dib0 - Mathbox for Norm Megill

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Theorem dib0 41752

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 6876	. . . 4 ⊢ (Base‘𝐾) ∈ V
2		resiexg 7889	. . . 4 ⊢ ((Base‘𝐾) ∈ V → ( I ↾ (Base‘𝐾)) ∈ V)
3	1, 2	ax-mp 5	. . 3 ⊢ ( I ↾ (Base‘𝐾)) ∈ V
4		fvex 6876	. . . 4 ⊢ ((LTrn‘𝐾)‘𝑊) ∈ V
5	4	mptex 7203	. . 3 ⊢ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) ∈ V
6	3, 5	xpsn 7119	. 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 39950	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
9	8	adantr 484	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐾 ∈ OP)
10		eqid 2761	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
11		dib0.z	. . . . . 6 ⊢ 0 = (0.‘𝐾)
12	10, 11	op0cl 39772	. . . . 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 40586	. . . . 5 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
16		eqid 2761	. . . . . 6 ⊢ (le‘𝐾) = (le‘𝐾)
17	10, 16, 11	op0le 39774	. . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑊 ∈ (Base‘𝐾)) → 0 (le‘𝐾)𝑊)
18	8, 15, 17	syl2an 605	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 (le‘𝐾)𝑊)
19		eqid 2761	. . . . 5 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
20		eqid 2761	. . . . 5 ⊢ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))
21		eqid 2761	. . . . 5 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
22		dib0.i	. . . . 5 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)
23	10, 16, 14, 19, 20, 21, 22	dibval2 41732	. . . 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 847	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐼‘ 0 ) = ((((DIsoA‘𝐾)‘𝑊)‘ 0 ) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))}))
25	10, 11, 14, 21	dia0 41640	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((DIsoA‘𝐾)‘𝑊)‘ 0 ) = {( I ↾ (Base‘𝐾))})
26	25	xpeq1d 5674	. . 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 2796	. 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 41699	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑂 = ⟨( I ↾ (Base‘𝐾)), (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩)
31	30	sneqd 4593	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → {𝑂} = {⟨( I ↾ (Base‘𝐾)), (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩})
32	6, 27, 31	3eqtr4a 2822	1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐼‘ 0 ) = {𝑂})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1559 ∈ wcel 2141 Vcvv 3453 {csn 4581 ⟨cop 4587 class class class wbr 5099 ↦ cmpt 5180 I cid 5539 × cxp 5643 ↾ cres 5647 ‘cfv 6517 Basecbs 17228 lecple 17276 0_gc0g 17451 0.cp0 18436 OPcops 39760 HLchlt 39938 LHypclh 40572 LTrncltrn 40689 DIsoAcdia 41616 DVecHcdvh 41666 DIsoBcdib 41726

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-cnex 11126 ax-resscn 11127 ax-1cn 11128 ax-icn 11129 ax-addcl 11130 ax-addrcl 11131 ax-mulcl 11132 ax-mulrcl 11133 ax-mulcom 11134 ax-addass 11135 ax-mulass 11136 ax-distr 11137 ax-i2m1 11138 ax-1ne0 11139 ax-1rid 11140 ax-rnegex 11141 ax-rrecex 11142 ax-cnre 11143 ax-pre-lttri 11144 ax-pre-lttrn 11145 ax-pre-ltadd 11146 ax-pre-mulgt0 11147 ax-riotaBAD 39541

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-iun 4950 df-iin 4951 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6284 df-ord 6345 df-on 6346 df-lim 6347 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-om 7843 df-1st 7966 df-2nd 7967 df-tpos 8201 df-undef 8248 df-frecs 8257 df-wrecs 8288 df-recs 8337 df-rdg 8376 df-1o 8432 df-er 8673 df-map 8805 df-en 8924 df-dom 8925 df-sdom 8926 df-fin 8927 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11413 df-neg 11414 df-nn 12208 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-n0 12479 df-z 12566 df-uz 12837 df-fz 13510 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17250 df-plusg 17282 df-mulr 17283 df-sca 17285 df-vsca 17286 df-0g 17453 df-proset 18309 df-poset 18328 df-plt 18343 df-lub 18359 df-glb 18360 df-join 18361 df-meet 18362 df-p0 18438 df-p1 18439 df-lat 18447 df-clat 18514 df-mgm 18657 df-sgrp 18736 df-mnd 18752 df-grp 18961 df-minusg 18962 df-cmn 19805 df-abl 19806 df-mgp 20170 df-rng 20182 df-ur 20211 df-ring 20264 df-oppr 20365 df-dvdsr 20385 df-unit 20386 df-invr 20416 df-dvr 20429 df-drng 20760 df-lmod 20909 df-lvec 21150 df-oposet 39764 df-ol 39766 df-oml 39767 df-covers 39854 df-ats 39855 df-atl 39886 df-cvlat 39910 df-hlat 39939 df-llines 40086 df-lplanes 40087 df-lvols 40088 df-lines 40089 df-psubsp 40091 df-pmap 40092 df-padd 40384 df-lhyp 40576 df-laut 40577 df-ldil 40692 df-ltrn 40693 df-trl 40747 df-tendo 41343 df-edring 41345 df-disoa 41617 df-dvech 41667 df-dib 41727

This theorem is referenced by: dihvalcqat 41827 dih0 41868

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