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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dib0 | Structured version Visualization version GIF version |
Description: The value of partial isomorphism B at the lattice zero is the singleton of the zero vector i.e. the zero subspace. (Contributed by NM, 27-Mar-2014.) |
Ref | Expression |
---|---|
dib0.z | ⊢ 0 = (0.‘𝐾) |
dib0.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dib0.i | ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) |
dib0.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dib0.o | ⊢ 𝑂 = (0g‘𝑈) |
Ref | Expression |
---|---|
dib0 | ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐼‘ 0 ) = {𝑂}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6658 | . . . 4 ⊢ (Base‘𝐾) ∈ V | |
2 | resiexg 7601 | . . . 4 ⊢ ((Base‘𝐾) ∈ V → ( I ↾ (Base‘𝐾)) ∈ V) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ ( I ↾ (Base‘𝐾)) ∈ V |
4 | fvex 6658 | . . . 4 ⊢ ((LTrn‘𝐾)‘𝑊) ∈ V | |
5 | 4 | mptex 6963 | . . 3 ⊢ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) ∈ V |
6 | 3, 5 | xpsn 6880 | . 2 ⊢ ({( I ↾ (Base‘𝐾))} × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))}) = {〈( I ↾ (Base‘𝐾)), (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))〉} |
7 | id 22 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
8 | hlop 36658 | . . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
9 | 8 | adantr 484 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐾 ∈ OP) |
10 | eqid 2798 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
11 | dib0.z | . . . . . 6 ⊢ 0 = (0.‘𝐾) | |
12 | 10, 11 | op0cl 36480 | . . . . 5 ⊢ (𝐾 ∈ OP → 0 ∈ (Base‘𝐾)) |
13 | 9, 12 | syl 17 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 ∈ (Base‘𝐾)) |
14 | dib0.h | . . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾) | |
15 | 10, 14 | lhpbase 37294 | . . . . 5 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾)) |
16 | eqid 2798 | . . . . . 6 ⊢ (le‘𝐾) = (le‘𝐾) | |
17 | 10, 16, 11 | op0le 36482 | . . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑊 ∈ (Base‘𝐾)) → 0 (le‘𝐾)𝑊) |
18 | 8, 15, 17 | syl2an 598 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 (le‘𝐾)𝑊) |
19 | eqid 2798 | . . . . 5 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | |
20 | eqid 2798 | . . . . 5 ⊢ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) | |
21 | eqid 2798 | . . . . 5 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊) | |
22 | dib0.i | . . . . 5 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) | |
23 | 10, 16, 14, 19, 20, 21, 22 | dibval2 38440 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( 0 ∈ (Base‘𝐾) ∧ 0 (le‘𝐾)𝑊)) → (𝐼‘ 0 ) = ((((DIsoA‘𝐾)‘𝑊)‘ 0 ) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))})) |
24 | 7, 13, 18, 23 | syl12anc 835 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐼‘ 0 ) = ((((DIsoA‘𝐾)‘𝑊)‘ 0 ) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))})) |
25 | 10, 11, 14, 21 | dia0 38348 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((DIsoA‘𝐾)‘𝑊)‘ 0 ) = {( I ↾ (Base‘𝐾))}) |
26 | 25 | xpeq1d 5548 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((((DIsoA‘𝐾)‘𝑊)‘ 0 ) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))}) = ({( I ↾ (Base‘𝐾))} × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))})) |
27 | 24, 26 | eqtrd 2833 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐼‘ 0 ) = ({( I ↾ (Base‘𝐾))} × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))})) |
28 | dib0.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
29 | dib0.o | . . . 4 ⊢ 𝑂 = (0g‘𝑈) | |
30 | 10, 14, 19, 28, 29, 20 | dvh0g 38407 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑂 = 〈( I ↾ (Base‘𝐾)), (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))〉) |
31 | 30 | sneqd 4537 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → {𝑂} = {〈( I ↾ (Base‘𝐾)), (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))〉}) |
32 | 6, 27, 31 | 3eqtr4a 2859 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐼‘ 0 ) = {𝑂}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 {csn 4525 〈cop 4531 class class class wbr 5030 ↦ cmpt 5110 I cid 5424 × cxp 5517 ↾ cres 5521 ‘cfv 6324 Basecbs 16475 lecple 16564 0gc0g 16705 0.cp0 17639 OPcops 36468 HLchlt 36646 LHypclh 37280 LTrncltrn 37397 DIsoAcdia 38324 DVecHcdvh 38374 DIsoBcdib 38434 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-riotaBAD 36249 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-tpos 7875 df-undef 7922 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-0g 16707 df-proset 17530 df-poset 17548 df-plt 17560 df-lub 17576 df-glb 17577 df-join 17578 df-meet 17579 df-p0 17641 df-p1 17642 df-lat 17648 df-clat 17710 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-mgp 19233 df-ur 19245 df-ring 19292 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-dvr 19429 df-drng 19497 df-lmod 19629 df-lvec 19868 df-oposet 36472 df-ol 36474 df-oml 36475 df-covers 36562 df-ats 36563 df-atl 36594 df-cvlat 36618 df-hlat 36647 df-llines 36794 df-lplanes 36795 df-lvols 36796 df-lines 36797 df-psubsp 36799 df-pmap 36800 df-padd 37092 df-lhyp 37284 df-laut 37285 df-ldil 37400 df-ltrn 37401 df-trl 37455 df-tendo 38051 df-edring 38053 df-disoa 38325 df-dvech 38375 df-dib 38435 |
This theorem is referenced by: dihvalcqat 38535 dih0 38576 |
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