dihprrnlem2 - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > dihprrnlem2		Structured version Visualization version GIF version

Theorem dihprrnlem2 37563

Description: Lemma for dihprrn 37564. (Contributed by NM, 29-Sep-2014.)

**Hypotheses**
Ref	Expression
dihprrn.h	⊢ 𝐻 = (LHyp‘𝐾)
dihprrn.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dihprrn.v	⊢ 𝑉 = (Base‘𝑈)
dihprrn.n	⊢ 𝑁 = (LSpan‘𝑈)
dihprrn.i	⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
dihprrn.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
dihprrn.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
dihprrn.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)
dihprrnlem2.o	⊢ 0 = (0_g‘𝑈)
dihprrnlem2.xz	⊢ (𝜑 → 𝑋 ≠ 0 )
dihprrnlem2.yz	⊢ (𝜑 → 𝑌 ≠ 0 )

**Assertion**
Ref	Expression
dihprrnlem2	⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ ran 𝐼)

**Proof of Theorem dihprrnlem2**
Step	Hyp	Ref	Expression
1		df-pr 4400	. . . 4 ⊢ {𝑋, 𝑌} = ({𝑋} ∪ {𝑌})
2	1	fveq2i 6449	. . 3 ⊢ (𝑁‘{𝑋, 𝑌}) = (𝑁‘({𝑋} ∪ {𝑌}))
3		dihprrn.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
4		eqid 2777	. . . . 5 ⊢ (join‘𝐾) = (join‘𝐾)
5		eqid 2777	. . . . 5 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
6		dihprrn.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
7		eqid 2777	. . . . 5 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈)
8		dihprrn.i	. . . . 5 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
9		dihprrn.k	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
10		dihprrn.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
11		dihprrnlem2.xz	. . . . . 6 ⊢ (𝜑 → 𝑋 ≠ 0 )
12		dihprrn.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑈)
13		dihprrnlem2.o	. . . . . . 7 ⊢ 0 = (0_g‘𝑈)
14		dihprrn.n	. . . . . . 7 ⊢ 𝑁 = (LSpan‘𝑈)
15	5, 3, 6, 12, 13, 14, 8	dihlspsnat 37471	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ) → (^◡𝐼‘(𝑁‘{𝑋})) ∈ (Atoms‘𝐾))
16	9, 10, 11, 15	syl3anc 1439	. . . . 5 ⊢ (𝜑 → (^◡𝐼‘(𝑁‘{𝑋})) ∈ (Atoms‘𝐾))
17		dihprrn.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
18		dihprrnlem2.yz	. . . . . 6 ⊢ (𝜑 → 𝑌 ≠ 0 )
19	5, 3, 6, 12, 13, 14, 8	dihlspsnat 37471	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝑉 ∧ 𝑌 ≠ 0 ) → (^◡𝐼‘(𝑁‘{𝑌})) ∈ (Atoms‘𝐾))
20	9, 17, 18, 19	syl3anc 1439	. . . . 5 ⊢ (𝜑 → (^◡𝐼‘(𝑁‘{𝑌})) ∈ (Atoms‘𝐾))
21	3, 4, 5, 6, 7, 8, 9, 16, 20	dihjat 37561	. . . 4 ⊢ (𝜑 → (𝐼‘((^◡𝐼‘(𝑁‘{𝑋}))(join‘𝐾)(^◡𝐼‘(𝑁‘{𝑌})))) = ((𝐼‘(^◡𝐼‘(𝑁‘{𝑋})))(LSSum‘𝑈)(𝐼‘(^◡𝐼‘(𝑁‘{𝑌})))))
22	3, 6, 12, 14, 8	dihlsprn 37469	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ ran 𝐼)
23	9, 10, 22	syl2anc 579	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ ran 𝐼)
24	3, 8	dihcnvid2 37411	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑋}) ∈ ran 𝐼) → (𝐼‘(^◡𝐼‘(𝑁‘{𝑋}))) = (𝑁‘{𝑋}))
25	9, 23, 24	syl2anc 579	. . . . 5 ⊢ (𝜑 → (𝐼‘(^◡𝐼‘(𝑁‘{𝑋}))) = (𝑁‘{𝑋}))
26	3, 6, 12, 14, 8	dihlsprn 37469	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ ran 𝐼)
27	9, 17, 26	syl2anc 579	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ ran 𝐼)
28	3, 8	dihcnvid2 37411	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑌}) ∈ ran 𝐼) → (𝐼‘(^◡𝐼‘(𝑁‘{𝑌}))) = (𝑁‘{𝑌}))
29	9, 27, 28	syl2anc 579	. . . . 5 ⊢ (𝜑 → (𝐼‘(^◡𝐼‘(𝑁‘{𝑌}))) = (𝑁‘{𝑌}))
30	25, 29	oveq12d 6940	. . . 4 ⊢ (𝜑 → ((𝐼‘(^◡𝐼‘(𝑁‘{𝑋})))(LSSum‘𝑈)(𝐼‘(^◡𝐼‘(𝑁‘{𝑌})))) = ((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑌})))
31	3, 6, 9	dvhlmod 37248	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
32	10	snssd 4571	. . . . 5 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
33	17	snssd 4571	. . . . 5 ⊢ (𝜑 → {𝑌} ⊆ 𝑉)
34	12, 14, 7	lsmsp2 19482	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ {𝑋} ⊆ 𝑉 ∧ {𝑌} ⊆ 𝑉) → ((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑌})) = (𝑁‘({𝑋} ∪ {𝑌})))
35	31, 32, 33, 34	syl3anc 1439	. . . 4 ⊢ (𝜑 → ((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑌})) = (𝑁‘({𝑋} ∪ {𝑌})))
36	21, 30, 35	3eqtrrd 2818	. . 3 ⊢ (𝜑 → (𝑁‘({𝑋} ∪ {𝑌})) = (𝐼‘((^◡𝐼‘(𝑁‘{𝑋}))(join‘𝐾)(^◡𝐼‘(𝑁‘{𝑌})))))
37	2, 36	syl5eq 2825	. 2 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = (𝐼‘((^◡𝐼‘(𝑁‘{𝑋}))(join‘𝐾)(^◡𝐼‘(𝑁‘{𝑌})))))
38	9	simpld 490	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ HL)
39	38	hllatd 35502	. . . 4 ⊢ (𝜑 → 𝐾 ∈ Lat)
40		eqid 2777	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
41	40, 3, 8	dihcnvcl 37409	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑋}) ∈ ran 𝐼) → (^◡𝐼‘(𝑁‘{𝑋})) ∈ (Base‘𝐾))
42	9, 23, 41	syl2anc 579	. . . 4 ⊢ (𝜑 → (^◡𝐼‘(𝑁‘{𝑋})) ∈ (Base‘𝐾))
43	40, 3, 8	dihcnvcl 37409	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑌}) ∈ ran 𝐼) → (^◡𝐼‘(𝑁‘{𝑌})) ∈ (Base‘𝐾))
44	9, 27, 43	syl2anc 579	. . . 4 ⊢ (𝜑 → (^◡𝐼‘(𝑁‘{𝑌})) ∈ (Base‘𝐾))
45	40, 4	latjcl 17437	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ (^◡𝐼‘(𝑁‘{𝑋})) ∈ (Base‘𝐾) ∧ (^◡𝐼‘(𝑁‘{𝑌})) ∈ (Base‘𝐾)) → ((^◡𝐼‘(𝑁‘{𝑋}))(join‘𝐾)(^◡𝐼‘(𝑁‘{𝑌}))) ∈ (Base‘𝐾))
46	39, 42, 44, 45	syl3anc 1439	. . 3 ⊢ (𝜑 → ((^◡𝐼‘(𝑁‘{𝑋}))(join‘𝐾)(^◡𝐼‘(𝑁‘{𝑌}))) ∈ (Base‘𝐾))
47	40, 3, 8	dihcl 37408	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((^◡𝐼‘(𝑁‘{𝑋}))(join‘𝐾)(^◡𝐼‘(𝑁‘{𝑌}))) ∈ (Base‘𝐾)) → (𝐼‘((^◡𝐼‘(𝑁‘{𝑋}))(join‘𝐾)(^◡𝐼‘(𝑁‘{𝑌})))) ∈ ran 𝐼)
48	9, 46, 47	syl2anc 579	. 2 ⊢ (𝜑 → (𝐼‘((^◡𝐼‘(𝑁‘{𝑋}))(join‘𝐾)(^◡𝐼‘(𝑁‘{𝑌})))) ∈ ran 𝐼)
49	37, 48	eqeltrd 2858	1 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ ran 𝐼)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2106 ≠ wne 2968 ∪ cun 3789 ⊆ wss 3791 {csn 4397 {cpr 4399 ^◡ccnv 5354 ran crn 5356 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 0_gc0g 16486 joincjn 17330 Latclat 17431 LSSumclsm 18433 LModclmod 19255 LSpanclspn 19366 Atomscatm 35401 HLchlt 35488 LHypclh 36122 DVecHcdvh 37216 DIsoHcdih 37366

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-riotaBAD 35091

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-tpos 7634 df-undef 7681 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-n0 11643 df-z 11729 df-uz 11993 df-fz 12644 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-sca 16354 df-vsca 16355 df-0g 16488 df-proset 17314 df-poset 17332 df-plt 17344 df-lub 17360 df-glb 17361 df-join 17362 df-meet 17363 df-p0 17425 df-p1 17426 df-lat 17432 df-clat 17494 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-submnd 17722 df-grp 17812 df-minusg 17813 df-sbg 17814 df-subg 17975 df-cntz 18133 df-lsm 18435 df-cmn 18581 df-abl 18582 df-mgp 18877 df-ur 18889 df-ring 18936 df-oppr 19010 df-dvdsr 19028 df-unit 19029 df-invr 19059 df-dvr 19070 df-drng 19141 df-lmod 19257 df-lss 19325 df-lsp 19367 df-lvec 19498 df-lsatoms 35114 df-oposet 35314 df-ol 35316 df-oml 35317 df-covers 35404 df-ats 35405 df-atl 35436 df-cvlat 35460 df-hlat 35489 df-llines 35636 df-lplanes 35637 df-lvols 35638 df-lines 35639 df-psubsp 35641 df-pmap 35642 df-padd 35934 df-lhyp 36126 df-laut 36127 df-ldil 36242 df-ltrn 36243 df-trl 36297 df-tgrp 36881 df-tendo 36893 df-edring 36895 df-dveca 37141 df-disoa 37167 df-dvech 37217 df-dib 37277 df-dic 37311 df-dih 37367 df-doch 37486 df-djh 37533

This theorem is referenced by: dihprrn 37564

W3C validator