dihatexv2 - Mathbox for Norm Megill

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Theorem dihatexv2 40816

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 2727	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		dihatexv2.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
3	1, 2	atbase 38765	. . 3 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
4	3	anim2i 615	. 2 ⊢ ((𝜑 ∧ 𝑄 ∈ 𝐴) → (𝜑 ∧ 𝑄 ∈ (Base‘𝐾)))
5		dihatexv2.k	. . . . . . 7 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
6	5	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
7		eldifi 4125	. . . . . . 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 40808	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ 𝑉) → (𝑁‘{𝑥}) ∈ ran 𝐼)
14	5, 7, 13	syl2an 594	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → (𝑁‘{𝑥}) ∈ ran 𝐼)
15	1, 8, 12	dihcnvcl 40748	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑥}) ∈ ran 𝐼) → (^◡𝐼‘(𝑁‘{𝑥})) ∈ (Base‘𝐾))
16	6, 14, 15	syl2anc 582	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → (^◡𝐼‘(𝑁‘{𝑥})) ∈ (Base‘𝐾))
17		eleq1a 2823	. . . . 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 3151	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ (𝑉 ∖ { 0 })𝑄 = (^◡𝐼‘(𝑁‘{𝑥})) → 𝑄 ∈ (Base‘𝐾)))
20	19	imdistani 567	. 2 ⊢ ((𝜑 ∧ ∃𝑥 ∈ (𝑉 ∖ { 0 })𝑄 = (^◡𝐼‘(𝑁‘{𝑥}))) → (𝜑 ∧ 𝑄 ∈ (Base‘𝐾)))
21		dihatexv2.o	. . . 4 ⊢ 0 = (0_g‘𝑈)
22	5	adantr 479	. . . 4 ⊢ ((𝜑 ∧ 𝑄 ∈ (Base‘𝐾)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
23		simpr 483	. . . 4 ⊢ ((𝜑 ∧ 𝑄 ∈ (Base‘𝐾)) → 𝑄 ∈ (Base‘𝐾))
24	1, 2, 8, 9, 10, 21, 11, 12, 22, 23	dihatexv 40815	. . 3 ⊢ ((𝜑 ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑄 ∈ 𝐴 ↔ ∃𝑥 ∈ (𝑉 ∖ { 0 })(𝐼‘𝑄) = (𝑁‘{𝑥})))
25	22	adantr 479	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑄 ∈ (Base‘𝐾)) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
26	22, 7, 13	syl2an 594	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑄 ∈ (Base‘𝐾)) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → (𝑁‘{𝑥}) ∈ ran 𝐼)
27	8, 12	dihcnvid2 40750	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑥}) ∈ ran 𝐼) → (𝐼‘(^◡𝐼‘(𝑁‘{𝑥}))) = (𝑁‘{𝑥}))
28	25, 26, 27	syl2anc 582	. . . . . 6 ⊢ (((𝜑 ∧ 𝑄 ∈ (Base‘𝐾)) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → (𝐼‘(^◡𝐼‘(𝑁‘{𝑥}))) = (𝑁‘{𝑥}))
29	28	eqeq2d 2738	. . . . 5 ⊢ (((𝜑 ∧ 𝑄 ∈ (Base‘𝐾)) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → ((𝐼‘𝑄) = (𝐼‘(^◡𝐼‘(𝑁‘{𝑥}))) ↔ (𝐼‘𝑄) = (𝑁‘{𝑥})))
30		simplr 767	. . . . . 6 ⊢ (((𝜑 ∧ 𝑄 ∈ (Base‘𝐾)) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → 𝑄 ∈ (Base‘𝐾))
31	25, 26, 15	syl2anc 582	. . . . . 6 ⊢ (((𝜑 ∧ 𝑄 ∈ (Base‘𝐾)) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → (^◡𝐼‘(𝑁‘{𝑥})) ∈ (Base‘𝐾))
32	1, 8, 12	dih11 40742	. . . . . 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 280	. . . 4 ⊢ (((𝜑 ∧ 𝑄 ∈ (Base‘𝐾)) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → ((𝐼‘𝑄) = (𝑁‘{𝑥}) ↔ 𝑄 = (^◡𝐼‘(𝑁‘{𝑥}))))
35	34	rexbidva 3172	. . 3 ⊢ ((𝜑 ∧ 𝑄 ∈ (Base‘𝐾)) → (∃𝑥 ∈ (𝑉 ∖ { 0 })(𝐼‘𝑄) = (𝑁‘{𝑥}) ↔ ∃𝑥 ∈ (𝑉 ∖ { 0 })𝑄 = (^◡𝐼‘(𝑁‘{𝑥}))))
36	24, 35	bitrd 278	. 2 ⊢ ((𝜑 ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑄 ∈ 𝐴 ↔ ∃𝑥 ∈ (𝑉 ∖ { 0 })𝑄 = (^◡𝐼‘(𝑁‘{𝑥}))))
37	4, 20, 36	pm5.21nd 800	1 ⊢ (𝜑 → (𝑄 ∈ 𝐴 ↔ ∃𝑥 ∈ (𝑉 ∖ { 0 })𝑄 = (^◡𝐼‘(𝑁‘{𝑥}))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∃wrex 3066 ∖ cdif 3944 {csn 4630 ^◡ccnv 5679 ran crn 5681 ‘cfv 6551 Basecbs 17185 0_gc0g 17426 LSpanclspn 20860 Atomscatm 38739 HLchlt 38826 LHypclh 39461 DVecHcdvh 40555 DIsoHcdih 40705

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 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 ax-riotaBAD 38429

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-iin 5001 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-1st 7997 df-2nd 7998 df-tpos 8236 df-undef 8283 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-er 8729 df-map 8851 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-n0 12509 df-z 12595 df-uz 12859 df-fz 13523 df-struct 17121 df-sets 17138 df-slot 17156 df-ndx 17168 df-base 17186 df-ress 17215 df-plusg 17251 df-mulr 17252 df-sca 17254 df-vsca 17255 df-0g 17428 df-proset 18292 df-poset 18310 df-plt 18327 df-lub 18343 df-glb 18344 df-join 18345 df-meet 18346 df-p0 18422 df-p1 18423 df-lat 18429 df-clat 18496 df-mgm 18605 df-sgrp 18684 df-mnd 18700 df-submnd 18746 df-grp 18898 df-minusg 18899 df-sbg 18900 df-subg 19083 df-cntz 19273 df-lsm 19596 df-cmn 19742 df-abl 19743 df-mgp 20080 df-rng 20098 df-ur 20127 df-ring 20180 df-oppr 20278 df-dvdsr 20301 df-unit 20302 df-invr 20332 df-dvr 20345 df-drng 20631 df-lmod 20750 df-lss 20821 df-lsp 20861 df-lvec 20993 df-lsatoms 38452 df-oposet 38652 df-ol 38654 df-oml 38655 df-covers 38742 df-ats 38743 df-atl 38774 df-cvlat 38798 df-hlat 38827 df-llines 38975 df-lplanes 38976 df-lvols 38977 df-lines 38978 df-psubsp 38980 df-pmap 38981 df-padd 39273 df-lhyp 39465 df-laut 39466 df-ldil 39581 df-ltrn 39582 df-trl 39636 df-tendo 40232 df-edring 40234 df-disoa 40506 df-dvech 40556 df-dib 40616 df-dic 40650 df-dih 40706

This theorem is referenced by: djhcvat42 40892

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