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Mirrors > Home > MPE Home > Th. List > Mathboxes > dihprrn | Structured version Visualization version GIF version |
Description: The span of a vector pair belongs to the range of isomorphism H i.e. is a closed subspace. (Contributed by NM, 29-Sep-2014.) |
Ref | Expression |
---|---|
dihprrn.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dihprrn.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dihprrn.v | ⊢ 𝑉 = (Base‘𝑈) |
dihprrn.n | ⊢ 𝑁 = (LSpan‘𝑈) |
dihprrn.i | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
dihprrn.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
dihprrn.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
dihprrn.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
Ref | Expression |
---|---|
dihprrn | ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ ran 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prcom 4499 | . . . . . 6 ⊢ {𝑋, 𝑌} = {𝑌, 𝑋} | |
2 | preq2 4501 | . . . . . 6 ⊢ (𝑋 = (0g‘𝑈) → {𝑌, 𝑋} = {𝑌, (0g‘𝑈)}) | |
3 | 1, 2 | syl5eq 2826 | . . . . 5 ⊢ (𝑋 = (0g‘𝑈) → {𝑋, 𝑌} = {𝑌, (0g‘𝑈)}) |
4 | 3 | fveq2d 6450 | . . . 4 ⊢ (𝑋 = (0g‘𝑈) → (𝑁‘{𝑋, 𝑌}) = (𝑁‘{𝑌, (0g‘𝑈)})) |
5 | dihprrn.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑈) | |
6 | eqid 2778 | . . . . 5 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
7 | dihprrn.n | . . . . 5 ⊢ 𝑁 = (LSpan‘𝑈) | |
8 | dihprrn.h | . . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾) | |
9 | dihprrn.u | . . . . . 6 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
10 | dihprrn.k | . . . . . 6 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
11 | 8, 9, 10 | dvhlmod 37264 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
12 | dihprrn.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
13 | 5, 6, 7, 11, 12 | lsppr0 19487 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑌, (0g‘𝑈)}) = (𝑁‘{𝑌})) |
14 | 4, 13 | sylan9eqr 2836 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = (0g‘𝑈)) → (𝑁‘{𝑋, 𝑌}) = (𝑁‘{𝑌})) |
15 | dihprrn.i | . . . . . 6 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
16 | 8, 9, 5, 7, 15 | dihlsprn 37485 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ ran 𝐼) |
17 | 10, 12, 16 | syl2anc 579 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ ran 𝐼) |
18 | 17 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = (0g‘𝑈)) → (𝑁‘{𝑌}) ∈ ran 𝐼) |
19 | 14, 18 | eqeltrd 2859 | . 2 ⊢ ((𝜑 ∧ 𝑋 = (0g‘𝑈)) → (𝑁‘{𝑋, 𝑌}) ∈ ran 𝐼) |
20 | preq2 4501 | . . . . 5 ⊢ (𝑌 = (0g‘𝑈) → {𝑋, 𝑌} = {𝑋, (0g‘𝑈)}) | |
21 | 20 | fveq2d 6450 | . . . 4 ⊢ (𝑌 = (0g‘𝑈) → (𝑁‘{𝑋, 𝑌}) = (𝑁‘{𝑋, (0g‘𝑈)})) |
22 | dihprrn.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
23 | 5, 6, 7, 11, 22 | lsppr0 19487 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋, (0g‘𝑈)}) = (𝑁‘{𝑋})) |
24 | 21, 23 | sylan9eqr 2836 | . . 3 ⊢ ((𝜑 ∧ 𝑌 = (0g‘𝑈)) → (𝑁‘{𝑋, 𝑌}) = (𝑁‘{𝑋})) |
25 | 8, 9, 5, 7, 15 | dihlsprn 37485 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ ran 𝐼) |
26 | 10, 22, 25 | syl2anc 579 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ ran 𝐼) |
27 | 26 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ 𝑌 = (0g‘𝑈)) → (𝑁‘{𝑋}) ∈ ran 𝐼) |
28 | 24, 27 | eqeltrd 2859 | . 2 ⊢ ((𝜑 ∧ 𝑌 = (0g‘𝑈)) → (𝑁‘{𝑋, 𝑌}) ∈ ran 𝐼) |
29 | 10 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ 𝑌 ≠ (0g‘𝑈))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
30 | 22 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ 𝑌 ≠ (0g‘𝑈))) → 𝑋 ∈ 𝑉) |
31 | 12 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ 𝑌 ≠ (0g‘𝑈))) → 𝑌 ∈ 𝑉) |
32 | simprl 761 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ 𝑌 ≠ (0g‘𝑈))) → 𝑋 ≠ (0g‘𝑈)) | |
33 | simprr 763 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ 𝑌 ≠ (0g‘𝑈))) → 𝑌 ≠ (0g‘𝑈)) | |
34 | 8, 9, 5, 7, 15, 29, 30, 31, 6, 32, 33 | dihprrnlem2 37579 | . 2 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ 𝑌 ≠ (0g‘𝑈))) → (𝑁‘{𝑋, 𝑌}) ∈ ran 𝐼) |
35 | 19, 28, 34 | pm2.61da2ne 3058 | 1 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ ran 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 {csn 4398 {cpr 4400 ran crn 5356 ‘cfv 6135 Basecbs 16255 0gc0g 16486 LSpanclspn 19366 HLchlt 35504 LHypclh 36138 DVecHcdvh 37232 DIsoHcdih 37382 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-riotaBAD 35107 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-tpos 7634 df-undef 7681 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-n0 11643 df-z 11729 df-uz 11993 df-fz 12644 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-sca 16354 df-vsca 16355 df-0g 16488 df-proset 17314 df-poset 17332 df-plt 17344 df-lub 17360 df-glb 17361 df-join 17362 df-meet 17363 df-p0 17425 df-p1 17426 df-lat 17432 df-clat 17494 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-submnd 17722 df-grp 17812 df-minusg 17813 df-sbg 17814 df-subg 17975 df-cntz 18133 df-lsm 18435 df-cmn 18581 df-abl 18582 df-mgp 18877 df-ur 18889 df-ring 18936 df-oppr 19010 df-dvdsr 19028 df-unit 19029 df-invr 19059 df-dvr 19070 df-drng 19141 df-lmod 19257 df-lss 19325 df-lsp 19367 df-lvec 19498 df-lsatoms 35130 df-oposet 35330 df-ol 35332 df-oml 35333 df-covers 35420 df-ats 35421 df-atl 35452 df-cvlat 35476 df-hlat 35505 df-llines 35652 df-lplanes 35653 df-lvols 35654 df-lines 35655 df-psubsp 35657 df-pmap 35658 df-padd 35950 df-lhyp 36142 df-laut 36143 df-ldil 36258 df-ltrn 36259 df-trl 36313 df-tgrp 36897 df-tendo 36909 df-edring 36911 df-dveca 37157 df-disoa 37183 df-dvech 37233 df-dib 37293 df-dic 37327 df-dih 37383 df-doch 37502 df-djh 37549 |
This theorem is referenced by: djhlsmat 37581 lclkrlem2v 37682 lcfrlem23 37719 |
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