dihprrnlem1N - Mathbox for Norm Megill

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Theorem dihprrnlem1N 39438

Description: Lemma for dihprrn 39440, showing one of 4 cases. (Contributed by NM, 30-Aug-2014.) (New usage is discouraged.)

**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)
dihprrnlem1.l	⊢ ≤ = (le‘𝐾)
dihprrnlem1.o	⊢ 0 = (0_g‘𝑈)
dihprrnlem1.nz	⊢ (𝜑 → 𝑌 ≠ 0 )
dihprrnlem1.x	⊢ (𝜑 → (^◡𝐼‘(𝑁‘{𝑋})) ≤ 𝑊)
dihprrnlem1.y	⊢ (𝜑 → ¬ (^◡𝐼‘(𝑁‘{𝑌})) ≤ 𝑊)

**Assertion**
Ref	Expression
dihprrnlem1N	⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ ran 𝐼)

**Proof of Theorem dihprrnlem1N**
Step	Hyp	Ref	Expression
1		df-pr 4564	. . . 4 ⊢ {𝑋, 𝑌} = ({𝑋} ∪ {𝑌})
2	1	fveq2i 6777	. . 3 ⊢ (𝑁‘{𝑋, 𝑌}) = (𝑁‘({𝑋} ∪ {𝑌}))
3		eqid 2738	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
4		dihprrnlem1.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
5		dihprrn.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
6		eqid 2738	. . . . 5 ⊢ (join‘𝐾) = (join‘𝐾)
7		eqid 2738	. . . . 5 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
8		dihprrn.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
9		eqid 2738	. . . . 5 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈)
10		dihprrn.i	. . . . 5 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
11		dihprrn.k	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
12		dihprrn.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
13		dihprrn.v	. . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑈)
14		dihprrn.n	. . . . . . . . 9 ⊢ 𝑁 = (LSpan‘𝑈)
15	5, 8, 13, 14, 10	dihlsprn 39345	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ ran 𝐼)
16	11, 12, 15	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ ran 𝐼)
17	3, 5, 10	dihcnvcl 39285	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑋}) ∈ ran 𝐼) → (^◡𝐼‘(𝑁‘{𝑋})) ∈ (Base‘𝐾))
18	11, 16, 17	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (^◡𝐼‘(𝑁‘{𝑋})) ∈ (Base‘𝐾))
19		dihprrnlem1.x	. . . . . 6 ⊢ (𝜑 → (^◡𝐼‘(𝑁‘{𝑋})) ≤ 𝑊)
20	18, 19	jca 512	. . . . 5 ⊢ (𝜑 → ((^◡𝐼‘(𝑁‘{𝑋})) ∈ (Base‘𝐾) ∧ (^◡𝐼‘(𝑁‘{𝑋})) ≤ 𝑊))
21		dihprrn.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
22		dihprrnlem1.nz	. . . . . . 7 ⊢ (𝜑 → 𝑌 ≠ 0 )
23		dihprrnlem1.o	. . . . . . . 8 ⊢ 0 = (0_g‘𝑈)
24	7, 5, 8, 13, 23, 14, 10	dihlspsnat 39347	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝑉 ∧ 𝑌 ≠ 0 ) → (^◡𝐼‘(𝑁‘{𝑌})) ∈ (Atoms‘𝐾))
25	11, 21, 22, 24	syl3anc 1370	. . . . . 6 ⊢ (𝜑 → (^◡𝐼‘(𝑁‘{𝑌})) ∈ (Atoms‘𝐾))
26		dihprrnlem1.y	. . . . . 6 ⊢ (𝜑 → ¬ (^◡𝐼‘(𝑁‘{𝑌})) ≤ 𝑊)
27	25, 26	jca 512	. . . . 5 ⊢ (𝜑 → ((^◡𝐼‘(𝑁‘{𝑌})) ∈ (Atoms‘𝐾) ∧ ¬ (^◡𝐼‘(𝑁‘{𝑌})) ≤ 𝑊))
28	3, 4, 5, 6, 7, 8, 9, 10, 11, 20, 27	dihjatc 39431	. . . 4 ⊢ (𝜑 → (𝐼‘((^◡𝐼‘(𝑁‘{𝑋}))(join‘𝐾)(^◡𝐼‘(𝑁‘{𝑌})))) = ((𝐼‘(^◡𝐼‘(𝑁‘{𝑋})))(LSSum‘𝑈)(𝐼‘(^◡𝐼‘(𝑁‘{𝑌})))))
29	5, 10	dihcnvid2 39287	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑋}) ∈ ran 𝐼) → (𝐼‘(^◡𝐼‘(𝑁‘{𝑋}))) = (𝑁‘{𝑋}))
30	11, 16, 29	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝐼‘(^◡𝐼‘(𝑁‘{𝑋}))) = (𝑁‘{𝑋}))
31	5, 8, 13, 14, 10	dihlsprn 39345	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ ran 𝐼)
32	11, 21, 31	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ ran 𝐼)
33	5, 10	dihcnvid2 39287	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑌}) ∈ ran 𝐼) → (𝐼‘(^◡𝐼‘(𝑁‘{𝑌}))) = (𝑁‘{𝑌}))
34	11, 32, 33	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝐼‘(^◡𝐼‘(𝑁‘{𝑌}))) = (𝑁‘{𝑌}))
35	30, 34	oveq12d 7293	. . . 4 ⊢ (𝜑 → ((𝐼‘(^◡𝐼‘(𝑁‘{𝑋})))(LSSum‘𝑈)(𝐼‘(^◡𝐼‘(𝑁‘{𝑌})))) = ((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑌})))
36	5, 8, 11	dvhlmod 39124	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
37	12	snssd 4742	. . . . 5 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
38	21	snssd 4742	. . . . 5 ⊢ (𝜑 → {𝑌} ⊆ 𝑉)
39	13, 14, 9	lsmsp2 20349	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ {𝑋} ⊆ 𝑉 ∧ {𝑌} ⊆ 𝑉) → ((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑌})) = (𝑁‘({𝑋} ∪ {𝑌})))
40	36, 37, 38, 39	syl3anc 1370	. . . 4 ⊢ (𝜑 → ((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑌})) = (𝑁‘({𝑋} ∪ {𝑌})))
41	28, 35, 40	3eqtrrd 2783	. . 3 ⊢ (𝜑 → (𝑁‘({𝑋} ∪ {𝑌})) = (𝐼‘((^◡𝐼‘(𝑁‘{𝑋}))(join‘𝐾)(^◡𝐼‘(𝑁‘{𝑌})))))
42	2, 41	eqtrid 2790	. 2 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = (𝐼‘((^◡𝐼‘(𝑁‘{𝑋}))(join‘𝐾)(^◡𝐼‘(𝑁‘{𝑌})))))
43	11	simpld 495	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ HL)
44	43	hllatd 37378	. . . 4 ⊢ (𝜑 → 𝐾 ∈ Lat)
45	3, 5, 10	dihcnvcl 39285	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑌}) ∈ ran 𝐼) → (^◡𝐼‘(𝑁‘{𝑌})) ∈ (Base‘𝐾))
46	11, 32, 45	syl2anc 584	. . . 4 ⊢ (𝜑 → (^◡𝐼‘(𝑁‘{𝑌})) ∈ (Base‘𝐾))
47	3, 6	latjcl 18157	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ (^◡𝐼‘(𝑁‘{𝑋})) ∈ (Base‘𝐾) ∧ (^◡𝐼‘(𝑁‘{𝑌})) ∈ (Base‘𝐾)) → ((^◡𝐼‘(𝑁‘{𝑋}))(join‘𝐾)(^◡𝐼‘(𝑁‘{𝑌}))) ∈ (Base‘𝐾))
48	44, 18, 46, 47	syl3anc 1370	. . 3 ⊢ (𝜑 → ((^◡𝐼‘(𝑁‘{𝑋}))(join‘𝐾)(^◡𝐼‘(𝑁‘{𝑌}))) ∈ (Base‘𝐾))
49	3, 5, 10	dihcl 39284	. . 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)
50	11, 48, 49	syl2anc 584	. 2 ⊢ (𝜑 → (𝐼‘((^◡𝐼‘(𝑁‘{𝑋}))(join‘𝐾)(^◡𝐼‘(𝑁‘{𝑌})))) ∈ ran 𝐼)
51	42, 50	eqeltrd 2839	1 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ ran 𝐼)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∪ cun 3885 ⊆ wss 3887 {csn 4561 {cpr 4563 class class class wbr 5074 ^◡ccnv 5588 ran crn 5590 ‘cfv 6433 (class class class)co 7275 Basecbs 16912 lecple 16969 0_gc0g 17150 joincjn 18029 Latclat 18149 LSSumclsm 19239 LModclmod 20123 LSpanclspn 20233 Atomscatm 37277 HLchlt 37364 LHypclh 37998 DVecHcdvh 39092 DIsoHcdih 39242

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-riotaBAD 36967

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-tpos 8042 df-undef 8089 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-sca 16978 df-vsca 16979 df-0g 17152 df-proset 18013 df-poset 18031 df-plt 18048 df-lub 18064 df-glb 18065 df-join 18066 df-meet 18067 df-p0 18143 df-p1 18144 df-lat 18150 df-clat 18217 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-grp 18580 df-minusg 18581 df-sbg 18582 df-subg 18752 df-cntz 18923 df-lsm 19241 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-oppr 19862 df-dvdsr 19883 df-unit 19884 df-invr 19914 df-dvr 19925 df-drng 19993 df-lmod 20125 df-lss 20194 df-lsp 20234 df-lvec 20365 df-lsatoms 36990 df-oposet 37190 df-ol 37192 df-oml 37193 df-covers 37280 df-ats 37281 df-atl 37312 df-cvlat 37336 df-hlat 37365 df-llines 37512 df-lplanes 37513 df-lvols 37514 df-lines 37515 df-psubsp 37517 df-pmap 37518 df-padd 37810 df-lhyp 38002 df-laut 38003 df-ldil 38118 df-ltrn 38119 df-trl 38173 df-tendo 38769 df-edring 38771 df-disoa 39043 df-dvech 39093 df-dib 39153 df-dic 39187 df-dih 39243

This theorem is referenced by: (None)

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