dochord - Mathbox for Norm Megill

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Theorem dochord 41364

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 2729	. . . . . 6 ⊢ ((DVecH‘𝐾)‘𝑊) = ((DVecH‘𝐾)‘𝑊)
6		doch11.i	. . . . . 6 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
7		eqid 2729	. . . . . 6 ⊢ (Base‘((DVecH‘𝐾)‘𝑊)) = (Base‘((DVecH‘𝐾)‘𝑊))
8	4, 5, 6, 7	dihrnss 41272	. . . . 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 41359	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ⊆ (Base‘((DVecH‘𝐾)‘𝑊)) ∧ 𝑋 ⊆ 𝑌) → ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋))
14	2, 10, 11, 13	syl3anc 1373	. 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 41272	. . . . . . . 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 41347	. . . . . . 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 41272	. . . . . 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 41359	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑋) ⊆ (Base‘((DVecH‘𝐾)‘𝑊)) ∧ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) → ( ⊥ ‘( ⊥ ‘𝑋)) ⊆ ( ⊥ ‘( ⊥ ‘𝑌)))
26	15, 23, 24, 25	syl3anc 1373	. . 3 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) → ( ⊥ ‘( ⊥ ‘𝑋)) ⊆ ( ⊥ ‘( ⊥ ‘𝑌)))
27	4, 6, 12	dochoc 41361	. . . . 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 41361	. . . . 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 4001	. 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 1540 ∈ wcel 2109 ⊆ wss 3914 ran crn 5639 ‘cfv 6511 Basecbs 17179 HLchlt 39343 LHypclh 39978 DVecHcdvh 41072 DIsoHcdih 41222 ocHcoch 41341

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-riotaBAD 38946

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-tpos 8205 df-undef 8252 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-er 8671 df-map 8801 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-n0 12443 df-z 12530 df-uz 12794 df-fz 13469 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-sca 17236 df-vsca 17237 df-0g 17404 df-proset 18255 df-poset 18274 df-plt 18289 df-lub 18305 df-glb 18306 df-join 18307 df-meet 18308 df-p0 18384 df-p1 18385 df-lat 18391 df-clat 18458 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-grp 18868 df-minusg 18869 df-sbg 18870 df-subg 19055 df-cntz 19249 df-lsm 19566 df-cmn 19712 df-abl 19713 df-mgp 20050 df-rng 20062 df-ur 20091 df-ring 20144 df-oppr 20246 df-dvdsr 20266 df-unit 20267 df-invr 20297 df-dvr 20310 df-drng 20640 df-lmod 20768 df-lss 20838 df-lsp 20878 df-lvec 21010 df-oposet 39169 df-ol 39171 df-oml 39172 df-covers 39259 df-ats 39260 df-atl 39291 df-cvlat 39315 df-hlat 39344 df-llines 39492 df-lplanes 39493 df-lvols 39494 df-lines 39495 df-psubsp 39497 df-pmap 39498 df-padd 39790 df-lhyp 39982 df-laut 39983 df-ldil 40098 df-ltrn 40099 df-trl 40153 df-tendo 40749 df-edring 40751 df-disoa 41023 df-dvech 41073 df-dib 41133 df-dic 41167 df-dih 41223 df-doch 41342

This theorem is referenced by: dochord2N 41365 dochord3 41366 doch11 41367 dochsordN 41368 dochsatshpb 41446 hdmapoc 41925

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