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Mirrors > Home > MPE Home > Th. List > Mathboxes > lclkrlem2t | Structured version Visualization version GIF version |
Description: Lemma for lclkr 38668. We eliminate all hypotheses with 𝐵 here. (Contributed by NM, 18-Jan-2015.) |
Ref | Expression |
---|---|
lclkrlem2m.v | ⊢ 𝑉 = (Base‘𝑈) |
lclkrlem2m.t | ⊢ · = ( ·𝑠 ‘𝑈) |
lclkrlem2m.s | ⊢ 𝑆 = (Scalar‘𝑈) |
lclkrlem2m.q | ⊢ × = (.r‘𝑆) |
lclkrlem2m.z | ⊢ 0 = (0g‘𝑆) |
lclkrlem2m.i | ⊢ 𝐼 = (invr‘𝑆) |
lclkrlem2m.m | ⊢ − = (-g‘𝑈) |
lclkrlem2m.f | ⊢ 𝐹 = (LFnl‘𝑈) |
lclkrlem2m.d | ⊢ 𝐷 = (LDual‘𝑈) |
lclkrlem2m.p | ⊢ + = (+g‘𝐷) |
lclkrlem2m.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
lclkrlem2m.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
lclkrlem2m.e | ⊢ (𝜑 → 𝐸 ∈ 𝐹) |
lclkrlem2m.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
lclkrlem2n.n | ⊢ 𝑁 = (LSpan‘𝑈) |
lclkrlem2n.l | ⊢ 𝐿 = (LKer‘𝑈) |
lclkrlem2o.h | ⊢ 𝐻 = (LHyp‘𝐾) |
lclkrlem2o.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
lclkrlem2o.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
lclkrlem2o.a | ⊢ ⊕ = (LSSum‘𝑈) |
lclkrlem2o.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
lclkrlem2q.le | ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋})) |
lclkrlem2q.lg | ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌})) |
lclkrlem2t.n | ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑌) ≠ 0 ) |
Ref | Expression |
---|---|
lclkrlem2t | ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lclkrlem2m.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
2 | lclkrlem2m.t | . . 3 ⊢ · = ( ·𝑠 ‘𝑈) | |
3 | lclkrlem2m.s | . . 3 ⊢ 𝑆 = (Scalar‘𝑈) | |
4 | lclkrlem2m.q | . . 3 ⊢ × = (.r‘𝑆) | |
5 | lclkrlem2m.z | . . 3 ⊢ 0 = (0g‘𝑆) | |
6 | lclkrlem2m.i | . . 3 ⊢ 𝐼 = (invr‘𝑆) | |
7 | lclkrlem2m.m | . . 3 ⊢ − = (-g‘𝑈) | |
8 | lclkrlem2m.f | . . 3 ⊢ 𝐹 = (LFnl‘𝑈) | |
9 | lclkrlem2m.d | . . 3 ⊢ 𝐷 = (LDual‘𝑈) | |
10 | lclkrlem2m.p | . . 3 ⊢ + = (+g‘𝐷) | |
11 | lclkrlem2m.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
12 | 11 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) = (0g‘𝑈)) → 𝑋 ∈ 𝑉) |
13 | lclkrlem2m.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
14 | 13 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) = (0g‘𝑈)) → 𝑌 ∈ 𝑉) |
15 | lclkrlem2m.e | . . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝐹) | |
16 | 15 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) = (0g‘𝑈)) → 𝐸 ∈ 𝐹) |
17 | lclkrlem2m.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
18 | 17 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) = (0g‘𝑈)) → 𝐺 ∈ 𝐹) |
19 | lclkrlem2n.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑈) | |
20 | lclkrlem2n.l | . . 3 ⊢ 𝐿 = (LKer‘𝑈) | |
21 | lclkrlem2o.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
22 | lclkrlem2o.o | . . 3 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
23 | lclkrlem2o.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
24 | lclkrlem2o.a | . . 3 ⊢ ⊕ = (LSSum‘𝑈) | |
25 | lclkrlem2o.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
26 | 25 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) = (0g‘𝑈)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
27 | lclkrlem2q.le | . . . 4 ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋})) | |
28 | 27 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) = (0g‘𝑈)) → (𝐿‘𝐸) = ( ⊥ ‘{𝑋})) |
29 | lclkrlem2q.lg | . . . 4 ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌})) | |
30 | 29 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) = (0g‘𝑈)) → (𝐿‘𝐺) = ( ⊥ ‘{𝑌})) |
31 | eqid 2821 | . . 3 ⊢ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) = (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) | |
32 | lclkrlem2t.n | . . . 4 ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑌) ≠ 0 ) | |
33 | 32 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) = (0g‘𝑈)) → ((𝐸 + 𝐺)‘𝑌) ≠ 0 ) |
34 | simpr 487 | . . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) = (0g‘𝑈)) → (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) = (0g‘𝑈)) | |
35 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 19, 20, 21, 22, 23, 24, 26, 28, 30, 31, 33, 34 | lclkrlem2s 38660 | . 2 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) = (0g‘𝑈)) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))) |
36 | 11 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) ≠ (0g‘𝑈)) → 𝑋 ∈ 𝑉) |
37 | 13 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) ≠ (0g‘𝑈)) → 𝑌 ∈ 𝑉) |
38 | 15 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) ≠ (0g‘𝑈)) → 𝐸 ∈ 𝐹) |
39 | 17 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) ≠ (0g‘𝑈)) → 𝐺 ∈ 𝐹) |
40 | 25 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) ≠ (0g‘𝑈)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
41 | 27 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) ≠ (0g‘𝑈)) → (𝐿‘𝐸) = ( ⊥ ‘{𝑋})) |
42 | 29 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) ≠ (0g‘𝑈)) → (𝐿‘𝐺) = ( ⊥ ‘{𝑌})) |
43 | 32 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) ≠ (0g‘𝑈)) → ((𝐸 + 𝐺)‘𝑌) ≠ 0 ) |
44 | simpr 487 | . . 3 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) ≠ (0g‘𝑈)) → (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) ≠ (0g‘𝑈)) | |
45 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 36, 37, 38, 39, 19, 20, 21, 22, 23, 24, 40, 41, 42, 31, 43, 44 | lclkrlem2q 38658 | . 2 ⊢ ((𝜑 ∧ (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌)) ≠ (0g‘𝑈)) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))) |
46 | 35, 45 | pm2.61dane 3104 | 1 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 {csn 4566 ‘cfv 6354 (class class class)co 7155 Basecbs 16482 +gcplusg 16564 .rcmulr 16565 Scalarcsca 16567 ·𝑠 cvsca 16568 0gc0g 16712 -gcsg 18104 LSSumclsm 18758 invrcinvr 19420 LSpanclspn 19742 LFnlclfn 36192 LKerclk 36220 LDualcld 36258 HLchlt 36485 LHypclh 37119 DVecHcdvh 38213 ocHcoch 38482 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-riotaBAD 36088 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-om 7580 df-1st 7688 df-2nd 7689 df-tpos 7891 df-undef 7938 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-map 8407 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-n0 11897 df-z 11981 df-uz 12243 df-fz 12892 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-sca 16580 df-vsca 16581 df-0g 16714 df-mre 16856 df-mrc 16857 df-acs 16859 df-proset 17537 df-poset 17555 df-plt 17567 df-lub 17583 df-glb 17584 df-join 17585 df-meet 17586 df-p0 17648 df-p1 17649 df-lat 17655 df-clat 17717 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-submnd 17956 df-grp 18105 df-minusg 18106 df-sbg 18107 df-subg 18275 df-cntz 18446 df-oppg 18473 df-lsm 18760 df-cmn 18907 df-abl 18908 df-mgp 19239 df-ur 19251 df-ring 19298 df-oppr 19372 df-dvdsr 19390 df-unit 19391 df-invr 19421 df-dvr 19432 df-drng 19503 df-lmod 19635 df-lss 19703 df-lsp 19743 df-lvec 19874 df-lsatoms 36111 df-lshyp 36112 df-lcv 36154 df-lfl 36193 df-lkr 36221 df-ldual 36259 df-oposet 36311 df-ol 36313 df-oml 36314 df-covers 36401 df-ats 36402 df-atl 36433 df-cvlat 36457 df-hlat 36486 df-llines 36633 df-lplanes 36634 df-lvols 36635 df-lines 36636 df-psubsp 36638 df-pmap 36639 df-padd 36931 df-lhyp 37123 df-laut 37124 df-ldil 37239 df-ltrn 37240 df-trl 37294 df-tgrp 37878 df-tendo 37890 df-edring 37892 df-dveca 38138 df-disoa 38164 df-dvech 38214 df-dib 38274 df-dic 38308 df-dih 38364 df-doch 38483 df-djh 38530 |
This theorem is referenced by: lclkrlem2u 38662 lclkrlem2x 38665 |
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