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 6201 | . . . 4 ⊢ (Base‘𝐾) ∈ V | |
2 | resiexg 7102 | . . . 4 ⊢ ((Base‘𝐾) ∈ V → ( I ↾ (Base‘𝐾)) ∈ V) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ ( I ↾ (Base‘𝐾)) ∈ V |
4 | fvex 6201 | . . . 4 ⊢ ((LTrn‘𝐾)‘𝑊) ∈ V | |
5 | 4 | mptex 6486 | . . 3 ⊢ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) ∈ V |
6 | 3, 5 | xpsn 6407 | . 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 34649 | . . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
9 | 8 | adantr 481 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐾 ∈ OP) |
10 | eqid 2622 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
11 | dib0.z | . . . . . 6 ⊢ 0 = (0.‘𝐾) | |
12 | 10, 11 | op0cl 34471 | . . . . 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 35284 | . . . . 5 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾)) |
16 | eqid 2622 | . . . . . 6 ⊢ (le‘𝐾) = (le‘𝐾) | |
17 | 10, 16, 11 | op0le 34473 | . . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑊 ∈ (Base‘𝐾)) → 0 (le‘𝐾)𝑊) |
18 | 8, 15, 17 | syl2an 494 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 (le‘𝐾)𝑊) |
19 | eqid 2622 | . . . . 5 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | |
20 | eqid 2622 | . . . . 5 ⊢ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) | |
21 | eqid 2622 | . . . . 5 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊) | |
22 | dib0.i | . . . . 5 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) | |
23 | 10, 16, 14, 19, 20, 21, 22 | dibval2 36433 | . . . 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 1324 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐼‘ 0 ) = ((((DIsoA‘𝐾)‘𝑊)‘ 0 ) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))})) |
25 | 10, 11, 14, 21 | dia0 36341 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((DIsoA‘𝐾)‘𝑊)‘ 0 ) = {( I ↾ (Base‘𝐾))}) |
26 | 25 | xpeq1d 5138 | . . 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 2656 | . 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 36400 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑂 = 〈( I ↾ (Base‘𝐾)), (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))〉) |
31 | 30 | sneqd 4189 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → {𝑂} = {〈( I ↾ (Base‘𝐾)), (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))〉}) |
32 | 6, 27, 31 | 3eqtr4a 2682 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐼‘ 0 ) = {𝑂}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 {csn 4177 〈cop 4183 class class class wbr 4653 ↦ cmpt 4729 I cid 5023 × cxp 5112 ↾ cres 5116 ‘cfv 5888 Basecbs 15857 lecple 15948 0gc0g 16100 0.cp0 17037 OPcops 34459 HLchlt 34637 LHypclh 35270 LTrncltrn 35387 DIsoAcdia 36317 DVecHcdvh 36367 DIsoBcdib 36427 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-riotaBAD 34239 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-undef 7399 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-0g 16102 df-preset 16928 df-poset 16946 df-plt 16958 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-p0 17039 df-p1 17040 df-lat 17046 df-clat 17108 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-mgp 18490 df-ur 18502 df-ring 18549 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-drng 18749 df-lmod 18865 df-lvec 19103 df-oposet 34463 df-ol 34465 df-oml 34466 df-covers 34553 df-ats 34554 df-atl 34585 df-cvlat 34609 df-hlat 34638 df-llines 34784 df-lplanes 34785 df-lvols 34786 df-lines 34787 df-psubsp 34789 df-pmap 34790 df-padd 35082 df-lhyp 35274 df-laut 35275 df-ldil 35390 df-ltrn 35391 df-trl 35446 df-tendo 36043 df-edring 36045 df-disoa 36318 df-dvech 36368 df-dib 36428 |
This theorem is referenced by: dihvalcqat 36528 dih0 36569 |
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