dihprrnlem2 - Mathbox for Norm Megill

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Theorem dihprrnlem2 39465

Description: Lemma for dihprrn 39466. (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 4567	. . . 4 ⊢ {𝑋, 𝑌} = ({𝑋} ∪ {𝑌})
2	1	fveq2i 6795	. . 3 ⊢ (𝑁‘{𝑋, 𝑌}) = (𝑁‘({𝑋} ∪ {𝑌}))
3		dihprrn.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
4		eqid 2733	. . . . 5 ⊢ (join‘𝐾) = (join‘𝐾)
5		eqid 2733	. . . . 5 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
6		dihprrn.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
7		eqid 2733	. . . . 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 39373	. . . . . 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 39373	. . . . . 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 39463	. . . 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 39371	. . . . . . 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 39313	. . . . . 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 39371	. . . . . . 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 39313	. . . . . 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 7313	. . . 4 ⊢ (𝜑 → ((𝐼‘(^◡𝐼‘(𝑁‘{𝑋})))(LSSum‘𝑈)(𝐼‘(^◡𝐼‘(𝑁‘{𝑌})))) = ((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑌})))
31	3, 6, 9	dvhlmod 39150	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
32	10	snssd 4745	. . . . 5 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
33	17	snssd 4745	. . . . 5 ⊢ (𝜑 → {𝑌} ⊆ 𝑉)
34	12, 14, 7	lsmsp2 20377	. . . . 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 2778	. . 3 ⊢ (𝜑 → (𝑁‘({𝑋} ∪ {𝑌})) = (𝐼‘((^◡𝐼‘(𝑁‘{𝑋}))(join‘𝐾)(^◡𝐼‘(𝑁‘{𝑌})))))
37	2, 36	eqtrid 2785	. 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 37404	. . . 4 ⊢ (𝜑 → 𝐾 ∈ Lat)
40		eqid 2733	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
41	40, 3, 8	dihcnvcl 39311	. . . . 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 39311	. . . . 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 18185	. . . 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 39310	. . 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 2834	1 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ ran 𝐼)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2101 ≠ wne 2938 ∪ cun 3887 ⊆ wss 3889 {csn 4564 {cpr 4566 ^◡ccnv 5590 ran crn 5592 ‘cfv 6447 (class class class)co 7295 Basecbs 16940 0_gc0g 17178 joincjn 18057 Latclat 18177 LSSumclsm 19267 LModclmod 20151 LSpanclspn 20261 Atomscatm 37303 HLchlt 37390 LHypclh 38024 DVecHcdvh 39118 DIsoHcdih 39268

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-rep 5212 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7608 ax-cnex 10955 ax-resscn 10956 ax-1cn 10957 ax-icn 10958 ax-addcl 10959 ax-addrcl 10960 ax-mulcl 10961 ax-mulrcl 10962 ax-mulcom 10963 ax-addass 10964 ax-mulass 10965 ax-distr 10966 ax-i2m1 10967 ax-1ne0 10968 ax-1rid 10969 ax-rnegex 10970 ax-rrecex 10971 ax-cnre 10972 ax-pre-lttri 10973 ax-pre-lttrn 10974 ax-pre-ltadd 10975 ax-pre-mulgt0 10976 ax-riotaBAD 36993

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3222 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3908 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-uni 4842 df-int 4883 df-iun 4929 df-iin 4930 df-br 5078 df-opab 5140 df-mpt 5161 df-tr 5195 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-riota 7252 df-ov 7298 df-oprab 7299 df-mpo 7300 df-om 7733 df-1st 7851 df-2nd 7852 df-tpos 8062 df-undef 8109 df-frecs 8117 df-wrecs 8148 df-recs 8222 df-rdg 8261 df-1o 8317 df-er 8518 df-map 8637 df-en 8754 df-dom 8755 df-sdom 8756 df-fin 8757 df-pnf 11039 df-mnf 11040 df-xr 11041 df-ltxr 11042 df-le 11043 df-sub 11235 df-neg 11236 df-nn 12002 df-2 12064 df-3 12065 df-4 12066 df-5 12067 df-6 12068 df-n0 12262 df-z 12348 df-uz 12611 df-fz 13268 df-struct 16876 df-sets 16893 df-slot 16911 df-ndx 16923 df-base 16941 df-ress 16970 df-plusg 17003 df-mulr 17004 df-sca 17006 df-vsca 17007 df-0g 17180 df-proset 18041 df-poset 18059 df-plt 18076 df-lub 18092 df-glb 18093 df-join 18094 df-meet 18095 df-p0 18171 df-p1 18172 df-lat 18178 df-clat 18245 df-mgm 18354 df-sgrp 18403 df-mnd 18414 df-submnd 18459 df-grp 18608 df-minusg 18609 df-sbg 18610 df-subg 18780 df-cntz 18951 df-lsm 19269 df-cmn 19416 df-abl 19417 df-mgp 19749 df-ur 19766 df-ring 19813 df-oppr 19890 df-dvdsr 19911 df-unit 19912 df-invr 19942 df-dvr 19953 df-drng 20021 df-lmod 20153 df-lss 20222 df-lsp 20262 df-lvec 20393 df-lsatoms 37016 df-oposet 37216 df-ol 37218 df-oml 37219 df-covers 37306 df-ats 37307 df-atl 37338 df-cvlat 37362 df-hlat 37391 df-llines 37538 df-lplanes 37539 df-lvols 37540 df-lines 37541 df-psubsp 37543 df-pmap 37544 df-padd 37836 df-lhyp 38028 df-laut 38029 df-ldil 38144 df-ltrn 38145 df-trl 38199 df-tgrp 38783 df-tendo 38795 df-edring 38797 df-dveca 39043 df-disoa 39069 df-dvech 39119 df-dib 39179 df-dic 39213 df-dih 39269 df-doch 39388 df-djh 39435

This theorem is referenced by: dihprrn 39466

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