dochval2 - Mathbox for Norm Megill

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Theorem dochval2 42377

Description: Subspace orthocomplement for DVecH vector space. (Contributed by NM, 14-Apr-2014.)

**Hypotheses**
Ref	Expression
dochval2.o	⊢ ⊥ = (oc‘𝐾)
dochval2.h	⊢ 𝐻 = (LHyp‘𝐾)
dochval2.i	⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
dochval2.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dochval2.v	⊢ 𝑉 = (Base‘𝑈)
dochval2.n	⊢ 𝑁 = ((ocH‘𝐾)‘𝑊)

**Assertion**
Ref	Expression
dochval2	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝑁‘𝑋) = (𝐼‘( ⊥ ‘(^◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}))))

Distinct variable groups: 𝑧,𝐻 𝑧,𝐼 𝑧,𝐾 𝑧,𝑉 𝑧,𝑊 𝑧,𝑋 Allowed substitution hints: 𝑈(𝑧) 𝑁(𝑧) ⊥ (𝑧)

Proof of Theorem dochval2 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2761	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2761	. . 3 ⊢ (glb‘𝐾) = (glb‘𝐾)
3		dochval2.o	. . 3 ⊢ ⊥ = (oc‘𝐾)
4		dochval2.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
5		dochval2.i	. . 3 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
6		dochval2.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
7		dochval2.v	. . 3 ⊢ 𝑉 = (Base‘𝑈)
8		dochval2.n	. . 3 ⊢ 𝑁 = ((ocH‘𝐾)‘𝑊)
9	1, 2, 3, 4, 5, 6, 7, 8	dochval 42376	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝑁‘𝑋) = (𝐼‘( ⊥ ‘((glb‘𝐾)‘{𝑥 ∈ (Base‘𝐾) ∣ 𝑋 ⊆ (𝐼‘𝑥)}))))
10		hlclat 40383	. . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat)
11	10	ad2antrr 739	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → 𝐾 ∈ CLat)
12		ssrab2 4028	. . . . . . 7 ⊢ {𝑥 ∈ (Base‘𝐾) ∣ 𝑋 ⊆ (𝐼‘𝑥)} ⊆ (Base‘𝐾)
13	1, 2	clatglbcl 18659	. . . . . . 7 ⊢ ((𝐾 ∈ CLat ∧ {𝑥 ∈ (Base‘𝐾) ∣ 𝑋 ⊆ (𝐼‘𝑥)} ⊆ (Base‘𝐾)) → ((glb‘𝐾)‘{𝑥 ∈ (Base‘𝐾) ∣ 𝑋 ⊆ (𝐼‘𝑥)}) ∈ (Base‘𝐾))
14	11, 12, 13	sylancl 598	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ((glb‘𝐾)‘{𝑥 ∈ (Base‘𝐾) ∣ 𝑋 ⊆ (𝐼‘𝑥)}) ∈ (Base‘𝐾))
15	1, 4, 5	dihcnvid1 42297	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((glb‘𝐾)‘{𝑥 ∈ (Base‘𝐾) ∣ 𝑋 ⊆ (𝐼‘𝑥)}) ∈ (Base‘𝐾)) → (^◡𝐼‘(𝐼‘((glb‘𝐾)‘{𝑥 ∈ (Base‘𝐾) ∣ 𝑋 ⊆ (𝐼‘𝑥)}))) = ((glb‘𝐾)‘{𝑥 ∈ (Base‘𝐾) ∣ 𝑋 ⊆ (𝐼‘𝑥)}))
16	14, 15	syldan 603	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (^◡𝐼‘(𝐼‘((glb‘𝐾)‘{𝑥 ∈ (Base‘𝐾) ∣ 𝑋 ⊆ (𝐼‘𝑥)}))) = ((glb‘𝐾)‘{𝑥 ∈ (Base‘𝐾) ∣ 𝑋 ⊆ (𝐼‘𝑥)}))
17	1, 2, 4, 5, 6, 7	dihglb2 42367	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝐼‘((glb‘𝐾)‘{𝑥 ∈ (Base‘𝐾) ∣ 𝑋 ⊆ (𝐼‘𝑥)})) = ∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})
18	17	fveq2d 6881	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (^◡𝐼‘(𝐼‘((glb‘𝐾)‘{𝑥 ∈ (Base‘𝐾) ∣ 𝑋 ⊆ (𝐼‘𝑥)}))) = (^◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}))
19	16, 18	eqtr3d 2798	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ((glb‘𝐾)‘{𝑥 ∈ (Base‘𝐾) ∣ 𝑋 ⊆ (𝐼‘𝑥)}) = (^◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}))
20	19	fveq2d 6881	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ( ⊥ ‘((glb‘𝐾)‘{𝑥 ∈ (Base‘𝐾) ∣ 𝑋 ⊆ (𝐼‘𝑥)})) = ( ⊥ ‘(^◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})))
21	20	fveq2d 6881	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝐼‘( ⊥ ‘((glb‘𝐾)‘{𝑥 ∈ (Base‘𝐾) ∣ 𝑋 ⊆ (𝐼‘𝑥)}))) = (𝐼‘( ⊥ ‘(^◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}))))
22	9, 21	eqtrd 2796	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝑁‘𝑋) = (𝐼‘( ⊥ ‘(^◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 {crab 3413 ⊆ wss 3899 ∩ cint 4907 ^◡ccnv 5650 ran crn 5652 ‘cfv 6531 Basecbs 17367 occoc 17416 glbcglb 18464 CLatccla 18652 HLchlt 40375 LHypclh 41009 DVecHcdvh 42103 DIsoHcdih 42253 ocHcoch 42372

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-cnex 11237 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258 ax-riotaBAD 39978

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7867 df-1st 7990 df-2nd 7991 df-tpos 8227 df-undef 8274 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8460 df-er 8701 df-map 8833 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-nn 12317 df-2 12386 df-3 12387 df-4 12388 df-5 12389 df-6 12390 df-n0 12588 df-z 12675 df-uz 12947 df-fz 13621 df-struct 17305 df-sets 17322 df-slot 17340 df-ndx 17352 df-base 17368 df-ress 17389 df-plusg 17421 df-mulr 17422 df-sca 17424 df-vsca 17425 df-0g 17592 df-proset 18448 df-poset 18467 df-plt 18482 df-lub 18498 df-glb 18499 df-join 18500 df-meet 18501 df-p0 18577 df-p1 18578 df-lat 18586 df-clat 18653 df-mgm 18796 df-sgrp 18888 df-mnd 18904 df-submnd 18959 df-grp 19127 df-minusg 19128 df-sbg 19129 df-subg 19313 df-cntz 19511 df-lsm 19830 df-cmn 19976 df-abl 19977 df-mgp 20341 df-rng 20355 df-ur 20388 df-ring 20441 df-oppr 20547 df-dvdsr 20567 df-unit 20568 df-invr 20598 df-dvr 20611 df-drng 20962 df-lmod 21117 df-lss 21187 df-lsp 21227 df-lvec 21358 df-lsatoms 40001 df-oposet 40201 df-ol 40203 df-oml 40204 df-covers 40291 df-ats 40292 df-atl 40323 df-cvlat 40347 df-hlat 40376 df-llines 40523 df-lplanes 40524 df-lvols 40525 df-lines 40526 df-psubsp 40528 df-pmap 40529 df-padd 40821 df-lhyp 41013 df-laut 41014 df-ldil 41129 df-ltrn 41130 df-trl 41184 df-tendo 41780 df-edring 41782 df-disoa 42054 df-dvech 42104 df-dib 42164 df-dic 42198 df-dih 42254 df-doch 42373

This theorem is used by: doch2val2 42389 dochocss 42391

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