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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 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 484 | . . . 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 38966 | . . 3 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐻) → (¬ 𝑋 ∈ 𝑈 ↔ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉)) |
13 | 2, 12 | mpbid 232 | . 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 21123 | . . . . . . . . . . 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 20998 | . . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (𝑁‘𝑈) = 𝑈) |
23 | 20, 14, 22 | syl2anc 584 | . . . . . . . . 9 ⊢ (𝜑 → (𝑁‘𝑈) = 𝑈) |
24 | 23 | uneq1d 4177 | . . . . . . . 8 ⊢ (𝜑 → ((𝑁‘𝑈) ∪ (𝑁‘{𝑋})) = (𝑈 ∪ (𝑁‘{𝑋}))) |
25 | 24 | fveq2d 6911 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘((𝑁‘𝑈) ∪ (𝑁‘{𝑋}))) = (𝑁‘(𝑈 ∪ (𝑁‘{𝑋})))) |
26 | 3, 21 | lssss 20952 | . . . . . . . . 9 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ⊆ 𝑉) |
27 | 14, 26 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ⊆ 𝑉) |
28 | 10 | snssd 4814 | . . . . . . . 8 ⊢ (𝜑 → {𝑋} ⊆ 𝑉) |
29 | 3, 4 | lspun 21003 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉 ∧ {𝑋} ⊆ 𝑉) → (𝑁‘(𝑈 ∪ {𝑋})) = (𝑁‘((𝑁‘𝑈) ∪ (𝑁‘{𝑋})))) |
30 | 20, 27, 28, 29 | syl3anc 1370 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘(𝑈 ∪ {𝑋})) = (𝑁‘((𝑁‘𝑈) ∪ (𝑁‘{𝑋})))) |
31 | 3, 21, 4 | lspsncl 20993 | . . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ 𝑆) |
32 | 20, 10, 31 | syl2anc 584 | . . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ 𝑆) |
33 | 21, 4, 5 | lsmsp 21103 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ (𝑁‘{𝑋}) ∈ 𝑆) → (𝑈 ⊕ (𝑁‘{𝑋})) = (𝑁‘(𝑈 ∪ (𝑁‘{𝑋})))) |
34 | 20, 14, 32, 33 | syl3anc 1370 | . . . . . . 7 ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑋})) = (𝑁‘(𝑈 ∪ (𝑁‘{𝑋})))) |
35 | 25, 30, 34 | 3eqtr4rd 2786 | . . . . . 6 ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑋})) = (𝑁‘(𝑈 ∪ {𝑋}))) |
36 | 35 | eqeq1d 2737 | . . . . 5 ⊢ (𝜑 → ((𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉 ↔ (𝑁‘(𝑈 ∪ {𝑋})) = 𝑉)) |
37 | 36 | biimpa 476 | . . . 4 ⊢ ((𝜑 ∧ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) → (𝑁‘(𝑈 ∪ {𝑋})) = 𝑉) |
38 | sneq 4641 | . . . . . . 7 ⊢ (𝑣 = 𝑋 → {𝑣} = {𝑋}) | |
39 | 38 | uneq2d 4178 | . . . . . 6 ⊢ (𝑣 = 𝑋 → (𝑈 ∪ {𝑣}) = (𝑈 ∪ {𝑋})) |
40 | 39 | fveqeq2d 6915 | . . . . 5 ⊢ (𝑣 = 𝑋 → ((𝑁‘(𝑈 ∪ {𝑣})) = 𝑉 ↔ (𝑁‘(𝑈 ∪ {𝑋})) = 𝑉)) |
41 | 40 | rspcev 3622 | . . . 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 480 | . . . 4 ⊢ ((𝜑 ∧ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) → 𝑊 ∈ LVec) |
44 | 3, 4, 21, 6 | islshp 38961 | . . . 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 1341 | . 2 ⊢ ((𝜑 ∧ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉) → 𝑈 ∈ 𝐻) |
47 | 13, 46 | impbida 801 | 1 ⊢ (𝜑 → (𝑈 ∈ 𝐻 ↔ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∃wrex 3068 ∪ cun 3961 ⊆ wss 3963 {csn 4631 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 LSSumclsm 19667 LModclmod 20875 LSubSpclss 20947 LSpanclspn 20987 LVecclvec 21119 LSHypclsh 38957 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-cntz 19348 df-lsm 19669 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-drng 20748 df-lmod 20877 df-lss 20948 df-lsp 20988 df-lvec 21120 df-lshyp 38959 |
This theorem is referenced by: (None) |
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