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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dihssxp | Structured version Visualization version GIF version |
Description: An isomorphism H value is included in the vector space (expressed as 𝑇 × 𝐸). (Contributed by NM, 26-Sep-2014.) |
Ref | Expression |
---|---|
dihssxp.b | ⊢ 𝐵 = (Base‘𝐾) |
dihssxp.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dihssxp.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
dihssxp.e | ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) |
dihssxp.i | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
dihssxp.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
dihssxp.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
Ref | Expression |
---|---|
dihssxp | ⊢ (𝜑 → (𝐼‘𝑋) ⊆ (𝑇 × 𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dihssxp.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | dihssxp.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
3 | dihssxp.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
4 | dihssxp.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | dihssxp.i | . . . 4 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
6 | eqid 2778 | . . . 4 ⊢ ((DVecH‘𝐾)‘𝑊) = ((DVecH‘𝐾)‘𝑊) | |
7 | eqid 2778 | . . . 4 ⊢ (Base‘((DVecH‘𝐾)‘𝑊)) = (Base‘((DVecH‘𝐾)‘𝑊)) | |
8 | 3, 4, 5, 6, 7 | dihss 37414 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐼‘𝑋) ⊆ (Base‘((DVecH‘𝐾)‘𝑊))) |
9 | 1, 2, 8 | syl2anc 579 | . 2 ⊢ (𝜑 → (𝐼‘𝑋) ⊆ (Base‘((DVecH‘𝐾)‘𝑊))) |
10 | dihssxp.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
11 | dihssxp.e | . . . 4 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
12 | 4, 10, 11, 6, 7 | dvhvbase 37250 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘((DVecH‘𝐾)‘𝑊)) = (𝑇 × 𝐸)) |
13 | 1, 12 | syl 17 | . 2 ⊢ (𝜑 → (Base‘((DVecH‘𝐾)‘𝑊)) = (𝑇 × 𝐸)) |
14 | 9, 13 | sseqtrd 3860 | 1 ⊢ (𝜑 → (𝐼‘𝑋) ⊆ (𝑇 × 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ⊆ wss 3792 × cxp 5355 ‘cfv 6137 Basecbs 16266 HLchlt 35513 LHypclh 36147 LTrncltrn 36264 TEndoctendo 36915 DVecHcdvh 37241 DIsoHcdih 37391 |
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 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-riotaBAD 35116 |
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 4674 df-int 4713 df-iun 4757 df-iin 4758 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-tpos 7636 df-undef 7683 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-map 8144 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11380 df-2 11443 df-3 11444 df-4 11445 df-5 11446 df-6 11447 df-n0 11648 df-z 11734 df-uz 11998 df-fz 12649 df-struct 16268 df-ndx 16269 df-slot 16270 df-base 16272 df-sets 16273 df-ress 16274 df-plusg 16362 df-mulr 16363 df-sca 16365 df-vsca 16366 df-0g 16499 df-proset 17325 df-poset 17343 df-plt 17355 df-lub 17371 df-glb 17372 df-join 17373 df-meet 17374 df-p0 17436 df-p1 17437 df-lat 17443 df-clat 17505 df-mgm 17639 df-sgrp 17681 df-mnd 17692 df-submnd 17733 df-grp 17823 df-minusg 17824 df-sbg 17825 df-subg 17986 df-cntz 18144 df-lsm 18446 df-cmn 18592 df-abl 18593 df-mgp 18888 df-ur 18900 df-ring 18947 df-oppr 19021 df-dvdsr 19039 df-unit 19040 df-invr 19070 df-dvr 19081 df-drng 19152 df-lmod 19268 df-lss 19336 df-lsp 19378 df-lvec 19509 df-oposet 35339 df-ol 35341 df-oml 35342 df-covers 35429 df-ats 35430 df-atl 35461 df-cvlat 35485 df-hlat 35514 df-llines 35661 df-lplanes 35662 df-lvols 35663 df-lines 35664 df-psubsp 35666 df-pmap 35667 df-padd 35959 df-lhyp 36151 df-laut 36152 df-ldil 36267 df-ltrn 36268 df-trl 36322 df-tendo 36918 df-edring 36920 df-disoa 37192 df-dvech 37242 df-dib 37302 df-dic 37336 df-dih 37392 |
This theorem is referenced by: dihopcl 37416 |
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