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Mirrors > Home > MPE Home > Th. List > Mathboxes > dihcl | Structured version Visualization version GIF version |
Description: Closure of isomorphism H. (Contributed by NM, 8-Mar-2014.) |
Ref | Expression |
---|---|
dihfn.b | ⊢ 𝐵 = (Base‘𝐾) |
dihfn.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dihfn.i | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
dihcl | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐼‘𝑋) ∈ ran 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dihfn.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
2 | dihfn.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | dihfn.i | . . . . 5 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
4 | eqid 2772 | . . . . 5 ⊢ ((DVecH‘𝐾)‘𝑊) = ((DVecH‘𝐾)‘𝑊) | |
5 | eqid 2772 | . . . . 5 ⊢ (LSubSp‘((DVecH‘𝐾)‘𝑊)) = (LSubSp‘((DVecH‘𝐾)‘𝑊)) | |
6 | 1, 2, 3, 4, 5 | dihf11 37848 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐼:𝐵–1-1→(LSubSp‘((DVecH‘𝐾)‘𝑊))) |
7 | 6 | adantr 473 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) → 𝐼:𝐵–1-1→(LSubSp‘((DVecH‘𝐾)‘𝑊))) |
8 | f1fn 6399 | . . 3 ⊢ (𝐼:𝐵–1-1→(LSubSp‘((DVecH‘𝐾)‘𝑊)) → 𝐼 Fn 𝐵) | |
9 | 7, 8 | syl 17 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) → 𝐼 Fn 𝐵) |
10 | fnfvelrn 6667 | . 2 ⊢ ((𝐼 Fn 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝐼‘𝑋) ∈ ran 𝐼) | |
11 | 9, 10 | sylancom 579 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐼‘𝑋) ∈ ran 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ran crn 5402 Fn wfn 6177 –1-1→wf1 6179 ‘cfv 6182 Basecbs 16333 LSubSpclss 19419 HLchlt 35931 LHypclh 36565 DVecHcdvh 37659 DIsoHcdih 37809 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10385 ax-resscn 10386 ax-1cn 10387 ax-icn 10388 ax-addcl 10389 ax-addrcl 10390 ax-mulcl 10391 ax-mulrcl 10392 ax-mulcom 10393 ax-addass 10394 ax-mulass 10395 ax-distr 10396 ax-i2m1 10397 ax-1ne0 10398 ax-1rid 10399 ax-rnegex 10400 ax-rrecex 10401 ax-cnre 10402 ax-pre-lttri 10403 ax-pre-lttrn 10404 ax-pre-ltadd 10405 ax-pre-mulgt0 10406 ax-riotaBAD 35534 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-iin 4789 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5306 df-eprel 5311 df-po 5320 df-so 5321 df-fr 5360 df-we 5362 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7495 df-2nd 7496 df-tpos 7689 df-undef 7736 df-wrecs 7744 df-recs 7806 df-rdg 7844 df-1o 7899 df-oadd 7903 df-er 8083 df-map 8202 df-en 8301 df-dom 8302 df-sdom 8303 df-fin 8304 df-pnf 10470 df-mnf 10471 df-xr 10472 df-ltxr 10473 df-le 10474 df-sub 10666 df-neg 10667 df-nn 11434 df-2 11497 df-3 11498 df-4 11499 df-5 11500 df-6 11501 df-n0 11702 df-z 11788 df-uz 12053 df-fz 12703 df-struct 16335 df-ndx 16336 df-slot 16337 df-base 16339 df-sets 16340 df-ress 16341 df-plusg 16428 df-mulr 16429 df-sca 16431 df-vsca 16432 df-0g 16565 df-proset 17390 df-poset 17408 df-plt 17420 df-lub 17436 df-glb 17437 df-join 17438 df-meet 17439 df-p0 17501 df-p1 17502 df-lat 17508 df-clat 17570 df-mgm 17704 df-sgrp 17746 df-mnd 17757 df-submnd 17798 df-grp 17888 df-minusg 17889 df-sbg 17890 df-subg 18054 df-cntz 18212 df-lsm 18516 df-cmn 18662 df-abl 18663 df-mgp 18957 df-ur 18969 df-ring 19016 df-oppr 19090 df-dvdsr 19108 df-unit 19109 df-invr 19139 df-dvr 19150 df-drng 19221 df-lmod 19352 df-lss 19420 df-lsp 19460 df-lvec 19591 df-oposet 35757 df-ol 35759 df-oml 35760 df-covers 35847 df-ats 35848 df-atl 35879 df-cvlat 35903 df-hlat 35932 df-llines 36079 df-lplanes 36080 df-lvols 36081 df-lines 36082 df-psubsp 36084 df-pmap 36085 df-padd 36377 df-lhyp 36569 df-laut 36570 df-ldil 36685 df-ltrn 36686 df-trl 36740 df-tendo 37336 df-edring 37338 df-disoa 37610 df-dvech 37660 df-dib 37720 df-dic 37754 df-dih 37810 |
This theorem is referenced by: dih1rn 37868 dihintcl 37925 dihmeetcl 37926 dochcl 37934 dochvalr2 37943 dochoc 37948 djhljjN 37983 dihprrnlem1N 38005 dihprrnlem2 38006 dihjat3 38013 dihjat5N 38018 |
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