dochord - Mathbox for Norm Megill

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Theorem dochord 41373

Description: Ordering law for orthocomplement. (Contributed by NM, 12-Aug-2014.)

**Hypotheses**
Ref	Expression
doch11.h	⊢ 𝐻 = (LHyp‘𝐾)
doch11.i	⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
doch11.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
doch11.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
doch11.x	⊢ (𝜑 → 𝑋 ∈ ran 𝐼)
doch11.y	⊢ (𝜑 → 𝑌 ∈ ran 𝐼)

**Assertion**
Ref	Expression
dochord	⊢ (𝜑 → (𝑋 ⊆ 𝑌 ↔ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)))

**Proof of Theorem dochord**
Step	Hyp	Ref	Expression
1		doch11.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2	1	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 ⊆ 𝑌) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
3		doch11.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ ran 𝐼)
4		doch11.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
5		eqid 2736	. . . . . 6 ⊢ ((DVecH‘𝐾)‘𝑊) = ((DVecH‘𝐾)‘𝑊)
6		doch11.i	. . . . . 6 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
7		eqid 2736	. . . . . 6 ⊢ (Base‘((DVecH‘𝐾)‘𝑊)) = (Base‘((DVecH‘𝐾)‘𝑊))
8	4, 5, 6, 7	dihrnss 41281	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ ran 𝐼) → 𝑌 ⊆ (Base‘((DVecH‘𝐾)‘𝑊)))
9	1, 3, 8	syl2anc 584	. . . 4 ⊢ (𝜑 → 𝑌 ⊆ (Base‘((DVecH‘𝐾)‘𝑊)))
10	9	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 ⊆ 𝑌) → 𝑌 ⊆ (Base‘((DVecH‘𝐾)‘𝑊)))
11		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝑋 ⊆ 𝑌) → 𝑋 ⊆ 𝑌)
12		doch11.o	. . . 4 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
13	4, 5, 7, 12	dochss 41368	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ⊆ (Base‘((DVecH‘𝐾)‘𝑊)) ∧ 𝑋 ⊆ 𝑌) → ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋))
14	2, 10, 11, 13	syl3anc 1372	. 2 ⊢ ((𝜑 ∧ 𝑋 ⊆ 𝑌) → ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋))
15	1	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
16		doch11.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ran 𝐼)
17	4, 5, 6, 7	dihrnss 41281	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → 𝑋 ⊆ (Base‘((DVecH‘𝐾)‘𝑊)))
18	1, 16, 17	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → 𝑋 ⊆ (Base‘((DVecH‘𝐾)‘𝑊)))
19	4, 6, 5, 7, 12	dochcl 41356	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ (Base‘((DVecH‘𝐾)‘𝑊))) → ( ⊥ ‘𝑋) ∈ ran 𝐼)
20	1, 18, 19	syl2anc 584	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘𝑋) ∈ ran 𝐼)
21	4, 5, 6, 7	dihrnss 41281	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑋) ∈ ran 𝐼) → ( ⊥ ‘𝑋) ⊆ (Base‘((DVecH‘𝐾)‘𝑊)))
22	1, 20, 21	syl2anc 584	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘𝑋) ⊆ (Base‘((DVecH‘𝐾)‘𝑊)))
23	22	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) → ( ⊥ ‘𝑋) ⊆ (Base‘((DVecH‘𝐾)‘𝑊)))
24		simpr 484	. . . 4 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) → ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋))
25	4, 5, 7, 12	dochss 41368	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑋) ⊆ (Base‘((DVecH‘𝐾)‘𝑊)) ∧ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) → ( ⊥ ‘( ⊥ ‘𝑋)) ⊆ ( ⊥ ‘( ⊥ ‘𝑌)))
26	15, 23, 24, 25	syl3anc 1372	. . 3 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) → ( ⊥ ‘( ⊥ ‘𝑋)) ⊆ ( ⊥ ‘( ⊥ ‘𝑌)))
27	4, 6, 12	dochoc 41370	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
28	1, 16, 27	syl2anc 584	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
29	28	adantr 480	. . 3 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
30	4, 6, 12	dochoc 41370	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ ran 𝐼) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌)
31	1, 3, 30	syl2anc 584	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌)
32	31	adantr 480	. . 3 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌)
33	26, 29, 32	3sstr3d 4037	. 2 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) → 𝑋 ⊆ 𝑌)
34	14, 33	impbida 800	1 ⊢ (𝜑 → (𝑋 ⊆ 𝑌 ↔ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ⊆ wss 3950 ran crn 5685 ‘cfv 6560 Basecbs 17248 HLchlt 39352 LHypclh 39987 DVecHcdvh 41081 DIsoHcdih 41231 ocHcoch 41350

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-riotaBAD 38955

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-tpos 8252 df-undef 8299 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-er 8746 df-map 8869 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-n0 12529 df-z 12616 df-uz 12880 df-fz 13549 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-0g 17487 df-proset 18341 df-poset 18360 df-plt 18376 df-lub 18392 df-glb 18393 df-join 18394 df-meet 18395 df-p0 18471 df-p1 18472 df-lat 18478 df-clat 18545 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-submnd 18798 df-grp 18955 df-minusg 18956 df-sbg 18957 df-subg 19142 df-cntz 19336 df-lsm 19655 df-cmn 19801 df-abl 19802 df-mgp 20139 df-rng 20151 df-ur 20180 df-ring 20233 df-oppr 20335 df-dvdsr 20358 df-unit 20359 df-invr 20389 df-dvr 20402 df-drng 20732 df-lmod 20861 df-lss 20931 df-lsp 20971 df-lvec 21103 df-oposet 39178 df-ol 39180 df-oml 39181 df-covers 39268 df-ats 39269 df-atl 39300 df-cvlat 39324 df-hlat 39353 df-llines 39501 df-lplanes 39502 df-lvols 39503 df-lines 39504 df-psubsp 39506 df-pmap 39507 df-padd 39799 df-lhyp 39991 df-laut 39992 df-ldil 40107 df-ltrn 40108 df-trl 40162 df-tendo 40758 df-edring 40760 df-disoa 41032 df-dvech 41082 df-dib 41142 df-dic 41176 df-dih 41232 df-doch 41351

This theorem is referenced by: dochord2N 41374 dochord3 41375 doch11 41376 dochsordN 41377 dochsatshpb 41455 hdmapoc 41934

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