 ∧ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) → 𝑈 ∈ 𝐻) |
48 | 13, 47 | impbida 895 | 1 ⊢ (𝜑 → (𝑈 ∈ 𝐻 ↔ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 383 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 ∃wrex 2942 ∪ cun 3605 ⊆ wss 3607 {csn 4210 ‘cfv 5926 (class class class)co 6690 Basecbs 15904 LSSumclsm 18095 LModclmod 18911 LSubSpclss 18980 LSpanclspn 19019 LVecclvec 19150 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-tpos 7397 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-3 11118 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 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-cntz 17796 df-lsm 18097 df-cmn 18241 df-abl 18242 df-mgp 18536 df-ur 18548 df-ring 18595 df-oppr 18669 df-dvdsr 18687 df-unit 18688 df-invr 18718 df-drng 18797 df-lmod 18913 df-lss 18981 df-lsp 19020 df-lvec 19151 df-lshyp 34582 |
This theorem is referenced by: (None) |
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