dochord - Mathbox for Norm Megill

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Theorem dochord 41863

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 481	. . 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 2740	. . . . . 6 ⊢ ((DVecH‘𝐾)‘𝑊) = ((DVecH‘𝐾)‘𝑊)
6		doch11.i	. . . . . 6 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
7		eqid 2740	. . . . . 6 ⊢ (Base‘((DVecH‘𝐾)‘𝑊)) = (Base‘((DVecH‘𝐾)‘𝑊))
8	4, 5, 6, 7	dihrnss 41771	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ ran 𝐼) → 𝑌 ⊆ (Base‘((DVecH‘𝐾)‘𝑊)))
9	1, 3, 8	syl2anc 590	. . . 4 ⊢ (𝜑 → 𝑌 ⊆ (Base‘((DVecH‘𝐾)‘𝑊)))
10	9	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑋 ⊆ 𝑌) → 𝑌 ⊆ (Base‘((DVecH‘𝐾)‘𝑊)))
11		simpr 485	. . 3 ⊢ ((𝜑 ∧ 𝑋 ⊆ 𝑌) → 𝑋 ⊆ 𝑌)
12		doch11.o	. . . 4 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
13	4, 5, 7, 12	dochss 41858	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ⊆ (Base‘((DVecH‘𝐾)‘𝑊)) ∧ 𝑋 ⊆ 𝑌) → ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋))
14	2, 10, 11, 13	syl3anc 1379	. 2 ⊢ ((𝜑 ∧ 𝑋 ⊆ 𝑌) → ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋))
15	1	adantr 481	. . . 4 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
16		doch11.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ran 𝐼)
17	4, 5, 6, 7	dihrnss 41771	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → 𝑋 ⊆ (Base‘((DVecH‘𝐾)‘𝑊)))
18	1, 16, 17	syl2anc 590	. . . . . . 7 ⊢ (𝜑 → 𝑋 ⊆ (Base‘((DVecH‘𝐾)‘𝑊)))
19	4, 6, 5, 7, 12	dochcl 41846	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ (Base‘((DVecH‘𝐾)‘𝑊))) → ( ⊥ ‘𝑋) ∈ ran 𝐼)
20	1, 18, 19	syl2anc 590	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘𝑋) ∈ ran 𝐼)
21	4, 5, 6, 7	dihrnss 41771	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑋) ∈ ran 𝐼) → ( ⊥ ‘𝑋) ⊆ (Base‘((DVecH‘𝐾)‘𝑊)))
22	1, 20, 21	syl2anc 590	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘𝑋) ⊆ (Base‘((DVecH‘𝐾)‘𝑊)))
23	22	adantr 481	. . . 4 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) → ( ⊥ ‘𝑋) ⊆ (Base‘((DVecH‘𝐾)‘𝑊)))
24		simpr 485	. . . 4 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) → ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋))
25	4, 5, 7, 12	dochss 41858	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑋) ⊆ (Base‘((DVecH‘𝐾)‘𝑊)) ∧ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) → ( ⊥ ‘( ⊥ ‘𝑋)) ⊆ ( ⊥ ‘( ⊥ ‘𝑌)))
26	15, 23, 24, 25	syl3anc 1379	. . 3 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) → ( ⊥ ‘( ⊥ ‘𝑋)) ⊆ ( ⊥ ‘( ⊥ ‘𝑌)))
27	4, 6, 12	dochoc 41860	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
28	1, 16, 27	syl2anc 590	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
29	28	adantr 481	. . 3 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
30	4, 6, 12	dochoc 41860	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ ran 𝐼) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌)
31	1, 3, 30	syl2anc 590	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌)
32	31	adantr 481	. . 3 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌)
33	26, 29, 32	3sstr3d 3976	. 2 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) → 𝑋 ⊆ 𝑌)
34	14, 33	impbida 806	1 ⊢ (𝜑 → (𝑋 ⊆ 𝑌 ↔ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ⊆ wss 3890 ran crn 5626 ‘cfv 6492 Basecbs 17177 HLchlt 39843 LHypclh 40477 DVecHcdvh 41571 DIsoHcdih 41721 ocHcoch 41840

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113 ax-riotaBAD 39446

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-iin 4931 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-om 7814 df-1st 7938 df-2nd 7939 df-tpos 8173 df-undef 8220 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-er 8640 df-map 8772 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-n0 12436 df-z 12523 df-uz 12787 df-fz 13460 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17178 df-ress 17199 df-plusg 17231 df-mulr 17232 df-sca 17234 df-vsca 17235 df-0g 17402 df-proset 18258 df-poset 18277 df-plt 18292 df-lub 18308 df-glb 18309 df-join 18310 df-meet 18311 df-p0 18387 df-p1 18388 df-lat 18396 df-clat 18463 df-mgm 18606 df-sgrp 18685 df-mnd 18701 df-submnd 18750 df-grp 18910 df-minusg 18911 df-sbg 18912 df-subg 19097 df-cntz 19290 df-lsm 19609 df-cmn 19755 df-abl 19756 df-mgp 20120 df-rng 20132 df-ur 20161 df-ring 20214 df-oppr 20315 df-dvdsr 20335 df-unit 20336 df-invr 20366 df-dvr 20379 df-drng 20710 df-lmod 20859 df-lss 20929 df-lsp 20969 df-lvec 21100 df-oposet 39669 df-ol 39671 df-oml 39672 df-covers 39759 df-ats 39760 df-atl 39791 df-cvlat 39815 df-hlat 39844 df-llines 39991 df-lplanes 39992 df-lvols 39993 df-lines 39994 df-psubsp 39996 df-pmap 39997 df-padd 40289 df-lhyp 40481 df-laut 40482 df-ldil 40597 df-ltrn 40598 df-trl 40652 df-tendo 41248 df-edring 41250 df-disoa 41522 df-dvech 41572 df-dib 41632 df-dic 41666 df-dih 41722 df-doch 41841

This theorem is referenced by: dochord2N 41864 dochord3 41865 doch11 41866 dochsordN 41867 dochsatshpb 41945 hdmapoc 42424

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