dihatexv2 - Mathbox for Norm Megill

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Theorem dihatexv2 40721

Description: There is a nonzero vector that maps to every lattice atom. (Contributed by NM, 17-Aug-2014.)

**Hypotheses**
Ref	Expression
dihatexv2.a	⊢ 𝐴 = (Atoms‘𝐾)
dihatexv2.h	⊢ 𝐻 = (LHyp‘𝐾)
dihatexv2.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dihatexv2.v	⊢ 𝑉 = (Base‘𝑈)
dihatexv2.o	⊢ 0 = (0_g‘𝑈)
dihatexv2.n	⊢ 𝑁 = (LSpan‘𝑈)
dihatexv2.i	⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
dihatexv2.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))

**Assertion**
Ref	Expression
dihatexv2	⊢ (𝜑 → (𝑄 ∈ 𝐴 ↔ ∃𝑥 ∈ (𝑉 ∖ { 0 })𝑄 = (^◡𝐼‘(𝑁‘{𝑥}))))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐼 𝑥,𝐾 𝑥,𝑁 𝑥,𝑄 𝑥,𝑉 𝑥,𝑊 𝜑,𝑥 Allowed substitution hints: 𝑈(𝑥) 𝐻(𝑥) 0 (𝑥)

**Proof of Theorem dihatexv2**
Step	Hyp	Ref	Expression
1		eqid 2726	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		dihatexv2.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
3	1, 2	atbase 38670	. . 3 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
4	3	anim2i 616	. 2 ⊢ ((𝜑 ∧ 𝑄 ∈ 𝐴) → (𝜑 ∧ 𝑄 ∈ (Base‘𝐾)))
5		dihatexv2.k	. . . . . . 7 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
6	5	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
7		eldifi 4121	. . . . . . 7 ⊢ (𝑥 ∈ (𝑉 ∖ { 0 }) → 𝑥 ∈ 𝑉)
8		dihatexv2.h	. . . . . . . 8 ⊢ 𝐻 = (LHyp‘𝐾)
9		dihatexv2.u	. . . . . . . 8 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
10		dihatexv2.v	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝑈)
11		dihatexv2.n	. . . . . . . 8 ⊢ 𝑁 = (LSpan‘𝑈)
12		dihatexv2.i	. . . . . . . 8 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
13	8, 9, 10, 11, 12	dihlsprn 40713	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ 𝑉) → (𝑁‘{𝑥}) ∈ ran 𝐼)
14	5, 7, 13	syl2an 595	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → (𝑁‘{𝑥}) ∈ ran 𝐼)
15	1, 8, 12	dihcnvcl 40653	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑥}) ∈ ran 𝐼) → (^◡𝐼‘(𝑁‘{𝑥})) ∈ (Base‘𝐾))
16	6, 14, 15	syl2anc 583	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → (^◡𝐼‘(𝑁‘{𝑥})) ∈ (Base‘𝐾))
17		eleq1a 2822	. . . . 5 ⊢ ((^◡𝐼‘(𝑁‘{𝑥})) ∈ (Base‘𝐾) → (𝑄 = (^◡𝐼‘(𝑁‘{𝑥})) → 𝑄 ∈ (Base‘𝐾)))
18	16, 17	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → (𝑄 = (^◡𝐼‘(𝑁‘{𝑥})) → 𝑄 ∈ (Base‘𝐾)))
19	18	rexlimdva 3149	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ (𝑉 ∖ { 0 })𝑄 = (^◡𝐼‘(𝑁‘{𝑥})) → 𝑄 ∈ (Base‘𝐾)))
20	19	imdistani 568	. 2 ⊢ ((𝜑 ∧ ∃𝑥 ∈ (𝑉 ∖ { 0 })𝑄 = (^◡𝐼‘(𝑁‘{𝑥}))) → (𝜑 ∧ 𝑄 ∈ (Base‘𝐾)))
21		dihatexv2.o	. . . 4 ⊢ 0 = (0_g‘𝑈)
22	5	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑄 ∈ (Base‘𝐾)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
23		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑄 ∈ (Base‘𝐾)) → 𝑄 ∈ (Base‘𝐾))
24	1, 2, 8, 9, 10, 21, 11, 12, 22, 23	dihatexv 40720	. . 3 ⊢ ((𝜑 ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑄 ∈ 𝐴 ↔ ∃𝑥 ∈ (𝑉 ∖ { 0 })(𝐼‘𝑄) = (𝑁‘{𝑥})))
25	22	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑄 ∈ (Base‘𝐾)) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
26	22, 7, 13	syl2an 595	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑄 ∈ (Base‘𝐾)) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → (𝑁‘{𝑥}) ∈ ran 𝐼)
27	8, 12	dihcnvid2 40655	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑥}) ∈ ran 𝐼) → (𝐼‘(^◡𝐼‘(𝑁‘{𝑥}))) = (𝑁‘{𝑥}))
28	25, 26, 27	syl2anc 583	. . . . . 6 ⊢ (((𝜑 ∧ 𝑄 ∈ (Base‘𝐾)) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → (𝐼‘(^◡𝐼‘(𝑁‘{𝑥}))) = (𝑁‘{𝑥}))
29	28	eqeq2d 2737	. . . . 5 ⊢ (((𝜑 ∧ 𝑄 ∈ (Base‘𝐾)) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → ((𝐼‘𝑄) = (𝐼‘(^◡𝐼‘(𝑁‘{𝑥}))) ↔ (𝐼‘𝑄) = (𝑁‘{𝑥})))
30		simplr 766	. . . . . 6 ⊢ (((𝜑 ∧ 𝑄 ∈ (Base‘𝐾)) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → 𝑄 ∈ (Base‘𝐾))
31	25, 26, 15	syl2anc 583	. . . . . 6 ⊢ (((𝜑 ∧ 𝑄 ∈ (Base‘𝐾)) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → (^◡𝐼‘(𝑁‘{𝑥})) ∈ (Base‘𝐾))
32	1, 8, 12	dih11 40647	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ (Base‘𝐾) ∧ (^◡𝐼‘(𝑁‘{𝑥})) ∈ (Base‘𝐾)) → ((𝐼‘𝑄) = (𝐼‘(^◡𝐼‘(𝑁‘{𝑥}))) ↔ 𝑄 = (^◡𝐼‘(𝑁‘{𝑥}))))
33	25, 30, 31, 32	syl3anc 1368	. . . . 5 ⊢ (((𝜑 ∧ 𝑄 ∈ (Base‘𝐾)) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → ((𝐼‘𝑄) = (𝐼‘(^◡𝐼‘(𝑁‘{𝑥}))) ↔ 𝑄 = (^◡𝐼‘(𝑁‘{𝑥}))))
34	29, 33	bitr3d 281	. . . 4 ⊢ (((𝜑 ∧ 𝑄 ∈ (Base‘𝐾)) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → ((𝐼‘𝑄) = (𝑁‘{𝑥}) ↔ 𝑄 = (^◡𝐼‘(𝑁‘{𝑥}))))
35	34	rexbidva 3170	. . 3 ⊢ ((𝜑 ∧ 𝑄 ∈ (Base‘𝐾)) → (∃𝑥 ∈ (𝑉 ∖ { 0 })(𝐼‘𝑄) = (𝑁‘{𝑥}) ↔ ∃𝑥 ∈ (𝑉 ∖ { 0 })𝑄 = (^◡𝐼‘(𝑁‘{𝑥}))))
36	24, 35	bitrd 279	. 2 ⊢ ((𝜑 ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑄 ∈ 𝐴 ↔ ∃𝑥 ∈ (𝑉 ∖ { 0 })𝑄 = (^◡𝐼‘(𝑁‘{𝑥}))))
37	4, 20, 36	pm5.21nd 799	1 ⊢ (𝜑 → (𝑄 ∈ 𝐴 ↔ ∃𝑥 ∈ (𝑉 ∖ { 0 })𝑄 = (^◡𝐼‘(𝑁‘{𝑥}))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∃wrex 3064 ∖ cdif 3940 {csn 4623 ^◡ccnv 5668 ran crn 5670 ‘cfv 6536 Basecbs 17151 0_gc0g 17392 LSpanclspn 20816 Atomscatm 38644 HLchlt 38731 LHypclh 39366 DVecHcdvh 40460 DIsoHcdih 40610

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-riotaBAD 38334

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-tpos 8209 df-undef 8256 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-n0 12474 df-z 12560 df-uz 12824 df-fz 13488 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 19048 df-cntz 19231 df-lsm 19554 df-cmn 19700 df-abl 19701 df-mgp 20038 df-rng 20056 df-ur 20085 df-ring 20138 df-oppr 20234 df-dvdsr 20257 df-unit 20258 df-invr 20288 df-dvr 20301 df-drng 20587 df-lmod 20706 df-lss 20777 df-lsp 20817 df-lvec 20949 df-lsatoms 38357 df-oposet 38557 df-ol 38559 df-oml 38560 df-covers 38647 df-ats 38648 df-atl 38679 df-cvlat 38703 df-hlat 38732 df-llines 38880 df-lplanes 38881 df-lvols 38882 df-lines 38883 df-psubsp 38885 df-pmap 38886 df-padd 39178 df-lhyp 39370 df-laut 39371 df-ldil 39486 df-ltrn 39487 df-trl 39541 df-tendo 40137 df-edring 40139 df-disoa 40411 df-dvech 40461 df-dib 40521 df-dic 40555 df-dih 40611

This theorem is referenced by: djhcvat42 40797

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