dihprrnlem2 - Mathbox for Norm Megill

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Theorem dihprrnlem2 37231

Description: Lemma for dihprrn 37232. (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 4319	. . . 4 ⊢ {𝑋, 𝑌} = ({𝑋} ∪ {𝑌})
2	1	fveq2i 6335	. . 3 ⊢ (𝑁‘{𝑋, 𝑌}) = (𝑁‘({𝑋} ∪ {𝑌}))
3		dihprrn.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
4		eqid 2771	. . . . 5 ⊢ (join‘𝐾) = (join‘𝐾)
5		eqid 2771	. . . . 5 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
6		dihprrn.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
7		eqid 2771	. . . . 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 37139	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ) → (^◡𝐼‘(𝑁‘{𝑋})) ∈ (Atoms‘𝐾))
16	9, 10, 11, 15	syl3anc 1476	. . . . 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 37139	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝑉 ∧ 𝑌 ≠ 0 ) → (^◡𝐼‘(𝑁‘{𝑌})) ∈ (Atoms‘𝐾))
20	9, 17, 18, 19	syl3anc 1476	. . . . 5 ⊢ (𝜑 → (^◡𝐼‘(𝑁‘{𝑌})) ∈ (Atoms‘𝐾))
21	3, 4, 5, 6, 7, 8, 9, 16, 20	dihjat 37229	. . . 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 37137	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ ran 𝐼)
23	9, 10, 22	syl2anc 565	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ ran 𝐼)
24	3, 8	dihcnvid2 37079	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑋}) ∈ ran 𝐼) → (𝐼‘(^◡𝐼‘(𝑁‘{𝑋}))) = (𝑁‘{𝑋}))
25	9, 23, 24	syl2anc 565	. . . . 5 ⊢ (𝜑 → (𝐼‘(^◡𝐼‘(𝑁‘{𝑋}))) = (𝑁‘{𝑋}))
26	3, 6, 12, 14, 8	dihlsprn 37137	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ ran 𝐼)
27	9, 17, 26	syl2anc 565	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ ran 𝐼)
28	3, 8	dihcnvid2 37079	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑌}) ∈ ran 𝐼) → (𝐼‘(^◡𝐼‘(𝑁‘{𝑌}))) = (𝑁‘{𝑌}))
29	9, 27, 28	syl2anc 565	. . . . 5 ⊢ (𝜑 → (𝐼‘(^◡𝐼‘(𝑁‘{𝑌}))) = (𝑁‘{𝑌}))
30	25, 29	oveq12d 6810	. . . 4 ⊢ (𝜑 → ((𝐼‘(^◡𝐼‘(𝑁‘{𝑋})))(LSSum‘𝑈)(𝐼‘(^◡𝐼‘(𝑁‘{𝑌})))) = ((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑌})))
31	3, 6, 9	dvhlmod 36916	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
32	10	snssd 4475	. . . . 5 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
33	17	snssd 4475	. . . . 5 ⊢ (𝜑 → {𝑌} ⊆ 𝑉)
34	12, 14, 7	lsmsp2 19299	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ {𝑋} ⊆ 𝑉 ∧ {𝑌} ⊆ 𝑉) → ((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑌})) = (𝑁‘({𝑋} ∪ {𝑌})))
35	31, 32, 33, 34	syl3anc 1476	. . . 4 ⊢ (𝜑 → ((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑌})) = (𝑁‘({𝑋} ∪ {𝑌})))
36	21, 30, 35	3eqtrrd 2810	. . 3 ⊢ (𝜑 → (𝑁‘({𝑋} ∪ {𝑌})) = (𝐼‘((^◡𝐼‘(𝑁‘{𝑋}))(join‘𝐾)(^◡𝐼‘(𝑁‘{𝑌})))))
37	2, 36	syl5eq 2817	. 2 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = (𝐼‘((^◡𝐼‘(𝑁‘{𝑋}))(join‘𝐾)(^◡𝐼‘(𝑁‘{𝑌})))))
38	9	simpld 476	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ HL)
39		hllat 35168	. . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
40	38, 39	syl 17	. . . 4 ⊢ (𝜑 → 𝐾 ∈ Lat)
41		eqid 2771	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
42	41, 3, 8	dihcnvcl 37077	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑋}) ∈ ran 𝐼) → (^◡𝐼‘(𝑁‘{𝑋})) ∈ (Base‘𝐾))
43	9, 23, 42	syl2anc 565	. . . 4 ⊢ (𝜑 → (^◡𝐼‘(𝑁‘{𝑋})) ∈ (Base‘𝐾))
44	41, 3, 8	dihcnvcl 37077	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑌}) ∈ ran 𝐼) → (^◡𝐼‘(𝑁‘{𝑌})) ∈ (Base‘𝐾))
45	9, 27, 44	syl2anc 565	. . . 4 ⊢ (𝜑 → (^◡𝐼‘(𝑁‘{𝑌})) ∈ (Base‘𝐾))
46	41, 4	latjcl 17258	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ (^◡𝐼‘(𝑁‘{𝑋})) ∈ (Base‘𝐾) ∧ (^◡𝐼‘(𝑁‘{𝑌})) ∈ (Base‘𝐾)) → ((^◡𝐼‘(𝑁‘{𝑋}))(join‘𝐾)(^◡𝐼‘(𝑁‘{𝑌}))) ∈ (Base‘𝐾))
47	40, 43, 45, 46	syl3anc 1476	. . 3 ⊢ (𝜑 → ((^◡𝐼‘(𝑁‘{𝑋}))(join‘𝐾)(^◡𝐼‘(𝑁‘{𝑌}))) ∈ (Base‘𝐾))
48	41, 3, 8	dihcl 37076	. . 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)
49	9, 47, 48	syl2anc 565	. 2 ⊢ (𝜑 → (𝐼‘((^◡𝐼‘(𝑁‘{𝑋}))(join‘𝐾)(^◡𝐼‘(𝑁‘{𝑌})))) ∈ ran 𝐼)
50	37, 49	eqeltrd 2850	1 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ ran 𝐼)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 ∪ cun 3721 ⊆ wss 3723 {csn 4316 {cpr 4318 ^◡ccnv 5248 ran crn 5250 ‘cfv 6031 (class class class)co 6792 Basecbs 16063 0_gc0g 16307 joincjn 17151 Latclat 17252 LSSumclsm 18255 LModclmod 19072 LSpanclspn 19183 Atomscatm 35068 HLchlt 35155 LHypclh 35788 DVecHcdvh 36884 DIsoHcdih 37034

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7095 ax-cnex 10193 ax-resscn 10194 ax-1cn 10195 ax-icn 10196 ax-addcl 10197 ax-addrcl 10198 ax-mulcl 10199 ax-mulrcl 10200 ax-mulcom 10201 ax-addass 10202 ax-mulass 10203 ax-distr 10204 ax-i2m1 10205 ax-1ne0 10206 ax-1rid 10207 ax-rnegex 10208 ax-rrecex 10209 ax-cnre 10210 ax-pre-lttri 10211 ax-pre-lttrn 10212 ax-pre-ltadd 10213 ax-pre-mulgt0 10214 ax-riotaBAD 34757

This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-iin 4657 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-om 7212 df-1st 7314 df-2nd 7315 df-tpos 7503 df-undef 7550 df-wrecs 7558 df-recs 7620 df-rdg 7658 df-1o 7712 df-oadd 7716 df-er 7895 df-map 8010 df-en 8109 df-dom 8110 df-sdom 8111 df-fin 8112 df-pnf 10277 df-mnf 10278 df-xr 10279 df-ltxr 10280 df-le 10281 df-sub 10469 df-neg 10470 df-nn 11222 df-2 11280 df-3 11281 df-4 11282 df-5 11283 df-6 11284 df-n0 11494 df-z 11579 df-uz 11888 df-fz 12533 df-struct 16065 df-ndx 16066 df-slot 16067 df-base 16069 df-sets 16070 df-ress 16071 df-plusg 16161 df-mulr 16162 df-sca 16164 df-vsca 16165 df-0g 16309 df-preset 17135 df-poset 17153 df-plt 17165 df-lub 17181 df-glb 17182 df-join 17183 df-meet 17184 df-p0 17246 df-p1 17247 df-lat 17253 df-clat 17315 df-mgm 17449 df-sgrp 17491 df-mnd 17502 df-submnd 17543 df-grp 17632 df-minusg 17633 df-sbg 17634 df-subg 17798 df-cntz 17956 df-lsm 18257 df-cmn 18401 df-abl 18402 df-mgp 18697 df-ur 18709 df-ring 18756 df-oppr 18830 df-dvdsr 18848 df-unit 18849 df-invr 18879 df-dvr 18890 df-drng 18958 df-lmod 19074 df-lss 19142 df-lsp 19184 df-lvec 19315 df-lsatoms 34781 df-oposet 34981 df-ol 34983 df-oml 34984 df-covers 35071 df-ats 35072 df-atl 35103 df-cvlat 35127 df-hlat 35156 df-llines 35302 df-lplanes 35303 df-lvols 35304 df-lines 35305 df-psubsp 35307 df-pmap 35308 df-padd 35600 df-lhyp 35792 df-laut 35793 df-ldil 35908 df-ltrn 35909 df-trl 35964 df-tgrp 36548 df-tendo 36560 df-edring 36562 df-dveca 36808 df-disoa 36835 df-dvech 36885 df-dib 36945 df-dic 36979 df-dih 37035 df-doch 37154 df-djh 37201

This theorem is referenced by: dihprrn 37232

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