dia2dimlem9 - Mathbox for Norm Megill

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Theorem dia2dimlem9 41074

Description: Lemma for dia2dim 41079. 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 41028	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑌 ∈ LVec)
5		lveclmod 21105	. . . . . . 7 ⊢ (𝑌 ∈ LVec → 𝑌 ∈ LMod)
6		dia2dimlem9.s	. . . . . . . 8 ⊢ 𝑆 = (LSubSp‘𝑌)
7	6	lsssssubg 20956	. . . . . . 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 494	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
11		eqid 2737	. . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾)
12		dia2dimlem9.a	. . . . . . . . 9 ⊢ 𝐴 = (Atoms‘𝐾)
13	11, 12	atbase 39290	. . . . . . . 8 ⊢ (𝑈 ∈ 𝐴 → 𝑈 ∈ (Base‘𝐾))
14	10, 13	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾))
15	9	simprd 495	. . . . . . 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 41048	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ (Base‘𝐾) ∧ 𝑈 ≤ 𝑊)) → (𝐼‘𝑈) ∈ 𝑆)
19	1, 14, 15, 18	syl12anc 837	. . . . . 6 ⊢ (𝜑 → (𝐼‘𝑈) ∈ 𝑆)
20	8, 19	sseldd 3984	. . . . 5 ⊢ (𝜑 → (𝐼‘𝑈) ∈ (SubGrp‘𝑌))
21		dia2dimlem9.v	. . . . . . . . 9 ⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))
22	21	simpld 494	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ 𝐴)
23	11, 12	atbase 39290	. . . . . . . 8 ⊢ (𝑉 ∈ 𝐴 → 𝑉 ∈ (Base‘𝐾))
24	22, 23	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ (Base‘𝐾))
25	21	simprd 495	. . . . . . 7 ⊢ (𝜑 → 𝑉 ≤ 𝑊)
26	11, 16, 2, 3, 17, 6	dialss 41048	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑉 ∈ (Base‘𝐾) ∧ 𝑉 ≤ 𝑊)) → (𝐼‘𝑉) ∈ 𝑆)
27	1, 24, 25, 26	syl12anc 837	. . . . . 6 ⊢ (𝜑 → (𝐼‘𝑉) ∈ 𝑆)
28	8, 27	sseldd 3984	. . . . 5 ⊢ (𝜑 → (𝐼‘𝑉) ∈ (SubGrp‘𝑌))
29		dia2dimlem9.pl	. . . . . 6 ⊢ ⊕ = (LSSum‘𝑌)
30	29	lsmub1 19675	. . . . 5 ⊢ (((𝐼‘𝑈) ∈ (SubGrp‘𝑌) ∧ (𝐼‘𝑉) ∈ (SubGrp‘𝑌)) → (𝐼‘𝑈) ⊆ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))
31	20, 28, 30	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐼‘𝑈) ⊆ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))
32	31	adantr 480	. . 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 41065	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ (𝐼‘(𝑅‘𝐹)))
37	1, 33, 36	syl2anc 584	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐼‘(𝑅‘𝐹)))
38	37	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑅‘𝐹) = 𝑈) → 𝐹 ∈ (𝐼‘(𝑅‘𝐹)))
39		fveq2 6906	. . . . 5 ⊢ ((𝑅‘𝐹) = 𝑈 → (𝐼‘(𝑅‘𝐹)) = (𝐼‘𝑈))
40	39	adantl 481	. . . 4 ⊢ ((𝜑 ∧ (𝑅‘𝐹) = 𝑈) → (𝐼‘(𝑅‘𝐹)) = (𝐼‘𝑈))
41	38, 40	eleqtrd 2843	. . 3 ⊢ ((𝜑 ∧ (𝑅‘𝐹) = 𝑈) → 𝐹 ∈ (𝐼‘𝑈))
42	32, 41	sseldd 3984	. 2 ⊢ ((𝜑 ∧ (𝑅‘𝐹) = 𝑈) → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))
43	20	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑅‘𝐹) = 𝑉) → (𝐼‘𝑈) ∈ (SubGrp‘𝑌))
44	28	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑅‘𝐹) = 𝑉) → (𝐼‘𝑉) ∈ (SubGrp‘𝑌))
45	29	lsmub2 19676	. . . 4 ⊢ (((𝐼‘𝑈) ∈ (SubGrp‘𝑌) ∧ (𝐼‘𝑉) ∈ (SubGrp‘𝑌)) → (𝐼‘𝑉) ⊆ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))
46	43, 44, 45	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ (𝑅‘𝐹) = 𝑉) → (𝐼‘𝑉) ⊆ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))
47	37	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑅‘𝐹) = 𝑉) → 𝐹 ∈ (𝐼‘(𝑅‘𝐹)))
48		fveq2 6906	. . . . 5 ⊢ ((𝑅‘𝐹) = 𝑉 → (𝐼‘(𝑅‘𝐹)) = (𝐼‘𝑉))
49	48	adantl 481	. . . 4 ⊢ ((𝜑 ∧ (𝑅‘𝐹) = 𝑉) → (𝐼‘(𝑅‘𝐹)) = (𝐼‘𝑉))
50	47, 49	eleqtrd 2843	. . 3 ⊢ ((𝜑 ∧ (𝑅‘𝐹) = 𝑉) → 𝐹 ∈ (𝐼‘𝑉))
51	46, 50	sseldd 3984	. 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 480	. . 3 ⊢ ((𝜑 ∧ ((𝑅‘𝐹) ≠ 𝑈 ∧ (𝑅‘𝐹) ≠ 𝑉)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
56	9	adantr 480	. . 3 ⊢ ((𝜑 ∧ ((𝑅‘𝐹) ≠ 𝑈 ∧ (𝑅‘𝐹) ≠ 𝑉)) → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊))
57	21	adantr 480	. . 3 ⊢ ((𝜑 ∧ ((𝑅‘𝐹) ≠ 𝑈 ∧ (𝑅‘𝐹) ≠ 𝑉)) → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))
58	33	adantr 480	. . 3 ⊢ ((𝜑 ∧ ((𝑅‘𝐹) ≠ 𝑈 ∧ (𝑅‘𝐹) ≠ 𝑉)) → 𝐹 ∈ 𝑇)
59		dia2dimlem9.rf	. . . 4 ⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉))
60	59	adantr 480	. . 3 ⊢ ((𝜑 ∧ ((𝑅‘𝐹) ≠ 𝑈 ∧ (𝑅‘𝐹) ≠ 𝑉)) → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉))
61		dia2dimlem9.uv	. . . 4 ⊢ (𝜑 → 𝑈 ≠ 𝑉)
62	61	adantr 480	. . 3 ⊢ ((𝜑 ∧ ((𝑅‘𝐹) ≠ 𝑈 ∧ (𝑅‘𝐹) ≠ 𝑉)) → 𝑈 ≠ 𝑉)
63		simprl 771	. . 3 ⊢ ((𝜑 ∧ ((𝑅‘𝐹) ≠ 𝑈 ∧ (𝑅‘𝐹) ≠ 𝑉)) → (𝑅‘𝐹) ≠ 𝑈)
64		simprr 773	. . 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 41073	. 2 ⊢ ((𝜑 ∧ ((𝑅‘𝐹) ≠ 𝑈 ∧ (𝑅‘𝐹) ≠ 𝑉)) → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))
66	42, 51, 65	pm2.61da2ne 3030	1 ⊢ (𝜑 → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ⊆ wss 3951 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 lecple 17304 joincjn 18357 meetcmee 18358 SubGrpcsubg 19138 LSSumclsm 19652 LModclmod 20858 LSubSpclss 20929 LSpanclspn 20969 LVecclvec 21101 Atomscatm 39264 HLchlt 39351 LHypclh 39986 LTrncltrn 40103 trLctrl 40160 DVecAcdveca 41004 DIsoAcdia 41030

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-riotaBAD 38954

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-undef 8298 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-0g 17486 df-proset 18340 df-poset 18359 df-plt 18375 df-lub 18391 df-glb 18392 df-join 18393 df-meet 18394 df-p0 18470 df-p1 18471 df-lat 18477 df-clat 18544 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-cntz 19335 df-lsm 19654 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-dvr 20401 df-drng 20731 df-lmod 20860 df-lss 20930 df-lsp 20970 df-lvec 21102 df-oposet 39177 df-ol 39179 df-oml 39180 df-covers 39267 df-ats 39268 df-atl 39299 df-cvlat 39323 df-hlat 39352 df-llines 39500 df-lplanes 39501 df-lvols 39502 df-lines 39503 df-psubsp 39505 df-pmap 39506 df-padd 39798 df-lhyp 39990 df-laut 39991 df-ldil 40106 df-ltrn 40107 df-trl 40161 df-tgrp 40745 df-tendo 40757 df-edring 40759 df-dveca 41005 df-disoa 41031

This theorem is referenced by: dia2dimlem11 41076

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