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Mirrors > Home > MPE Home > Th. List > Mathboxes > lclkrlem2j | Structured version Visualization version GIF version |
Description: Lemma for lclkr 38684. Kernel closure when 𝑌 is zero. (Contributed by NM, 18-Jan-2015.) |
Ref | Expression |
---|---|
lclkrlem2f.h | ⊢ 𝐻 = (LHyp‘𝐾) |
lclkrlem2f.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
lclkrlem2f.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
lclkrlem2f.v | ⊢ 𝑉 = (Base‘𝑈) |
lclkrlem2f.s | ⊢ 𝑆 = (Scalar‘𝑈) |
lclkrlem2f.q | ⊢ 𝑄 = (0g‘𝑆) |
lclkrlem2f.z | ⊢ 0 = (0g‘𝑈) |
lclkrlem2f.a | ⊢ ⊕ = (LSSum‘𝑈) |
lclkrlem2f.n | ⊢ 𝑁 = (LSpan‘𝑈) |
lclkrlem2f.f | ⊢ 𝐹 = (LFnl‘𝑈) |
lclkrlem2f.j | ⊢ 𝐽 = (LSHyp‘𝑈) |
lclkrlem2f.l | ⊢ 𝐿 = (LKer‘𝑈) |
lclkrlem2f.d | ⊢ 𝐷 = (LDual‘𝑈) |
lclkrlem2f.p | ⊢ + = (+g‘𝐷) |
lclkrlem2f.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
lclkrlem2f.b | ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 })) |
lclkrlem2f.e | ⊢ (𝜑 → 𝐸 ∈ 𝐹) |
lclkrlem2f.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
lclkrlem2f.le | ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋})) |
lclkrlem2f.lg | ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌})) |
lclkrlem2f.kb | ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝐵) = 𝑄) |
lclkrlem2f.nx | ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵}))) |
lclkrlem2j.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
lclkrlem2j.y | ⊢ (𝜑 → 𝑌 = 0 ) |
Ref | Expression |
---|---|
lclkrlem2j | ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lclkrlem2f.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | lclkrlem2j.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
3 | 2 | snssd 4742 | . . . 4 ⊢ (𝜑 → {𝑋} ⊆ 𝑉) |
4 | lclkrlem2f.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | eqid 2821 | . . . . 5 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊) | |
6 | lclkrlem2f.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
7 | lclkrlem2f.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑈) | |
8 | lclkrlem2f.o | . . . . 5 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
9 | 4, 5, 6, 7, 8 | dochcl 38504 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑋} ⊆ 𝑉) → ( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
10 | 1, 3, 9 | syl2anc 586 | . . 3 ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
11 | 4, 5, 8 | dochoc 38518 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑋}))) = ( ⊥ ‘{𝑋})) |
12 | 1, 10, 11 | syl2anc 586 | . 2 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑋}))) = ( ⊥ ‘{𝑋})) |
13 | lclkrlem2f.lg | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌})) | |
14 | lclkrlem2j.y | . . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 = 0 ) | |
15 | 14 | sneqd 4579 | . . . . . . . . . . . 12 ⊢ (𝜑 → {𝑌} = { 0 }) |
16 | 15 | fveq2d 6674 | . . . . . . . . . . 11 ⊢ (𝜑 → ( ⊥ ‘{𝑌}) = ( ⊥ ‘{ 0 })) |
17 | eqid 2821 | . . . . . . . . . . . . 13 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
18 | lclkrlem2f.z | . . . . . . . . . . . . 13 ⊢ 0 = (0g‘𝑈) | |
19 | 4, 6, 8, 17, 18 | doch0 38509 | . . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘{ 0 }) = (Base‘𝑈)) |
20 | 1, 19 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → ( ⊥ ‘{ 0 }) = (Base‘𝑈)) |
21 | 13, 16, 20 | 3eqtrd 2860 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐿‘𝐺) = (Base‘𝑈)) |
22 | 4, 6, 1 | dvhlmod 38261 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ LMod) |
23 | lclkrlem2f.g | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
24 | lclkrlem2f.s | . . . . . . . . . . . 12 ⊢ 𝑆 = (Scalar‘𝑈) | |
25 | lclkrlem2f.q | . . . . . . . . . . . 12 ⊢ 𝑄 = (0g‘𝑆) | |
26 | lclkrlem2f.f | . . . . . . . . . . . 12 ⊢ 𝐹 = (LFnl‘𝑈) | |
27 | lclkrlem2f.l | . . . . . . . . . . . 12 ⊢ 𝐿 = (LKer‘𝑈) | |
28 | 24, 25, 17, 26, 27 | lkr0f 36245 | . . . . . . . . . . 11 ⊢ ((𝑈 ∈ LMod ∧ 𝐺 ∈ 𝐹) → ((𝐿‘𝐺) = (Base‘𝑈) ↔ 𝐺 = ((Base‘𝑈) × {𝑄}))) |
29 | 22, 23, 28 | syl2anc 586 | . . . . . . . . . 10 ⊢ (𝜑 → ((𝐿‘𝐺) = (Base‘𝑈) ↔ 𝐺 = ((Base‘𝑈) × {𝑄}))) |
30 | 21, 29 | mpbid 234 | . . . . . . . . 9 ⊢ (𝜑 → 𝐺 = ((Base‘𝑈) × {𝑄})) |
31 | lclkrlem2f.d | . . . . . . . . . 10 ⊢ 𝐷 = (LDual‘𝑈) | |
32 | eqid 2821 | . . . . . . . . . 10 ⊢ (0g‘𝐷) = (0g‘𝐷) | |
33 | 17, 24, 25, 31, 32, 22 | ldual0v 36301 | . . . . . . . . 9 ⊢ (𝜑 → (0g‘𝐷) = ((Base‘𝑈) × {𝑄})) |
34 | 30, 33 | eqtr4d 2859 | . . . . . . . 8 ⊢ (𝜑 → 𝐺 = (0g‘𝐷)) |
35 | 34 | oveq2d 7172 | . . . . . . 7 ⊢ (𝜑 → (𝐸 + 𝐺) = (𝐸 + (0g‘𝐷))) |
36 | 31, 22 | lduallmod 36304 | . . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ LMod) |
37 | eqid 2821 | . . . . . . . . 9 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
38 | lclkrlem2f.e | . . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ 𝐹) | |
39 | 26, 31, 37, 22, 38 | ldualelvbase 36278 | . . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ (Base‘𝐷)) |
40 | lclkrlem2f.p | . . . . . . . . 9 ⊢ + = (+g‘𝐷) | |
41 | 37, 40, 32 | lmod0vrid 19665 | . . . . . . . 8 ⊢ ((𝐷 ∈ LMod ∧ 𝐸 ∈ (Base‘𝐷)) → (𝐸 + (0g‘𝐷)) = 𝐸) |
42 | 36, 39, 41 | syl2anc 586 | . . . . . . 7 ⊢ (𝜑 → (𝐸 + (0g‘𝐷)) = 𝐸) |
43 | 35, 42 | eqtrd 2856 | . . . . . 6 ⊢ (𝜑 → (𝐸 + 𝐺) = 𝐸) |
44 | 43 | fveq2d 6674 | . . . . 5 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) = (𝐿‘𝐸)) |
45 | lclkrlem2f.le | . . . . 5 ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋})) | |
46 | 44, 45 | eqtr2d 2857 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘{𝑋}) = (𝐿‘(𝐸 + 𝐺))) |
47 | 46 | fveq2d 6674 | . . 3 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘{𝑋})) = ( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) |
48 | 47 | fveq2d 6674 | . 2 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑋}))) = ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺))))) |
49 | 12, 48, 46 | 3eqtr3d 2864 | 1 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ∖ cdif 3933 ⊆ wss 3936 {csn 4567 × cxp 5553 ran crn 5556 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 +gcplusg 16565 Scalarcsca 16568 0gc0g 16713 LSSumclsm 18759 LModclmod 19634 LSpanclspn 19743 LSHypclsh 36126 LFnlclfn 36208 LKerclk 36236 LDualcld 36274 HLchlt 36501 LHypclh 37135 DVecHcdvh 38229 DIsoHcdih 38379 ocHcoch 38498 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-riotaBAD 36104 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-tpos 7892 df-undef 7939 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-sca 16581 df-vsca 16582 df-0g 16715 df-proset 17538 df-poset 17556 df-plt 17568 df-lub 17584 df-glb 17585 df-join 17586 df-meet 17587 df-p0 17649 df-p1 17650 df-lat 17656 df-clat 17718 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-grp 18106 df-minusg 18107 df-sbg 18108 df-subg 18276 df-cntz 18447 df-lsm 18761 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-oppr 19373 df-dvdsr 19391 df-unit 19392 df-invr 19422 df-dvr 19433 df-drng 19504 df-lmod 19636 df-lss 19704 df-lsp 19744 df-lvec 19875 df-lfl 36209 df-lkr 36237 df-ldual 36275 df-oposet 36327 df-ol 36329 df-oml 36330 df-covers 36417 df-ats 36418 df-atl 36449 df-cvlat 36473 df-hlat 36502 df-llines 36649 df-lplanes 36650 df-lvols 36651 df-lines 36652 df-psubsp 36654 df-pmap 36655 df-padd 36947 df-lhyp 37139 df-laut 37140 df-ldil 37255 df-ltrn 37256 df-trl 37310 df-tendo 37906 df-edring 37908 df-disoa 38180 df-dvech 38230 df-dib 38290 df-dic 38324 df-dih 38380 df-doch 38499 |
This theorem is referenced by: lclkrlem2k 38668 lclkrlem2l 38669 |
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