dia2dimlem9 - Mathbox for Norm Megill

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Theorem dia2dimlem9 40783

Description: Lemma for dia2dim 40788. Eliminate (𝑅‘𝐹) ≠ 𝑈, 𝑉 conditions. (Contributed by NM, 8-Sep-2014.)

**Hypotheses**
Ref	Expression
dia2dimlem9.l	⊢ ≤ = (le‘𝐾)
dia2dimlem9.j	⊢ ∨ = (join‘𝐾)
dia2dimlem9.m	⊢ ∧ = (meet‘𝐾)
dia2dimlem9.a	⊢ 𝐴 = (Atoms‘𝐾)
dia2dimlem9.h	⊢ 𝐻 = (LHyp‘𝐾)
dia2dimlem9.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
dia2dimlem9.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
dia2dimlem9.y	⊢ 𝑌 = ((DVecA‘𝐾)‘𝑊)
dia2dimlem9.s	⊢ 𝑆 = (LSubSp‘𝑌)
dia2dimlem9.pl	⊢ ⊕ = (LSSum‘𝑌)
dia2dimlem9.n	⊢ 𝑁 = (LSpan‘𝑌)
dia2dimlem9.i	⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)
dia2dimlem9.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
dia2dimlem9.u	⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊))
dia2dimlem9.v	⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))
dia2dimlem9.f	⊢ (𝜑 → 𝐹 ∈ 𝑇)
dia2dimlem9.rf	⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉))
dia2dimlem9.uv	⊢ (𝜑 → 𝑈 ≠ 𝑉)

**Assertion**
Ref	Expression
dia2dimlem9	⊢ (𝜑 → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))

**Proof of Theorem dia2dimlem9**
Step	Hyp	Ref	Expression
1		dia2dimlem9.k	. . . . . . 7 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		dia2dimlem9.h	. . . . . . . 8 ⊢ 𝐻 = (LHyp‘𝐾)
3		dia2dimlem9.y	. . . . . . . 8 ⊢ 𝑌 = ((DVecA‘𝐾)‘𝑊)
4	2, 3	dvalvec 40737	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑌 ∈ LVec)
5		lveclmod 21079	. . . . . . 7 ⊢ (𝑌 ∈ LVec → 𝑌 ∈ LMod)
6		dia2dimlem9.s	. . . . . . . 8 ⊢ 𝑆 = (LSubSp‘𝑌)
7	6	lsssssubg 20930	. . . . . . 7 ⊢ (𝑌 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑌))
8	1, 4, 5, 7	4syl 19	. . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑌))
9		dia2dimlem9.u	. . . . . . . . 9 ⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊))
10	9	simpld 493	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
11		eqid 2726	. . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾)
12		dia2dimlem9.a	. . . . . . . . 9 ⊢ 𝐴 = (Atoms‘𝐾)
13	11, 12	atbase 38999	. . . . . . . 8 ⊢ (𝑈 ∈ 𝐴 → 𝑈 ∈ (Base‘𝐾))
14	10, 13	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾))
15	9	simprd 494	. . . . . . 7 ⊢ (𝜑 → 𝑈 ≤ 𝑊)
16		dia2dimlem9.l	. . . . . . . 8 ⊢ ≤ = (le‘𝐾)
17		dia2dimlem9.i	. . . . . . . 8 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)
18	11, 16, 2, 3, 17, 6	dialss 40757	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ (Base‘𝐾) ∧ 𝑈 ≤ 𝑊)) → (𝐼‘𝑈) ∈ 𝑆)
19	1, 14, 15, 18	syl12anc 835	. . . . . 6 ⊢ (𝜑 → (𝐼‘𝑈) ∈ 𝑆)
20	8, 19	sseldd 3981	. . . . 5 ⊢ (𝜑 → (𝐼‘𝑈) ∈ (SubGrp‘𝑌))
21		dia2dimlem9.v	. . . . . . . . 9 ⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))
22	21	simpld 493	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ 𝐴)
23	11, 12	atbase 38999	. . . . . . . 8 ⊢ (𝑉 ∈ 𝐴 → 𝑉 ∈ (Base‘𝐾))
24	22, 23	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ (Base‘𝐾))
25	21	simprd 494	. . . . . . 7 ⊢ (𝜑 → 𝑉 ≤ 𝑊)
26	11, 16, 2, 3, 17, 6	dialss 40757	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑉 ∈ (Base‘𝐾) ∧ 𝑉 ≤ 𝑊)) → (𝐼‘𝑉) ∈ 𝑆)
27	1, 24, 25, 26	syl12anc 835	. . . . . 6 ⊢ (𝜑 → (𝐼‘𝑉) ∈ 𝑆)
28	8, 27	sseldd 3981	. . . . 5 ⊢ (𝜑 → (𝐼‘𝑉) ∈ (SubGrp‘𝑌))
29		dia2dimlem9.pl	. . . . . 6 ⊢ ⊕ = (LSSum‘𝑌)
30	29	lsmub1 19650	. . . . 5 ⊢ (((𝐼‘𝑈) ∈ (SubGrp‘𝑌) ∧ (𝐼‘𝑉) ∈ (SubGrp‘𝑌)) → (𝐼‘𝑈) ⊆ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))
31	20, 28, 30	syl2anc 582	. . . 4 ⊢ (𝜑 → (𝐼‘𝑈) ⊆ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))
32	31	adantr 479	. . 3 ⊢ ((𝜑 ∧ (𝑅‘𝐹) = 𝑈) → (𝐼‘𝑈) ⊆ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))
33		dia2dimlem9.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝑇)
34		dia2dimlem9.t	. . . . . . 7 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
35		dia2dimlem9.r	. . . . . . 7 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
36	2, 34, 35, 17	dia1dimid 40774	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ (𝐼‘(𝑅‘𝐹)))
37	1, 33, 36	syl2anc 582	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐼‘(𝑅‘𝐹)))
38	37	adantr 479	. . . 4 ⊢ ((𝜑 ∧ (𝑅‘𝐹) = 𝑈) → 𝐹 ∈ (𝐼‘(𝑅‘𝐹)))
39		fveq2 6892	. . . . 5 ⊢ ((𝑅‘𝐹) = 𝑈 → (𝐼‘(𝑅‘𝐹)) = (𝐼‘𝑈))
40	39	adantl 480	. . . 4 ⊢ ((𝜑 ∧ (𝑅‘𝐹) = 𝑈) → (𝐼‘(𝑅‘𝐹)) = (𝐼‘𝑈))
41	38, 40	eleqtrd 2828	. . 3 ⊢ ((𝜑 ∧ (𝑅‘𝐹) = 𝑈) → 𝐹 ∈ (𝐼‘𝑈))
42	32, 41	sseldd 3981	. 2 ⊢ ((𝜑 ∧ (𝑅‘𝐹) = 𝑈) → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))
43	20	adantr 479	. . . 4 ⊢ ((𝜑 ∧ (𝑅‘𝐹) = 𝑉) → (𝐼‘𝑈) ∈ (SubGrp‘𝑌))
44	28	adantr 479	. . . 4 ⊢ ((𝜑 ∧ (𝑅‘𝐹) = 𝑉) → (𝐼‘𝑉) ∈ (SubGrp‘𝑌))
45	29	lsmub2 19651	. . . 4 ⊢ (((𝐼‘𝑈) ∈ (SubGrp‘𝑌) ∧ (𝐼‘𝑉) ∈ (SubGrp‘𝑌)) → (𝐼‘𝑉) ⊆ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))
46	43, 44, 45	syl2anc 582	. . 3 ⊢ ((𝜑 ∧ (𝑅‘𝐹) = 𝑉) → (𝐼‘𝑉) ⊆ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))
47	37	adantr 479	. . . 4 ⊢ ((𝜑 ∧ (𝑅‘𝐹) = 𝑉) → 𝐹 ∈ (𝐼‘(𝑅‘𝐹)))
48		fveq2 6892	. . . . 5 ⊢ ((𝑅‘𝐹) = 𝑉 → (𝐼‘(𝑅‘𝐹)) = (𝐼‘𝑉))
49	48	adantl 480	. . . 4 ⊢ ((𝜑 ∧ (𝑅‘𝐹) = 𝑉) → (𝐼‘(𝑅‘𝐹)) = (𝐼‘𝑉))
50	47, 49	eleqtrd 2828	. . 3 ⊢ ((𝜑 ∧ (𝑅‘𝐹) = 𝑉) → 𝐹 ∈ (𝐼‘𝑉))
51	46, 50	sseldd 3981	. 2 ⊢ ((𝜑 ∧ (𝑅‘𝐹) = 𝑉) → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))
52		dia2dimlem9.j	. . 3 ⊢ ∨ = (join‘𝐾)
53		dia2dimlem9.m	. . 3 ⊢ ∧ = (meet‘𝐾)
54		dia2dimlem9.n	. . 3 ⊢ 𝑁 = (LSpan‘𝑌)
55	1	adantr 479	. . 3 ⊢ ((𝜑 ∧ ((𝑅‘𝐹) ≠ 𝑈 ∧ (𝑅‘𝐹) ≠ 𝑉)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
56	9	adantr 479	. . 3 ⊢ ((𝜑 ∧ ((𝑅‘𝐹) ≠ 𝑈 ∧ (𝑅‘𝐹) ≠ 𝑉)) → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊))
57	21	adantr 479	. . 3 ⊢ ((𝜑 ∧ ((𝑅‘𝐹) ≠ 𝑈 ∧ (𝑅‘𝐹) ≠ 𝑉)) → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))
58	33	adantr 479	. . 3 ⊢ ((𝜑 ∧ ((𝑅‘𝐹) ≠ 𝑈 ∧ (𝑅‘𝐹) ≠ 𝑉)) → 𝐹 ∈ 𝑇)
59		dia2dimlem9.rf	. . . 4 ⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉))
60	59	adantr 479	. . 3 ⊢ ((𝜑 ∧ ((𝑅‘𝐹) ≠ 𝑈 ∧ (𝑅‘𝐹) ≠ 𝑉)) → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉))
61		dia2dimlem9.uv	. . . 4 ⊢ (𝜑 → 𝑈 ≠ 𝑉)
62	61	adantr 479	. . 3 ⊢ ((𝜑 ∧ ((𝑅‘𝐹) ≠ 𝑈 ∧ (𝑅‘𝐹) ≠ 𝑉)) → 𝑈 ≠ 𝑉)
63		simprl 769	. . 3 ⊢ ((𝜑 ∧ ((𝑅‘𝐹) ≠ 𝑈 ∧ (𝑅‘𝐹) ≠ 𝑉)) → (𝑅‘𝐹) ≠ 𝑈)
64		simprr 771	. . 3 ⊢ ((𝜑 ∧ ((𝑅‘𝐹) ≠ 𝑈 ∧ (𝑅‘𝐹) ≠ 𝑉)) → (𝑅‘𝐹) ≠ 𝑉)
65	16, 52, 53, 12, 2, 34, 35, 3, 6, 29, 54, 17, 55, 56, 57, 58, 60, 62, 63, 64	dia2dimlem8 40782	. 2 ⊢ ((𝜑 ∧ ((𝑅‘𝐹) ≠ 𝑈 ∧ (𝑅‘𝐹) ≠ 𝑉)) → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))
66	42, 51, 65	pm2.61da2ne 3020	1 ⊢ (𝜑 → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ⊆ wss 3948 class class class wbr 5145 ‘cfv 6545 (class class class)co 7415 Basecbs 17207 lecple 17267 joincjn 18330 meetcmee 18331 SubGrpcsubg 19109 LSSumclsm 19627 LModclmod 20831 LSubSpclss 20903 LSpanclspn 20943 LVecclvec 21075 Atomscatm 38973 HLchlt 39060 LHypclh 39695 LTrncltrn 39812 trLctrl 39869 DVecAcdveca 40713 DIsoAcdia 40739

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7737 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 ax-riotaBAD 38663

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3365 df-reu 3366 df-rab 3421 df-v 3466 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3968 df-nul 4325 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4908 df-int 4949 df-iun 4997 df-iin 4998 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6497 df-fun 6547 df-fn 6548 df-f 6549 df-f1 6550 df-fo 6551 df-f1o 6552 df-fv 6553 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7994 df-2nd 7995 df-tpos 8232 df-undef 8279 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8848 df-en 8966 df-dom 8967 df-sdom 8968 df-fin 8969 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12258 df-2 12320 df-3 12321 df-4 12322 df-5 12323 df-6 12324 df-n0 12518 df-z 12604 df-uz 12868 df-fz 13532 df-struct 17143 df-sets 17160 df-slot 17178 df-ndx 17190 df-base 17208 df-ress 17237 df-plusg 17273 df-mulr 17274 df-sca 17276 df-vsca 17277 df-0g 17450 df-proset 18314 df-poset 18332 df-plt 18349 df-lub 18365 df-glb 18366 df-join 18367 df-meet 18368 df-p0 18444 df-p1 18445 df-lat 18451 df-clat 18518 df-mgm 18627 df-sgrp 18706 df-mnd 18722 df-submnd 18768 df-grp 18925 df-minusg 18926 df-sbg 18927 df-subg 19112 df-cntz 19306 df-lsm 19629 df-cmn 19775 df-abl 19776 df-mgp 20113 df-rng 20131 df-ur 20160 df-ring 20213 df-oppr 20311 df-dvdsr 20334 df-unit 20335 df-invr 20365 df-dvr 20378 df-drng 20704 df-lmod 20833 df-lss 20904 df-lsp 20944 df-lvec 21076 df-oposet 38886 df-ol 38888 df-oml 38889 df-covers 38976 df-ats 38977 df-atl 39008 df-cvlat 39032 df-hlat 39061 df-llines 39209 df-lplanes 39210 df-lvols 39211 df-lines 39212 df-psubsp 39214 df-pmap 39215 df-padd 39507 df-lhyp 39699 df-laut 39700 df-ldil 39815 df-ltrn 39816 df-trl 39870 df-tgrp 40454 df-tendo 40466 df-edring 40468 df-dveca 40714 df-disoa 40740

This theorem is referenced by: dia2dimlem11 40785

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