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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 483 | . . 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 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐻) → 𝑊 ∈ LVec) |
9 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐻) → 𝑈 ∈ 𝐻) | |
10 | lshpnel2.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
11 | 10 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐻) → 𝑋 ∈ 𝑉) |
12 | 3, 4, 5, 6, 8, 9, 11 | lshpnelb 36124 | . . 3 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐻) → (¬ 𝑋 ∈ 𝑈 ↔ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉)) |
13 | 2, 12 | mpbid 234 | . 2 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐻) → (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) |
14 | lshpnel2.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
15 | 14 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) → 𝑈 ∈ 𝑆) |
16 | lshpnel2.t | . . . 4 ⊢ (𝜑 → 𝑈 ≠ 𝑉) | |
17 | 16 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) → 𝑈 ≠ 𝑉) |
18 | 10 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) → 𝑋 ∈ 𝑉) |
19 | lveclmod 19881 | . . . . . . . . . . 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 19757 | . . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (𝑁‘𝑈) = 𝑈) |
23 | 20, 14, 22 | syl2anc 586 | . . . . . . . . 9 ⊢ (𝜑 → (𝑁‘𝑈) = 𝑈) |
24 | 23 | uneq1d 4141 | . . . . . . . 8 ⊢ (𝜑 → ((𝑁‘𝑈) ∪ (𝑁‘{𝑋})) = (𝑈 ∪ (𝑁‘{𝑋}))) |
25 | 24 | fveq2d 6677 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘((𝑁‘𝑈) ∪ (𝑁‘{𝑋}))) = (𝑁‘(𝑈 ∪ (𝑁‘{𝑋})))) |
26 | 3, 21 | lssss 19711 | . . . . . . . . 9 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ⊆ 𝑉) |
27 | 14, 26 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ⊆ 𝑉) |
28 | 10 | snssd 4745 | . . . . . . . 8 ⊢ (𝜑 → {𝑋} ⊆ 𝑉) |
29 | 3, 4 | lspun 19762 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉 ∧ {𝑋} ⊆ 𝑉) → (𝑁‘(𝑈 ∪ {𝑋})) = (𝑁‘((𝑁‘𝑈) ∪ (𝑁‘{𝑋})))) |
30 | 20, 27, 28, 29 | syl3anc 1367 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘(𝑈 ∪ {𝑋})) = (𝑁‘((𝑁‘𝑈) ∪ (𝑁‘{𝑋})))) |
31 | 3, 21, 4 | lspsncl 19752 | . . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ 𝑆) |
32 | 20, 10, 31 | syl2anc 586 | . . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ 𝑆) |
33 | 21, 4, 5 | lsmsp 19861 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ (𝑁‘{𝑋}) ∈ 𝑆) → (𝑈 ⊕ (𝑁‘{𝑋})) = (𝑁‘(𝑈 ∪ (𝑁‘{𝑋})))) |
34 | 20, 14, 32, 33 | syl3anc 1367 | . . . . . . 7 ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑋})) = (𝑁‘(𝑈 ∪ (𝑁‘{𝑋})))) |
35 | 25, 30, 34 | 3eqtr4rd 2870 | . . . . . 6 ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑋})) = (𝑁‘(𝑈 ∪ {𝑋}))) |
36 | 35 | eqeq1d 2826 | . . . . 5 ⊢ (𝜑 → ((𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉 ↔ (𝑁‘(𝑈 ∪ {𝑋})) = 𝑉)) |
37 | 36 | biimpa 479 | . . . 4 ⊢ ((𝜑 ∧ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) → (𝑁‘(𝑈 ∪ {𝑋})) = 𝑉) |
38 | sneq 4580 | . . . . . . 7 ⊢ (𝑣 = 𝑋 → {𝑣} = {𝑋}) | |
39 | 38 | uneq2d 4142 | . . . . . 6 ⊢ (𝑣 = 𝑋 → (𝑈 ∪ {𝑣}) = (𝑈 ∪ {𝑋})) |
40 | 39 | fveqeq2d 6681 | . . . . 5 ⊢ (𝑣 = 𝑋 → ((𝑁‘(𝑈 ∪ {𝑣})) = 𝑉 ↔ (𝑁‘(𝑈 ∪ {𝑋})) = 𝑉)) |
41 | 40 | rspcev 3626 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑁‘(𝑈 ∪ {𝑋})) = 𝑉) → ∃𝑣 ∈ 𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉) |
42 | 18, 37, 41 | syl2anc 586 | . . 3 ⊢ ((𝜑 ∧ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) → ∃𝑣 ∈ 𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉) |
43 | 7 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) → 𝑊 ∈ LVec) |
44 | 3, 4, 21, 6 | islshp 36119 | . . . 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 1338 | . 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 208 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 ∃wrex 3142 ∪ cun 3937 ⊆ wss 3939 {csn 4570 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 LSSumclsm 18762 LModclmod 19637 LSubSpclss 19706 LSpanclspn 19746 LVecclvec 19877 LSHypclsh 36115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-tpos 7895 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-0g 16718 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-submnd 17960 df-grp 18109 df-minusg 18110 df-sbg 18111 df-subg 18279 df-cntz 18450 df-lsm 18764 df-cmn 18911 df-abl 18912 df-mgp 19243 df-ur 19255 df-ring 19302 df-oppr 19376 df-dvdsr 19394 df-unit 19395 df-invr 19425 df-drng 19507 df-lmod 19639 df-lss 19707 df-lsp 19747 df-lvec 19878 df-lshyp 36117 |
This theorem is referenced by: (None) |
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