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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lshpinN | Structured version Visualization version GIF version |
Description: The intersection of two different hyperplanes is not a hyperplane. (Contributed by NM, 29-Oct-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
lshpin.h | ⊢ 𝐻 = (LSHyp‘𝑊) |
lshpin.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lshpin.t | ⊢ (𝜑 → 𝑇 ∈ 𝐻) |
lshpin.u | ⊢ (𝜑 → 𝑈 ∈ 𝐻) |
Ref | Expression |
---|---|
lshpinN | ⊢ (𝜑 → ((𝑇 ∩ 𝑈) ∈ 𝐻 ↔ 𝑇 = 𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inss1 4231 | . . . . 5 ⊢ (𝑇 ∩ 𝑈) ⊆ 𝑇 | |
2 | lshpin.h | . . . . . 6 ⊢ 𝐻 = (LSHyp‘𝑊) | |
3 | lshpin.w | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
4 | 3 | adantr 479 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑇 ∩ 𝑈) ∈ 𝐻) → 𝑊 ∈ LVec) |
5 | simpr 483 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑇 ∩ 𝑈) ∈ 𝐻) → (𝑇 ∩ 𝑈) ∈ 𝐻) | |
6 | lshpin.t | . . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ 𝐻) | |
7 | 6 | adantr 479 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑇 ∩ 𝑈) ∈ 𝐻) → 𝑇 ∈ 𝐻) |
8 | 2, 4, 5, 7 | lshpcmp 38492 | . . . . 5 ⊢ ((𝜑 ∧ (𝑇 ∩ 𝑈) ∈ 𝐻) → ((𝑇 ∩ 𝑈) ⊆ 𝑇 ↔ (𝑇 ∩ 𝑈) = 𝑇)) |
9 | 1, 8 | mpbii 232 | . . . 4 ⊢ ((𝜑 ∧ (𝑇 ∩ 𝑈) ∈ 𝐻) → (𝑇 ∩ 𝑈) = 𝑇) |
10 | inss2 4232 | . . . . 5 ⊢ (𝑇 ∩ 𝑈) ⊆ 𝑈 | |
11 | lshpin.u | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ 𝐻) | |
12 | 11 | adantr 479 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑇 ∩ 𝑈) ∈ 𝐻) → 𝑈 ∈ 𝐻) |
13 | 2, 4, 5, 12 | lshpcmp 38492 | . . . . 5 ⊢ ((𝜑 ∧ (𝑇 ∩ 𝑈) ∈ 𝐻) → ((𝑇 ∩ 𝑈) ⊆ 𝑈 ↔ (𝑇 ∩ 𝑈) = 𝑈)) |
14 | 10, 13 | mpbii 232 | . . . 4 ⊢ ((𝜑 ∧ (𝑇 ∩ 𝑈) ∈ 𝐻) → (𝑇 ∩ 𝑈) = 𝑈) |
15 | 9, 14 | eqtr3d 2770 | . . 3 ⊢ ((𝜑 ∧ (𝑇 ∩ 𝑈) ∈ 𝐻) → 𝑇 = 𝑈) |
16 | 15 | ex 411 | . 2 ⊢ (𝜑 → ((𝑇 ∩ 𝑈) ∈ 𝐻 → 𝑇 = 𝑈)) |
17 | inidm 4221 | . . . 4 ⊢ (𝑇 ∩ 𝑇) = 𝑇 | |
18 | 17, 6 | eqeltrid 2833 | . . 3 ⊢ (𝜑 → (𝑇 ∩ 𝑇) ∈ 𝐻) |
19 | ineq2 4208 | . . . 4 ⊢ (𝑇 = 𝑈 → (𝑇 ∩ 𝑇) = (𝑇 ∩ 𝑈)) | |
20 | 19 | eleq1d 2814 | . . 3 ⊢ (𝑇 = 𝑈 → ((𝑇 ∩ 𝑇) ∈ 𝐻 ↔ (𝑇 ∩ 𝑈) ∈ 𝐻)) |
21 | 18, 20 | syl5ibcom 244 | . 2 ⊢ (𝜑 → (𝑇 = 𝑈 → (𝑇 ∩ 𝑈) ∈ 𝐻)) |
22 | 16, 21 | impbid 211 | 1 ⊢ (𝜑 → ((𝑇 ∩ 𝑈) ∈ 𝐻 ↔ 𝑇 = 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∩ cin 3948 ⊆ wss 3949 ‘cfv 6553 LVecclvec 20994 LSHypclsh 38479 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-tpos 8238 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-0g 17430 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-submnd 18748 df-grp 18900 df-minusg 18901 df-sbg 18902 df-subg 19085 df-cntz 19275 df-lsm 19598 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-ring 20182 df-oppr 20280 df-dvdsr 20303 df-unit 20304 df-invr 20334 df-drng 20633 df-lmod 20752 df-lss 20823 df-lsp 20863 df-lvec 20995 df-lshyp 38481 |
This theorem is referenced by: (None) |
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