dihoml4c - Mathbox for Norm Megill

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Theorem dihoml4c 40550

Description: Version of dihoml4 40551 with closed subspaces. (Contributed by NM, 15-Jan-2015.)

**Hypotheses**
Ref	Expression
dihoml4c.h	⊢ 𝐻 = (LHyp‘𝐾)
dihoml4c.i	⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
dihoml4c.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
dihoml4c.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
dihoml4c.x	⊢ (𝜑 → 𝑋 ∈ ran 𝐼)
dihoml4c.y	⊢ (𝜑 → 𝑌 ∈ ran 𝐼)
dihoml4c.l	⊢ (𝜑 → 𝑋 ⊆ 𝑌)

**Assertion**
Ref	Expression
dihoml4c	⊢ (𝜑 → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌) = 𝑋)

**Proof of Theorem dihoml4c**
Step	Hyp	Ref	Expression
1		eqid 2732	. . . 4 ⊢ (meet‘𝐾) = (meet‘𝐾)
2		dihoml4c.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
3		dihoml4c.i	. . . 4 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
4		dihoml4c.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
5		inss1 4228	. . . . . 6 ⊢ (( ⊥ ‘𝑋) ∩ 𝑌) ⊆ ( ⊥ ‘𝑋)
6		dihoml4c.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ran 𝐼)
7		eqid 2732	. . . . . . . . 9 ⊢ ((DVecH‘𝐾)‘𝑊) = ((DVecH‘𝐾)‘𝑊)
8		eqid 2732	. . . . . . . . 9 ⊢ (Base‘((DVecH‘𝐾)‘𝑊)) = (Base‘((DVecH‘𝐾)‘𝑊))
9	2, 7, 3, 8	dihrnss 40452	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → 𝑋 ⊆ (Base‘((DVecH‘𝐾)‘𝑊)))
10	4, 6, 9	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → 𝑋 ⊆ (Base‘((DVecH‘𝐾)‘𝑊)))
11		dihoml4c.o	. . . . . . . 8 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
12	2, 7, 8, 11	dochssv 40529	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ (Base‘((DVecH‘𝐾)‘𝑊))) → ( ⊥ ‘𝑋) ⊆ (Base‘((DVecH‘𝐾)‘𝑊)))
13	4, 10, 12	syl2anc 584	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘𝑋) ⊆ (Base‘((DVecH‘𝐾)‘𝑊)))
14	5, 13	sstrid 3993	. . . . 5 ⊢ (𝜑 → (( ⊥ ‘𝑋) ∩ 𝑌) ⊆ (Base‘((DVecH‘𝐾)‘𝑊)))
15	2, 3, 7, 8, 11	dochcl 40527	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (( ⊥ ‘𝑋) ∩ 𝑌) ⊆ (Base‘((DVecH‘𝐾)‘𝑊))) → ( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∈ ran 𝐼)
16	4, 14, 15	syl2anc 584	. . . 4 ⊢ (𝜑 → ( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∈ ran 𝐼)
17		dihoml4c.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ ran 𝐼)
18	1, 2, 3, 4, 16, 17	dihmeet2 40520	. . 3 ⊢ (𝜑 → (^◡𝐼‘(( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌)) = ((^◡𝐼‘( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)))(meet‘𝐾)(^◡𝐼‘𝑌)))
19		eqid 2732	. . . . . 6 ⊢ (oc‘𝐾) = (oc‘𝐾)
20	2, 3, 7, 8, 11	dochcl 40527	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ (Base‘((DVecH‘𝐾)‘𝑊))) → ( ⊥ ‘𝑋) ∈ ran 𝐼)
21	4, 10, 20	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → ( ⊥ ‘𝑋) ∈ ran 𝐼)
22	2, 3	dihmeetcl 40519	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (( ⊥ ‘𝑋) ∈ ran 𝐼 ∧ 𝑌 ∈ ran 𝐼)) → (( ⊥ ‘𝑋) ∩ 𝑌) ∈ ran 𝐼)
23	4, 21, 17, 22	syl12anc 835	. . . . . 6 ⊢ (𝜑 → (( ⊥ ‘𝑋) ∩ 𝑌) ∈ ran 𝐼)
24	19, 2, 3, 11, 4, 23	dochvalr3 40537	. . . . 5 ⊢ (𝜑 → ((oc‘𝐾)‘(^◡𝐼‘(( ⊥ ‘𝑋) ∩ 𝑌))) = (^◡𝐼‘( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌))))
25	1, 2, 3, 4, 21, 17	dihmeet2 40520	. . . . . . 7 ⊢ (𝜑 → (^◡𝐼‘(( ⊥ ‘𝑋) ∩ 𝑌)) = ((^◡𝐼‘( ⊥ ‘𝑋))(meet‘𝐾)(^◡𝐼‘𝑌)))
26	19, 2, 3, 11, 4, 6	dochvalr3 40537	. . . . . . . 8 ⊢ (𝜑 → ((oc‘𝐾)‘(^◡𝐼‘𝑋)) = (^◡𝐼‘( ⊥ ‘𝑋)))
27	26	oveq1d 7426	. . . . . . 7 ⊢ (𝜑 → (((oc‘𝐾)‘(^◡𝐼‘𝑋))(meet‘𝐾)(^◡𝐼‘𝑌)) = ((^◡𝐼‘( ⊥ ‘𝑋))(meet‘𝐾)(^◡𝐼‘𝑌)))
28	25, 27	eqtr4d 2775	. . . . . 6 ⊢ (𝜑 → (^◡𝐼‘(( ⊥ ‘𝑋) ∩ 𝑌)) = (((oc‘𝐾)‘(^◡𝐼‘𝑋))(meet‘𝐾)(^◡𝐼‘𝑌)))
29	28	fveq2d 6895	. . . . 5 ⊢ (𝜑 → ((oc‘𝐾)‘(^◡𝐼‘(( ⊥ ‘𝑋) ∩ 𝑌))) = ((oc‘𝐾)‘(((oc‘𝐾)‘(^◡𝐼‘𝑋))(meet‘𝐾)(^◡𝐼‘𝑌))))
30	24, 29	eqtr3d 2774	. . . 4 ⊢ (𝜑 → (^◡𝐼‘( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌))) = ((oc‘𝐾)‘(((oc‘𝐾)‘(^◡𝐼‘𝑋))(meet‘𝐾)(^◡𝐼‘𝑌))))
31	30	oveq1d 7426	. . 3 ⊢ (𝜑 → ((^◡𝐼‘( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)))(meet‘𝐾)(^◡𝐼‘𝑌)) = (((oc‘𝐾)‘(((oc‘𝐾)‘(^◡𝐼‘𝑋))(meet‘𝐾)(^◡𝐼‘𝑌)))(meet‘𝐾)(^◡𝐼‘𝑌)))
32		dihoml4c.l	. . . . 5 ⊢ (𝜑 → 𝑋 ⊆ 𝑌)
33		eqid 2732	. . . . . 6 ⊢ (le‘𝐾) = (le‘𝐾)
34	33, 2, 3, 4, 6, 17	dihcnvord 40448	. . . . 5 ⊢ (𝜑 → ((^◡𝐼‘𝑋)(le‘𝐾)(^◡𝐼‘𝑌) ↔ 𝑋 ⊆ 𝑌))
35	32, 34	mpbird 256	. . . 4 ⊢ (𝜑 → (^◡𝐼‘𝑋)(le‘𝐾)(^◡𝐼‘𝑌))
36	4	simpld 495	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ HL)
37		hloml 38530	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OML)
38	36, 37	syl 17	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ OML)
39		eqid 2732	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
40	39, 2, 3	dihcnvcl 40445	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → (^◡𝐼‘𝑋) ∈ (Base‘𝐾))
41	4, 6, 40	syl2anc 584	. . . . 5 ⊢ (𝜑 → (^◡𝐼‘𝑋) ∈ (Base‘𝐾))
42	39, 2, 3	dihcnvcl 40445	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ ran 𝐼) → (^◡𝐼‘𝑌) ∈ (Base‘𝐾))
43	4, 17, 42	syl2anc 584	. . . . 5 ⊢ (𝜑 → (^◡𝐼‘𝑌) ∈ (Base‘𝐾))
44	39, 33, 1, 19	omllaw4 38419	. . . . 5 ⊢ ((𝐾 ∈ OML ∧ (^◡𝐼‘𝑋) ∈ (Base‘𝐾) ∧ (^◡𝐼‘𝑌) ∈ (Base‘𝐾)) → ((^◡𝐼‘𝑋)(le‘𝐾)(^◡𝐼‘𝑌) → (((oc‘𝐾)‘(((oc‘𝐾)‘(^◡𝐼‘𝑋))(meet‘𝐾)(^◡𝐼‘𝑌)))(meet‘𝐾)(^◡𝐼‘𝑌)) = (^◡𝐼‘𝑋)))
45	38, 41, 43, 44	syl3anc 1371	. . . 4 ⊢ (𝜑 → ((^◡𝐼‘𝑋)(le‘𝐾)(^◡𝐼‘𝑌) → (((oc‘𝐾)‘(((oc‘𝐾)‘(^◡𝐼‘𝑋))(meet‘𝐾)(^◡𝐼‘𝑌)))(meet‘𝐾)(^◡𝐼‘𝑌)) = (^◡𝐼‘𝑋)))
46	35, 45	mpd 15	. . 3 ⊢ (𝜑 → (((oc‘𝐾)‘(((oc‘𝐾)‘(^◡𝐼‘𝑋))(meet‘𝐾)(^◡𝐼‘𝑌)))(meet‘𝐾)(^◡𝐼‘𝑌)) = (^◡𝐼‘𝑋))
47	18, 31, 46	3eqtrd 2776	. 2 ⊢ (𝜑 → (^◡𝐼‘(( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌)) = (^◡𝐼‘𝑋))
48	2, 3	dihmeetcl 40519	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∈ ran 𝐼 ∧ 𝑌 ∈ ran 𝐼)) → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌) ∈ ran 𝐼)
49	4, 16, 17, 48	syl12anc 835	. . 3 ⊢ (𝜑 → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌) ∈ ran 𝐼)
50	2, 3, 4, 49, 6	dihcnv11 40449	. 2 ⊢ (𝜑 → ((^◡𝐼‘(( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌)) = (^◡𝐼‘𝑋) ↔ (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌) = 𝑋))
51	47, 50	mpbid 231	1 ⊢ (𝜑 → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌) = 𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∩ cin 3947 ⊆ wss 3948 class class class wbr 5148 ^◡ccnv 5675 ran crn 5677 ‘cfv 6543 (class class class)co 7411 Basecbs 17148 lecple 17208 occoc 17209 meetcmee 18269 OMLcoml 38348 HLchlt 38523 LHypclh 39158 DVecHcdvh 40252 DIsoHcdih 40402 ocHcoch 40521

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-riotaBAD 38126

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-tpos 8213 df-undef 8260 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13489 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-sca 17217 df-vsca 17218 df-0g 17391 df-proset 18252 df-poset 18270 df-plt 18287 df-lub 18303 df-glb 18304 df-join 18305 df-meet 18306 df-p0 18382 df-p1 18383 df-lat 18389 df-clat 18456 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18706 df-grp 18858 df-minusg 18859 df-sbg 18860 df-subg 19039 df-cntz 19222 df-lsm 19545 df-cmn 19691 df-abl 19692 df-mgp 20029 df-rng 20047 df-ur 20076 df-ring 20129 df-oppr 20225 df-dvdsr 20248 df-unit 20249 df-invr 20279 df-dvr 20292 df-drng 20502 df-lmod 20616 df-lss 20687 df-lsp 20727 df-lvec 20858 df-lsatoms 38149 df-oposet 38349 df-ol 38351 df-oml 38352 df-covers 38439 df-ats 38440 df-atl 38471 df-cvlat 38495 df-hlat 38524 df-llines 38672 df-lplanes 38673 df-lvols 38674 df-lines 38675 df-psubsp 38677 df-pmap 38678 df-padd 38970 df-lhyp 39162 df-laut 39163 df-ldil 39278 df-ltrn 39279 df-trl 39333 df-tendo 39929 df-edring 39931 df-disoa 40203 df-dvech 40253 df-dib 40313 df-dic 40347 df-dih 40403 df-doch 40522

This theorem is referenced by: dihoml4 40551

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