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Mirrors > Home > MPE Home > Th. List > Mathboxes > lclkrlem2j | Structured version Visualization version GIF version |
Description: Lemma for lclkr 41516. 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 4814 | . . . 4 ⊢ (𝜑 → {𝑋} ⊆ 𝑉) |
4 | lclkrlem2f.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | eqid 2735 | . . . . 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 41336 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑋} ⊆ 𝑉) → ( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
10 | 1, 3, 9 | syl2anc 584 | . . 3 ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
11 | 4, 5, 8 | dochoc 41350 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑋}))) = ( ⊥ ‘{𝑋})) |
12 | 1, 10, 11 | syl2anc 584 | . 2 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑋}))) = ( ⊥ ‘{𝑋})) |
13 | lclkrlem2f.lg | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌})) | |
14 | lclkrlem2j.y | . . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 = 0 ) | |
15 | 14 | sneqd 4643 | . . . . . . . . . . . 12 ⊢ (𝜑 → {𝑌} = { 0 }) |
16 | 15 | fveq2d 6911 | . . . . . . . . . . 11 ⊢ (𝜑 → ( ⊥ ‘{𝑌}) = ( ⊥ ‘{ 0 })) |
17 | eqid 2735 | . . . . . . . . . . . . 13 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
18 | lclkrlem2f.z | . . . . . . . . . . . . 13 ⊢ 0 = (0g‘𝑈) | |
19 | 4, 6, 8, 17, 18 | doch0 41341 | . . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘{ 0 }) = (Base‘𝑈)) |
20 | 1, 19 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → ( ⊥ ‘{ 0 }) = (Base‘𝑈)) |
21 | 13, 16, 20 | 3eqtrd 2779 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐿‘𝐺) = (Base‘𝑈)) |
22 | 4, 6, 1 | dvhlmod 41093 | . . . . . . . . . . 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 39076 | . . . . . . . . . . 11 ⊢ ((𝑈 ∈ LMod ∧ 𝐺 ∈ 𝐹) → ((𝐿‘𝐺) = (Base‘𝑈) ↔ 𝐺 = ((Base‘𝑈) × {𝑄}))) |
29 | 22, 23, 28 | syl2anc 584 | . . . . . . . . . 10 ⊢ (𝜑 → ((𝐿‘𝐺) = (Base‘𝑈) ↔ 𝐺 = ((Base‘𝑈) × {𝑄}))) |
30 | 21, 29 | mpbid 232 | . . . . . . . . 9 ⊢ (𝜑 → 𝐺 = ((Base‘𝑈) × {𝑄})) |
31 | lclkrlem2f.d | . . . . . . . . . 10 ⊢ 𝐷 = (LDual‘𝑈) | |
32 | eqid 2735 | . . . . . . . . . 10 ⊢ (0g‘𝐷) = (0g‘𝐷) | |
33 | 17, 24, 25, 31, 32, 22 | ldual0v 39132 | . . . . . . . . 9 ⊢ (𝜑 → (0g‘𝐷) = ((Base‘𝑈) × {𝑄})) |
34 | 30, 33 | eqtr4d 2778 | . . . . . . . 8 ⊢ (𝜑 → 𝐺 = (0g‘𝐷)) |
35 | 34 | oveq2d 7447 | . . . . . . 7 ⊢ (𝜑 → (𝐸 + 𝐺) = (𝐸 + (0g‘𝐷))) |
36 | 31, 22 | lduallmod 39135 | . . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ LMod) |
37 | eqid 2735 | . . . . . . . . 9 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
38 | lclkrlem2f.e | . . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ 𝐹) | |
39 | 26, 31, 37, 22, 38 | ldualelvbase 39109 | . . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ (Base‘𝐷)) |
40 | lclkrlem2f.p | . . . . . . . . 9 ⊢ + = (+g‘𝐷) | |
41 | 37, 40, 32 | lmod0vrid 20908 | . . . . . . . 8 ⊢ ((𝐷 ∈ LMod ∧ 𝐸 ∈ (Base‘𝐷)) → (𝐸 + (0g‘𝐷)) = 𝐸) |
42 | 36, 39, 41 | syl2anc 584 | . . . . . . 7 ⊢ (𝜑 → (𝐸 + (0g‘𝐷)) = 𝐸) |
43 | 35, 42 | eqtrd 2775 | . . . . . 6 ⊢ (𝜑 → (𝐸 + 𝐺) = 𝐸) |
44 | 43 | fveq2d 6911 | . . . . 5 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) = (𝐿‘𝐸)) |
45 | lclkrlem2f.le | . . . . 5 ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋})) | |
46 | 44, 45 | eqtr2d 2776 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘{𝑋}) = (𝐿‘(𝐸 + 𝐺))) |
47 | 46 | fveq2d 6911 | . . 3 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘{𝑋})) = ( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) |
48 | 47 | fveq2d 6911 | . 2 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑋}))) = ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺))))) |
49 | 12, 48, 46 | 3eqtr3d 2783 | 1 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1537 ∈ wcel 2106 ∖ cdif 3960 ⊆ wss 3963 {csn 4631 × cxp 5687 ran crn 5690 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 Scalarcsca 17301 0gc0g 17486 LSSumclsm 19667 LModclmod 20875 LSpanclspn 20987 LSHypclsh 38957 LFnlclfn 39039 LKerclk 39067 LDualcld 39105 HLchlt 39332 LHypclh 39967 DVecHcdvh 41061 DIsoHcdih 41211 ocHcoch 41330 |
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 ax-riotaBAD 38935 |
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-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 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-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-tpos 8250 df-undef 8297 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 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-4 12329 df-5 12330 df-6 12331 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-0g 17488 df-proset 18352 df-poset 18371 df-plt 18388 df-lub 18404 df-glb 18405 df-join 18406 df-meet 18407 df-p0 18483 df-p1 18484 df-lat 18490 df-clat 18557 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-dvr 20418 df-drng 20748 df-lmod 20877 df-lss 20948 df-lsp 20988 df-lvec 21120 df-lfl 39040 df-lkr 39068 df-ldual 39106 df-oposet 39158 df-ol 39160 df-oml 39161 df-covers 39248 df-ats 39249 df-atl 39280 df-cvlat 39304 df-hlat 39333 df-llines 39481 df-lplanes 39482 df-lvols 39483 df-lines 39484 df-psubsp 39486 df-pmap 39487 df-padd 39779 df-lhyp 39971 df-laut 39972 df-ldil 40087 df-ltrn 40088 df-trl 40142 df-tendo 40738 df-edring 40740 df-disoa 41012 df-dvech 41062 df-dib 41122 df-dic 41156 df-dih 41212 df-doch 41331 |
This theorem is referenced by: lclkrlem2k 41500 lclkrlem2l 41501 |
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