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 2730	. . . 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 4227	. . . . . 6 ⊢ (( ⊥ ‘𝑋) ∩ 𝑌) ⊆ ( ⊥ ‘𝑋)
6		dihoml4c.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ran 𝐼)
7		eqid 2730	. . . . . . . . 9 ⊢ ((DVecH‘𝐾)‘𝑊) = ((DVecH‘𝐾)‘𝑊)
8		eqid 2730	. . . . . . . . 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 582	. . . . . . 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 582	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘𝑋) ⊆ (Base‘((DVecH‘𝐾)‘𝑊)))
14	5, 13	sstrid 3992	. . . . 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 582	. . . 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 2730	. . . . . 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 582	. . . . . . 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 833	. . . . . 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 2773	. . . . . 6 ⊢ (𝜑 → (^◡𝐼‘(( ⊥ ‘𝑋) ∩ 𝑌)) = (((oc‘𝐾)‘(^◡𝐼‘𝑋))(meet‘𝐾)(^◡𝐼‘𝑌)))
29	28	fveq2d 6894	. . . . 5 ⊢ (𝜑 → ((oc‘𝐾)‘(^◡𝐼‘(( ⊥ ‘𝑋) ∩ 𝑌))) = ((oc‘𝐾)‘(((oc‘𝐾)‘(^◡𝐼‘𝑋))(meet‘𝐾)(^◡𝐼‘𝑌))))
30	24, 29	eqtr3d 2772	. . . 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 2730	. . . . . 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 493	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ HL)
37		hloml 38530	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OML)
38	36, 37	syl 17	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ OML)
39		eqid 2730	. . . . . . 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 582	. . . . 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 582	. . . . 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 1369	. . . 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 2774	. 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 833	. . 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 394 = wceq 1539 ∈ wcel 2104 ∩ cin 3946 ⊆ wss 3947 class class class wbr 5147 ^◡ccnv 5674 ran crn 5676 ‘cfv 6542 (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 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 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 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 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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