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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 474 | . . 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 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐻) → 𝑊 ∈ LVec) |
9 | simpr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐻) → 𝑈 ∈ 𝐻) | |
10 | lshpnel2.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
11 | 10 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐻) → 𝑋 ∈ 𝑉) |
12 | 3, 4, 5, 6, 8, 9, 11 | lshpnelb 35059 | . . 3 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐻) → (¬ 𝑋 ∈ 𝑈 ↔ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉)) |
13 | 2, 12 | mpbid 224 | . 2 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐻) → (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) |
14 | lshpnel2.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
15 | 14 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) → 𝑈 ∈ 𝑆) |
16 | lshpnel2.t | . . . 4 ⊢ (𝜑 → 𝑈 ≠ 𝑉) | |
17 | 16 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) → 𝑈 ≠ 𝑉) |
18 | 10 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) → 𝑋 ∈ 𝑉) |
19 | lveclmod 19465 | . . . . . . . . . . 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 19341 | . . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (𝑁‘𝑈) = 𝑈) |
23 | 20, 14, 22 | syl2anc 581 | . . . . . . . . 9 ⊢ (𝜑 → (𝑁‘𝑈) = 𝑈) |
24 | 23 | uneq1d 3993 | . . . . . . . 8 ⊢ (𝜑 → ((𝑁‘𝑈) ∪ (𝑁‘{𝑋})) = (𝑈 ∪ (𝑁‘{𝑋}))) |
25 | 24 | fveq2d 6437 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘((𝑁‘𝑈) ∪ (𝑁‘{𝑋}))) = (𝑁‘(𝑈 ∪ (𝑁‘{𝑋})))) |
26 | 3, 21 | lssss 19293 | . . . . . . . . 9 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ⊆ 𝑉) |
27 | 14, 26 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ⊆ 𝑉) |
28 | 10 | snssd 4558 | . . . . . . . 8 ⊢ (𝜑 → {𝑋} ⊆ 𝑉) |
29 | 3, 4 | lspun 19346 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉 ∧ {𝑋} ⊆ 𝑉) → (𝑁‘(𝑈 ∪ {𝑋})) = (𝑁‘((𝑁‘𝑈) ∪ (𝑁‘{𝑋})))) |
30 | 20, 27, 28, 29 | syl3anc 1496 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘(𝑈 ∪ {𝑋})) = (𝑁‘((𝑁‘𝑈) ∪ (𝑁‘{𝑋})))) |
31 | 3, 21, 4 | lspsncl 19336 | . . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ 𝑆) |
32 | 20, 10, 31 | syl2anc 581 | . . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ 𝑆) |
33 | 21, 4, 5 | lsmsp 19445 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ (𝑁‘{𝑋}) ∈ 𝑆) → (𝑈 ⊕ (𝑁‘{𝑋})) = (𝑁‘(𝑈 ∪ (𝑁‘{𝑋})))) |
34 | 20, 14, 32, 33 | syl3anc 1496 | . . . . . . 7 ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑋})) = (𝑁‘(𝑈 ∪ (𝑁‘{𝑋})))) |
35 | 25, 30, 34 | 3eqtr4rd 2872 | . . . . . 6 ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑋})) = (𝑁‘(𝑈 ∪ {𝑋}))) |
36 | 35 | eqeq1d 2827 | . . . . 5 ⊢ (𝜑 → ((𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉 ↔ (𝑁‘(𝑈 ∪ {𝑋})) = 𝑉)) |
37 | 36 | biimpa 470 | . . . 4 ⊢ ((𝜑 ∧ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) → (𝑁‘(𝑈 ∪ {𝑋})) = 𝑉) |
38 | sneq 4407 | . . . . . . 7 ⊢ (𝑣 = 𝑋 → {𝑣} = {𝑋}) | |
39 | 38 | uneq2d 3994 | . . . . . 6 ⊢ (𝑣 = 𝑋 → (𝑈 ∪ {𝑣}) = (𝑈 ∪ {𝑋})) |
40 | 39 | fveqeq2d 6441 | . . . . 5 ⊢ (𝑣 = 𝑋 → ((𝑁‘(𝑈 ∪ {𝑣})) = 𝑉 ↔ (𝑁‘(𝑈 ∪ {𝑋})) = 𝑉)) |
41 | 40 | rspcev 3526 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑁‘(𝑈 ∪ {𝑋})) = 𝑉) → ∃𝑣 ∈ 𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉) |
42 | 18, 37, 41 | syl2anc 581 | . . 3 ⊢ ((𝜑 ∧ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) → ∃𝑣 ∈ 𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉) |
43 | 7 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) → 𝑊 ∈ LVec) |
44 | 3, 4, 21, 6 | islshp 35054 | . . . 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 1448 | . 2 ⊢ ((𝜑 ∧ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) → 𝑈 ∈ 𝐻) |
47 | 13, 46 | impbida 837 | 1 ⊢ (𝜑 → (𝑈 ∈ 𝐻 ↔ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ≠ wne 2999 ∃wrex 3118 ∪ cun 3796 ⊆ wss 3798 {csn 4397 ‘cfv 6123 (class class class)co 6905 Basecbs 16222 LSSumclsm 18400 LModclmod 19219 LSubSpclss 19288 LSpanclspn 19330 LVecclvec 19461 LSHypclsh 35050 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-tpos 7617 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-3 11415 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-ress 16230 df-plusg 16318 df-mulr 16319 df-0g 16455 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-submnd 17689 df-grp 17779 df-minusg 17780 df-sbg 17781 df-subg 17942 df-cntz 18100 df-lsm 18402 df-cmn 18548 df-abl 18549 df-mgp 18844 df-ur 18856 df-ring 18903 df-oppr 18977 df-dvdsr 18995 df-unit 18996 df-invr 19026 df-drng 19105 df-lmod 19221 df-lss 19289 df-lsp 19331 df-lvec 19462 df-lshyp 35052 |
This theorem is referenced by: (None) |
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