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Mathbox for Norm Megill |
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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 481 | . . 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 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐻) → 𝑊 ∈ LVec) |
9 | simpr 485 | . . . 4 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐻) → 𝑈 ∈ 𝐻) | |
10 | lshpnel2.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
11 | 10 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐻) → 𝑋 ∈ 𝑉) |
12 | 3, 4, 5, 6, 8, 9, 11 | lshpnelb 37437 | . . 3 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐻) → (¬ 𝑋 ∈ 𝑈 ↔ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉)) |
13 | 2, 12 | mpbid 231 | . 2 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐻) → (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) |
14 | lshpnel2.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
15 | 14 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) → 𝑈 ∈ 𝑆) |
16 | lshpnel2.t | . . . 4 ⊢ (𝜑 → 𝑈 ≠ 𝑉) | |
17 | 16 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) → 𝑈 ≠ 𝑉) |
18 | 10 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) → 𝑋 ∈ 𝑉) |
19 | lveclmod 20565 | . . . . . . . . . . 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 20441 | . . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (𝑁‘𝑈) = 𝑈) |
23 | 20, 14, 22 | syl2anc 584 | . . . . . . . . 9 ⊢ (𝜑 → (𝑁‘𝑈) = 𝑈) |
24 | 23 | uneq1d 4122 | . . . . . . . 8 ⊢ (𝜑 → ((𝑁‘𝑈) ∪ (𝑁‘{𝑋})) = (𝑈 ∪ (𝑁‘{𝑋}))) |
25 | 24 | fveq2d 6846 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘((𝑁‘𝑈) ∪ (𝑁‘{𝑋}))) = (𝑁‘(𝑈 ∪ (𝑁‘{𝑋})))) |
26 | 3, 21 | lssss 20395 | . . . . . . . . 9 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ⊆ 𝑉) |
27 | 14, 26 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ⊆ 𝑉) |
28 | 10 | snssd 4769 | . . . . . . . 8 ⊢ (𝜑 → {𝑋} ⊆ 𝑉) |
29 | 3, 4 | lspun 20446 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉 ∧ {𝑋} ⊆ 𝑉) → (𝑁‘(𝑈 ∪ {𝑋})) = (𝑁‘((𝑁‘𝑈) ∪ (𝑁‘{𝑋})))) |
30 | 20, 27, 28, 29 | syl3anc 1371 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘(𝑈 ∪ {𝑋})) = (𝑁‘((𝑁‘𝑈) ∪ (𝑁‘{𝑋})))) |
31 | 3, 21, 4 | lspsncl 20436 | . . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ 𝑆) |
32 | 20, 10, 31 | syl2anc 584 | . . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ 𝑆) |
33 | 21, 4, 5 | lsmsp 20545 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ (𝑁‘{𝑋}) ∈ 𝑆) → (𝑈 ⊕ (𝑁‘{𝑋})) = (𝑁‘(𝑈 ∪ (𝑁‘{𝑋})))) |
34 | 20, 14, 32, 33 | syl3anc 1371 | . . . . . . 7 ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑋})) = (𝑁‘(𝑈 ∪ (𝑁‘{𝑋})))) |
35 | 25, 30, 34 | 3eqtr4rd 2787 | . . . . . 6 ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑋})) = (𝑁‘(𝑈 ∪ {𝑋}))) |
36 | 35 | eqeq1d 2738 | . . . . 5 ⊢ (𝜑 → ((𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉 ↔ (𝑁‘(𝑈 ∪ {𝑋})) = 𝑉)) |
37 | 36 | biimpa 477 | . . . 4 ⊢ ((𝜑 ∧ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) → (𝑁‘(𝑈 ∪ {𝑋})) = 𝑉) |
38 | sneq 4596 | . . . . . . 7 ⊢ (𝑣 = 𝑋 → {𝑣} = {𝑋}) | |
39 | 38 | uneq2d 4123 | . . . . . 6 ⊢ (𝑣 = 𝑋 → (𝑈 ∪ {𝑣}) = (𝑈 ∪ {𝑋})) |
40 | 39 | fveqeq2d 6850 | . . . . 5 ⊢ (𝑣 = 𝑋 → ((𝑁‘(𝑈 ∪ {𝑣})) = 𝑉 ↔ (𝑁‘(𝑈 ∪ {𝑋})) = 𝑉)) |
41 | 40 | rspcev 3581 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑁‘(𝑈 ∪ {𝑋})) = 𝑉) → ∃𝑣 ∈ 𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉) |
42 | 18, 37, 41 | syl2anc 584 | . . 3 ⊢ ((𝜑 ∧ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) → ∃𝑣 ∈ 𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉) |
43 | 7 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) → 𝑊 ∈ LVec) |
44 | 3, 4, 21, 6 | islshp 37432 | . . . 4 ⊢ (𝑊 ∈ LVec → (𝑈 ∈ 𝐻 ↔ (𝑈 ∈ 𝑆 ∧ 𝑈 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉))) |
45 | 43, 44 | syl 17 | . . 3 ⊢ ((𝜑 ∧ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) → (𝑈 ∈ 𝐻 ↔ (𝑈 ∈ 𝑆 ∧ 𝑈 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉))) |
46 | 15, 17, 42, 45 | mpbir3and 1342 | . 2 ⊢ ((𝜑 ∧ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) → 𝑈 ∈ 𝐻) |
47 | 13, 46 | impbida 799 | 1 ⊢ (𝜑 → (𝑈 ∈ 𝐻 ↔ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∃wrex 3073 ∪ cun 3908 ⊆ wss 3910 {csn 4586 ‘cfv 6496 (class class class)co 7356 Basecbs 17082 LSSumclsm 19414 LModclmod 20320 LSubSpclss 20390 LSpanclspn 20430 LVecclvec 20561 LSHypclsh 37428 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7802 df-1st 7920 df-2nd 7921 df-tpos 8156 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-er 8647 df-en 8883 df-dom 8884 df-sdom 8885 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-nn 12153 df-2 12215 df-3 12216 df-sets 17035 df-slot 17053 df-ndx 17065 df-base 17083 df-ress 17112 df-plusg 17145 df-mulr 17146 df-0g 17322 df-mgm 18496 df-sgrp 18545 df-mnd 18556 df-submnd 18601 df-grp 18750 df-minusg 18751 df-sbg 18752 df-subg 18923 df-cntz 19095 df-lsm 19416 df-cmn 19562 df-abl 19563 df-mgp 19895 df-ur 19912 df-ring 19964 df-oppr 20047 df-dvdsr 20068 df-unit 20069 df-invr 20099 df-drng 20185 df-lmod 20322 df-lss 20391 df-lsp 20431 df-lvec 20562 df-lshyp 37430 |
This theorem is referenced by: (None) |
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