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Mirrors > Home > MPE Home > Th. List > Mathboxes > dih0rn | Structured version Visualization version GIF version |
Description: The zero subspace belongs to the range of isomorphism H. (Contributed by NM, 27-Apr-2014.) |
Ref | Expression |
---|---|
dih0rn.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dih0rn.i | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
dih0rn.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dih0rn.o | ⊢ 0 = (0g‘𝑈) |
Ref | Expression |
---|---|
dih0rn | ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → { 0 } ∈ ran 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2731 | . . 3 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
2 | dih0rn.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | dih0rn.i | . . 3 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
4 | dih0rn.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
5 | dih0rn.o | . . 3 ⊢ 0 = (0g‘𝑈) | |
6 | 1, 2, 3, 4, 5 | dih0 40614 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐼‘(0.‘𝐾)) = { 0 }) |
7 | eqid 2731 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
8 | 7, 2, 3 | dihfn 40602 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐼 Fn (Base‘𝐾)) |
9 | hlop 38695 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
10 | 9 | adantr 480 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐾 ∈ OP) |
11 | 7, 1 | op0cl 38517 | . . . 4 ⊢ (𝐾 ∈ OP → (0.‘𝐾) ∈ (Base‘𝐾)) |
12 | 10, 11 | syl 17 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (0.‘𝐾) ∈ (Base‘𝐾)) |
13 | fnfvelrn 7082 | . . 3 ⊢ ((𝐼 Fn (Base‘𝐾) ∧ (0.‘𝐾) ∈ (Base‘𝐾)) → (𝐼‘(0.‘𝐾)) ∈ ran 𝐼) | |
14 | 8, 12, 13 | syl2anc 583 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐼‘(0.‘𝐾)) ∈ ran 𝐼) |
15 | 6, 14 | eqeltrrd 2833 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → { 0 } ∈ ran 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 {csn 4628 ran crn 5677 Fn wfn 6538 ‘cfv 6543 Basecbs 17151 0gc0g 17392 0.cp0 18386 OPcops 38505 HLchlt 38683 LHypclh 39318 DVecHcdvh 40412 DIsoHcdih 40562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-riotaBAD 38286 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-tpos 8217 df-undef 8264 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-n0 12480 df-z 12566 df-uz 12830 df-fz 13492 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-sca 17220 df-vsca 17221 df-0g 17394 df-proset 18258 df-poset 18276 df-plt 18293 df-lub 18309 df-glb 18310 df-join 18311 df-meet 18312 df-p0 18388 df-p1 18389 df-lat 18395 df-clat 18462 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-grp 18864 df-minusg 18865 df-sbg 18866 df-subg 19046 df-cntz 19229 df-lsm 19552 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-oppr 20232 df-dvdsr 20255 df-unit 20256 df-invr 20286 df-dvr 20299 df-drng 20585 df-lmod 20704 df-lss 20775 df-lsp 20815 df-lvec 20946 df-oposet 38509 df-ol 38511 df-oml 38512 df-covers 38599 df-ats 38600 df-atl 38631 df-cvlat 38655 df-hlat 38684 df-llines 38832 df-lplanes 38833 df-lvols 38834 df-lines 38835 df-psubsp 38837 df-pmap 38838 df-padd 39130 df-lhyp 39322 df-laut 39323 df-ldil 39438 df-ltrn 39439 df-trl 39493 df-tendo 40089 df-edring 40091 df-disoa 40363 df-dvech 40413 df-dib 40473 df-dic 40507 df-dih 40563 |
This theorem is referenced by: dih0sb 40619 dihlsprn 40665 doch0 40692 djh01 40746 djh02 40747 dochsatshp 40785 dochshpsat 40788 |
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