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Mirrors > Home > MPE Home > Th. List > Mathboxes > lclkrlem2y | Structured version Visualization version GIF version |
Description: Lemma for lclkr 39159. Restate the hypotheses for 𝐸 and 𝐺 to say their kernels are closed, in order to eliminate the generating vectors 𝑋 and 𝑌. (Contributed by NM, 18-Jan-2015.) |
Ref | Expression |
---|---|
lclkrlem2y.l | ⊢ 𝐿 = (LKer‘𝑈) |
lclkrlem2y.h | ⊢ 𝐻 = (LHyp‘𝐾) |
lclkrlem2y.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
lclkrlem2y.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
lclkrlem2y.f | ⊢ 𝐹 = (LFnl‘𝑈) |
lclkrlem2y.d | ⊢ 𝐷 = (LDual‘𝑈) |
lclkrlem2y.p | ⊢ + = (+g‘𝐷) |
lclkrlem2y.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
lclkrlem2y.e | ⊢ (𝜑 → 𝐸 ∈ 𝐹) |
lclkrlem2y.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
lclkrlem2y.le | ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐸))) = (𝐿‘𝐸)) |
lclkrlem2y.lg | ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)) |
Ref | Expression |
---|---|
lclkrlem2y | ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lclkrlem2y.lg | . . 3 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)) | |
2 | lclkrlem2y.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | lclkrlem2y.o | . . . 4 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
4 | lclkrlem2y.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
5 | eqid 2738 | . . . 4 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
6 | lclkrlem2y.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑈) | |
7 | lclkrlem2y.l | . . . 4 ⊢ 𝐿 = (LKer‘𝑈) | |
8 | lclkrlem2y.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
9 | lclkrlem2y.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
10 | 2, 3, 4, 5, 6, 7, 8, 9 | lcfl8a 39129 | . . 3 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ↔ ∃𝑦 ∈ (Base‘𝑈)(𝐿‘𝐺) = ( ⊥ ‘{𝑦}))) |
11 | 1, 10 | mpbid 235 | . 2 ⊢ (𝜑 → ∃𝑦 ∈ (Base‘𝑈)(𝐿‘𝐺) = ( ⊥ ‘{𝑦})) |
12 | lclkrlem2y.le | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐸))) = (𝐿‘𝐸)) | |
13 | lclkrlem2y.e | . . . . . 6 ⊢ (𝜑 → 𝐸 ∈ 𝐹) | |
14 | 2, 3, 4, 5, 6, 7, 8, 13 | lcfl8a 39129 | . . . . 5 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐸))) = (𝐿‘𝐸) ↔ ∃𝑥 ∈ (Base‘𝑈)(𝐿‘𝐸) = ( ⊥ ‘{𝑥}))) |
15 | 12, 14 | mpbid 235 | . . . 4 ⊢ (𝜑 → ∃𝑥 ∈ (Base‘𝑈)(𝐿‘𝐸) = ( ⊥ ‘{𝑥})) |
16 | lclkrlem2y.d | . . . . . . . 8 ⊢ 𝐷 = (LDual‘𝑈) | |
17 | lclkrlem2y.p | . . . . . . . 8 ⊢ + = (+g‘𝐷) | |
18 | 8 | 3ad2ant1 1134 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑈) ∧ (𝐿‘𝐸) = ( ⊥ ‘{𝑥}) ∧ 𝑦 ∈ (Base‘𝑈)) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑦})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
19 | simp21 1207 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑈) ∧ (𝐿‘𝐸) = ( ⊥ ‘{𝑥}) ∧ 𝑦 ∈ (Base‘𝑈)) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑦})) → 𝑥 ∈ (Base‘𝑈)) | |
20 | simp23 1209 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑈) ∧ (𝐿‘𝐸) = ( ⊥ ‘{𝑥}) ∧ 𝑦 ∈ (Base‘𝑈)) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑦})) → 𝑦 ∈ (Base‘𝑈)) | |
21 | 13 | 3ad2ant1 1134 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑈) ∧ (𝐿‘𝐸) = ( ⊥ ‘{𝑥}) ∧ 𝑦 ∈ (Base‘𝑈)) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑦})) → 𝐸 ∈ 𝐹) |
22 | 9 | 3ad2ant1 1134 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑈) ∧ (𝐿‘𝐸) = ( ⊥ ‘{𝑥}) ∧ 𝑦 ∈ (Base‘𝑈)) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑦})) → 𝐺 ∈ 𝐹) |
23 | simp22 1208 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑈) ∧ (𝐿‘𝐸) = ( ⊥ ‘{𝑥}) ∧ 𝑦 ∈ (Base‘𝑈)) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑦})) → (𝐿‘𝐸) = ( ⊥ ‘{𝑥})) | |
24 | simp3 1139 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑈) ∧ (𝐿‘𝐸) = ( ⊥ ‘{𝑥}) ∧ 𝑦 ∈ (Base‘𝑈)) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑦})) → (𝐿‘𝐺) = ( ⊥ ‘{𝑦})) | |
25 | 7, 2, 3, 4, 5, 6, 16, 17, 18, 19, 20, 21, 22, 23, 24 | lclkrlem2x 39156 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑈) ∧ (𝐿‘𝐸) = ( ⊥ ‘{𝑥}) ∧ 𝑦 ∈ (Base‘𝑈)) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑦})) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))) |
26 | 25 | 3exp 1120 | . . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (Base‘𝑈) ∧ (𝐿‘𝐸) = ( ⊥ ‘{𝑥}) ∧ 𝑦 ∈ (Base‘𝑈)) → ((𝐿‘𝐺) = ( ⊥ ‘{𝑦}) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))))) |
27 | 26 | 3expd 1354 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑈) → ((𝐿‘𝐸) = ( ⊥ ‘{𝑥}) → (𝑦 ∈ (Base‘𝑈) → ((𝐿‘𝐺) = ( ⊥ ‘{𝑦}) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))))))) |
28 | 27 | rexlimdv 3192 | . . . 4 ⊢ (𝜑 → (∃𝑥 ∈ (Base‘𝑈)(𝐿‘𝐸) = ( ⊥ ‘{𝑥}) → (𝑦 ∈ (Base‘𝑈) → ((𝐿‘𝐺) = ( ⊥ ‘{𝑦}) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))))) |
29 | 15, 28 | mpd 15 | . . 3 ⊢ (𝜑 → (𝑦 ∈ (Base‘𝑈) → ((𝐿‘𝐺) = ( ⊥ ‘{𝑦}) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))))) |
30 | 29 | rexlimdv 3192 | . 2 ⊢ (𝜑 → (∃𝑦 ∈ (Base‘𝑈)(𝐿‘𝐺) = ( ⊥ ‘{𝑦}) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))) |
31 | 11, 30 | mpd 15 | 1 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2113 ∃wrex 3054 {csn 4513 ‘cfv 6333 (class class class)co 7164 Basecbs 16579 +gcplusg 16661 LFnlclfn 36683 LKerclk 36711 LDualcld 36749 HLchlt 36976 LHypclh 37610 DVecHcdvh 38704 ocHcoch 38973 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 ax-riotaBAD 36579 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-int 4834 df-iun 4880 df-iin 4881 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-of 7419 df-om 7594 df-1st 7707 df-2nd 7708 df-tpos 7914 df-undef 7961 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-er 8313 df-map 8432 df-en 8549 df-dom 8550 df-sdom 8551 df-fin 8552 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-nn 11710 df-2 11772 df-3 11773 df-4 11774 df-5 11775 df-6 11776 df-n0 11970 df-z 12056 df-uz 12318 df-fz 12975 df-struct 16581 df-ndx 16582 df-slot 16583 df-base 16585 df-sets 16586 df-ress 16587 df-plusg 16674 df-mulr 16675 df-sca 16677 df-vsca 16678 df-0g 16811 df-mre 16953 df-mrc 16954 df-acs 16956 df-proset 17647 df-poset 17665 df-plt 17677 df-lub 17693 df-glb 17694 df-join 17695 df-meet 17696 df-p0 17758 df-p1 17759 df-lat 17765 df-clat 17827 df-mgm 17961 df-sgrp 18010 df-mnd 18021 df-submnd 18066 df-grp 18215 df-minusg 18216 df-sbg 18217 df-subg 18387 df-cntz 18558 df-oppg 18585 df-lsm 18872 df-cmn 19019 df-abl 19020 df-mgp 19352 df-ur 19364 df-ring 19411 df-oppr 19488 df-dvdsr 19506 df-unit 19507 df-invr 19537 df-dvr 19548 df-drng 19616 df-lmod 19748 df-lss 19816 df-lsp 19856 df-lvec 19987 df-lsatoms 36602 df-lshyp 36603 df-lcv 36645 df-lfl 36684 df-lkr 36712 df-ldual 36750 df-oposet 36802 df-ol 36804 df-oml 36805 df-covers 36892 df-ats 36893 df-atl 36924 df-cvlat 36948 df-hlat 36977 df-llines 37124 df-lplanes 37125 df-lvols 37126 df-lines 37127 df-psubsp 37129 df-pmap 37130 df-padd 37422 df-lhyp 37614 df-laut 37615 df-ldil 37730 df-ltrn 37731 df-trl 37785 df-tgrp 38369 df-tendo 38381 df-edring 38383 df-dveca 38629 df-disoa 38655 df-dvech 38705 df-dib 38765 df-dic 38799 df-dih 38855 df-doch 38974 df-djh 39021 |
This theorem is referenced by: lclkrlem2 39158 |
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