dihprrnlem2 - Mathbox for Norm Megill

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Theorem dihprrnlem2 39366

Description: Lemma for dihprrn 39367. (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 4561	. . . 4 ⊢ {𝑋, 𝑌} = ({𝑋} ∪ {𝑌})
2	1	fveq2i 6759	. . 3 ⊢ (𝑁‘{𝑋, 𝑌}) = (𝑁‘({𝑋} ∪ {𝑌}))
3		dihprrn.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
4		eqid 2738	. . . . 5 ⊢ (join‘𝐾) = (join‘𝐾)
5		eqid 2738	. . . . 5 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
6		dihprrn.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
7		eqid 2738	. . . . 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 39274	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ) → (^◡𝐼‘(𝑁‘{𝑋})) ∈ (Atoms‘𝐾))
16	9, 10, 11, 15	syl3anc 1369	. . . . 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 39274	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝑉 ∧ 𝑌 ≠ 0 ) → (^◡𝐼‘(𝑁‘{𝑌})) ∈ (Atoms‘𝐾))
20	9, 17, 18, 19	syl3anc 1369	. . . . 5 ⊢ (𝜑 → (^◡𝐼‘(𝑁‘{𝑌})) ∈ (Atoms‘𝐾))
21	3, 4, 5, 6, 7, 8, 9, 16, 20	dihjat 39364	. . . 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 39272	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ ran 𝐼)
23	9, 10, 22	syl2anc 583	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ ran 𝐼)
24	3, 8	dihcnvid2 39214	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑋}) ∈ ran 𝐼) → (𝐼‘(^◡𝐼‘(𝑁‘{𝑋}))) = (𝑁‘{𝑋}))
25	9, 23, 24	syl2anc 583	. . . . 5 ⊢ (𝜑 → (𝐼‘(^◡𝐼‘(𝑁‘{𝑋}))) = (𝑁‘{𝑋}))
26	3, 6, 12, 14, 8	dihlsprn 39272	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ ran 𝐼)
27	9, 17, 26	syl2anc 583	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ ran 𝐼)
28	3, 8	dihcnvid2 39214	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑌}) ∈ ran 𝐼) → (𝐼‘(^◡𝐼‘(𝑁‘{𝑌}))) = (𝑁‘{𝑌}))
29	9, 27, 28	syl2anc 583	. . . . 5 ⊢ (𝜑 → (𝐼‘(^◡𝐼‘(𝑁‘{𝑌}))) = (𝑁‘{𝑌}))
30	25, 29	oveq12d 7273	. . . 4 ⊢ (𝜑 → ((𝐼‘(^◡𝐼‘(𝑁‘{𝑋})))(LSSum‘𝑈)(𝐼‘(^◡𝐼‘(𝑁‘{𝑌})))) = ((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑌})))
31	3, 6, 9	dvhlmod 39051	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
32	10	snssd 4739	. . . . 5 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
33	17	snssd 4739	. . . . 5 ⊢ (𝜑 → {𝑌} ⊆ 𝑉)
34	12, 14, 7	lsmsp2 20264	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ {𝑋} ⊆ 𝑉 ∧ {𝑌} ⊆ 𝑉) → ((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑌})) = (𝑁‘({𝑋} ∪ {𝑌})))
35	31, 32, 33, 34	syl3anc 1369	. . . 4 ⊢ (𝜑 → ((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑌})) = (𝑁‘({𝑋} ∪ {𝑌})))
36	21, 30, 35	3eqtrrd 2783	. . 3 ⊢ (𝜑 → (𝑁‘({𝑋} ∪ {𝑌})) = (𝐼‘((^◡𝐼‘(𝑁‘{𝑋}))(join‘𝐾)(^◡𝐼‘(𝑁‘{𝑌})))))
37	2, 36	syl5eq 2791	. 2 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = (𝐼‘((^◡𝐼‘(𝑁‘{𝑋}))(join‘𝐾)(^◡𝐼‘(𝑁‘{𝑌})))))
38	9	simpld 494	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ HL)
39	38	hllatd 37305	. . . 4 ⊢ (𝜑 → 𝐾 ∈ Lat)
40		eqid 2738	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
41	40, 3, 8	dihcnvcl 39212	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑋}) ∈ ran 𝐼) → (^◡𝐼‘(𝑁‘{𝑋})) ∈ (Base‘𝐾))
42	9, 23, 41	syl2anc 583	. . . 4 ⊢ (𝜑 → (^◡𝐼‘(𝑁‘{𝑋})) ∈ (Base‘𝐾))
43	40, 3, 8	dihcnvcl 39212	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑌}) ∈ ran 𝐼) → (^◡𝐼‘(𝑁‘{𝑌})) ∈ (Base‘𝐾))
44	9, 27, 43	syl2anc 583	. . . 4 ⊢ (𝜑 → (^◡𝐼‘(𝑁‘{𝑌})) ∈ (Base‘𝐾))
45	40, 4	latjcl 18072	. . . 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 1369	. . 3 ⊢ (𝜑 → ((^◡𝐼‘(𝑁‘{𝑋}))(join‘𝐾)(^◡𝐼‘(𝑁‘{𝑌}))) ∈ (Base‘𝐾))
47	40, 3, 8	dihcl 39211	. . 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 583	. 2 ⊢ (𝜑 → (𝐼‘((^◡𝐼‘(𝑁‘{𝑋}))(join‘𝐾)(^◡𝐼‘(𝑁‘{𝑌})))) ∈ ran 𝐼)
49	37, 48	eqeltrd 2839	1 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ ran 𝐼)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∪ cun 3881 ⊆ wss 3883 {csn 4558 {cpr 4560 ^◡ccnv 5579 ran crn 5581 ‘cfv 6418 (class class class)co 7255 Basecbs 16840 0_gc0g 17067 joincjn 17944 Latclat 18064 LSSumclsm 19154 LModclmod 20038 LSpanclspn 20148 Atomscatm 37204 HLchlt 37291 LHypclh 37925 DVecHcdvh 39019 DIsoHcdih 39169

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-riotaBAD 36894

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-tpos 8013 df-undef 8060 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-n0 12164 df-z 12250 df-uz 12512 df-fz 13169 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-sca 16904 df-vsca 16905 df-0g 17069 df-proset 17928 df-poset 17946 df-plt 17963 df-lub 17979 df-glb 17980 df-join 17981 df-meet 17982 df-p0 18058 df-p1 18059 df-lat 18065 df-clat 18132 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-grp 18495 df-minusg 18496 df-sbg 18497 df-subg 18667 df-cntz 18838 df-lsm 19156 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-ring 19700 df-oppr 19777 df-dvdsr 19798 df-unit 19799 df-invr 19829 df-dvr 19840 df-drng 19908 df-lmod 20040 df-lss 20109 df-lsp 20149 df-lvec 20280 df-lsatoms 36917 df-oposet 37117 df-ol 37119 df-oml 37120 df-covers 37207 df-ats 37208 df-atl 37239 df-cvlat 37263 df-hlat 37292 df-llines 37439 df-lplanes 37440 df-lvols 37441 df-lines 37442 df-psubsp 37444 df-pmap 37445 df-padd 37737 df-lhyp 37929 df-laut 37930 df-ldil 38045 df-ltrn 38046 df-trl 38100 df-tgrp 38684 df-tendo 38696 df-edring 38698 df-dveca 38944 df-disoa 38970 df-dvech 39020 df-dib 39080 df-dic 39114 df-dih 39170 df-doch 39289 df-djh 39336

This theorem is referenced by: dihprrn 39367

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