dib0 - Mathbox for Norm Megill

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Theorem dib0 41420

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.)

**Hypotheses**
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 = (0_g‘𝑈)

**Assertion**
Ref	Expression
dib0	⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐼‘ 0 ) = {𝑂})

Proof of Theorem dib0 Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6847	. . . 4 ⊢ (Base‘𝐾) ∈ V
2		resiexg 7854	. . . 4 ⊢ ((Base‘𝐾) ∈ V → ( I ↾ (Base‘𝐾)) ∈ V)
3	1, 2	ax-mp 5	. . 3 ⊢ ( I ↾ (Base‘𝐾)) ∈ V
4		fvex 6847	. . . 4 ⊢ ((LTrn‘𝐾)‘𝑊) ∈ V
5	4	mptex 7169	. . 3 ⊢ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) ∈ V
6	3, 5	xpsn 7086	. 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 39618	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
9	8	adantr 480	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐾 ∈ OP)
10		eqid 2736	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
11		dib0.z	. . . . . 6 ⊢ 0 = (0.‘𝐾)
12	10, 11	op0cl 39440	. . . . 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 40254	. . . . 5 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
16		eqid 2736	. . . . . 6 ⊢ (le‘𝐾) = (le‘𝐾)
17	10, 16, 11	op0le 39442	. . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑊 ∈ (Base‘𝐾)) → 0 (le‘𝐾)𝑊)
18	8, 15, 17	syl2an 596	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 (le‘𝐾)𝑊)
19		eqid 2736	. . . . 5 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
20		eqid 2736	. . . . 5 ⊢ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))
21		eqid 2736	. . . . 5 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
22		dib0.i	. . . . 5 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)
23	10, 16, 14, 19, 20, 21, 22	dibval2 41400	. . . 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 836	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐼‘ 0 ) = ((((DIsoA‘𝐾)‘𝑊)‘ 0 ) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))}))
25	10, 11, 14, 21	dia0 41308	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((DIsoA‘𝐾)‘𝑊)‘ 0 ) = {( I ↾ (Base‘𝐾))})
26	25	xpeq1d 5653	. . 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 2771	. 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 = (0_g‘𝑈)
30	10, 14, 19, 28, 29, 20	dvh0g 41367	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑂 = ⟨( I ↾ (Base‘𝐾)), (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩)
31	30	sneqd 4592	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → {𝑂} = {⟨( I ↾ (Base‘𝐾)), (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩})
32	6, 27, 31	3eqtr4a 2797	1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐼‘ 0 ) = {𝑂})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 Vcvv 3440 {csn 4580 ⟨cop 4586 class class class wbr 5098 ↦ cmpt 5179 I cid 5518 × cxp 5622 ↾ cres 5626 ‘cfv 6492 Basecbs 17136 lecple 17184 0_gc0g 17359 0.cp0 18344 OPcops 39428 HLchlt 39606 LHypclh 40240 LTrncltrn 40357 DIsoAcdia 41284 DVecHcdvh 41334 DIsoBcdib 41394

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 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-riotaBAD 39209

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-tpos 8168 df-undef 8215 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-er 8635 df-map 8765 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-n0 12402 df-z 12489 df-uz 12752 df-fz 13424 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-sca 17193 df-vsca 17194 df-0g 17361 df-proset 18217 df-poset 18236 df-plt 18251 df-lub 18267 df-glb 18268 df-join 18269 df-meet 18270 df-p0 18346 df-p1 18347 df-lat 18355 df-clat 18422 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-grp 18866 df-minusg 18867 df-cmn 19711 df-abl 19712 df-mgp 20076 df-rng 20088 df-ur 20117 df-ring 20170 df-oppr 20273 df-dvdsr 20293 df-unit 20294 df-invr 20324 df-dvr 20337 df-drng 20664 df-lmod 20813 df-lvec 21055 df-oposet 39432 df-ol 39434 df-oml 39435 df-covers 39522 df-ats 39523 df-atl 39554 df-cvlat 39578 df-hlat 39607 df-llines 39754 df-lplanes 39755 df-lvols 39756 df-lines 39757 df-psubsp 39759 df-pmap 39760 df-padd 40052 df-lhyp 40244 df-laut 40245 df-ldil 40360 df-ltrn 40361 df-trl 40415 df-tendo 41011 df-edring 41013 df-disoa 41285 df-dvech 41335 df-dib 41395

This theorem is referenced by: dihvalcqat 41495 dih0 41536

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