hdmapglem7a - Mathbox for Norm Megill

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Theorem hdmapglem7a 42800

Description: Lemma for hdmapg 42803. (Contributed by NM, 14-Jun-2015.)

**Hypotheses**
Ref	Expression
hdmapglem7.h	⊢ 𝐻 = (LHyp‘𝐾)
hdmapglem7.e	⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩
hdmapglem7.o	⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
hdmapglem7.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmapglem7.v	⊢ 𝑉 = (Base‘𝑈)
hdmapglem7.p	⊢ + = (+_g‘𝑈)
hdmapglem7.q	⊢ · = ( ·_𝑠 ‘𝑈)
hdmapglem7.r	⊢ 𝑅 = (Scalar‘𝑈)
hdmapglem7.b	⊢ 𝐵 = (Base‘𝑅)
hdmapglem7.a	⊢ ⊕ = (LSSum‘𝑈)
hdmapglem7.n	⊢ 𝑁 = (LSpan‘𝑈)
hdmapglem7.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
hdmapglem7.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)

**Assertion**
Ref	Expression
hdmapglem7a	⊢ (𝜑 → ∃𝑢 ∈ (𝑂‘{𝐸})∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢))

Distinct variable groups: 𝑢,𝑘, + 𝐵,𝑘,𝑢 𝑘,𝐸,𝑢 𝑘,𝑁,𝑢 𝑘,𝑂,𝑢 · ,𝑘,𝑢 𝑅,𝑘 𝑈,𝑘,𝑢 𝑘,𝑉 𝑘,𝑋,𝑢 𝜑,𝑘,𝑢 Allowed substitution hints: ⊕ (𝑢, 𝑘) 𝑅(𝑢) 𝐻(𝑢, 𝑘) 𝐾(𝑢, 𝑘) 𝑉(𝑢) 𝑊(𝑢, 𝑘)

Proof of Theorem hdmapglem7a Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hdmapglem7.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
2		hdmapglem7.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
3		hdmapglem7.o	. . . . . 6 ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
4		hdmapglem7.u	. . . . . 6 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
5		hdmapglem7.v	. . . . . 6 ⊢ 𝑉 = (Base‘𝑈)
6		eqid 2760	. . . . . 6 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
7		hdmapglem7.a	. . . . . 6 ⊢ ⊕ = (LSSum‘𝑈)
8		hdmapglem7.k	. . . . . 6 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
9	2, 4, 8	dvhlmod 41983	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LMod)
10		eqid 2760	. . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾)
11		eqid 2760	. . . . . . . . 9 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
12		eqid 2760	. . . . . . . . 9 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
13		hdmapglem7.e	. . . . . . . . 9 ⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩
14	2, 10, 11, 4, 5, 12, 13, 8	dvheveccl 41985	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ {(0_g‘𝑈)}))
15	14	eldifad 3911	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ 𝑉)
16		hdmapglem7.n	. . . . . . . 8 ⊢ 𝑁 = (LSpan‘𝑈)
17	5, 6, 16	lspsncl 21161	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝐸 ∈ 𝑉) → (𝑁‘{𝐸}) ∈ (LSubSp‘𝑈))
18	9, 15, 17	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝐸}) ∈ (LSubSp‘𝑈))
19	15	snssd 4747	. . . . . . . . 9 ⊢ (𝜑 → {𝐸} ⊆ 𝑉)
20	2, 4, 3, 5, 16, 8, 19	dochocsp 42252	. . . . . . . 8 ⊢ (𝜑 → (𝑂‘(𝑁‘{𝐸})) = (𝑂‘{𝐸}))
21	20	fveq2d 6882	. . . . . . 7 ⊢ (𝜑 → (𝑂‘(𝑂‘(𝑁‘{𝐸}))) = (𝑂‘(𝑂‘{𝐸})))
22	2, 4, 3, 5, 16, 8, 15	dochocsn 42254	. . . . . . 7 ⊢ (𝜑 → (𝑂‘(𝑂‘{𝐸})) = (𝑁‘{𝐸}))
23	21, 22	eqtrd 2795	. . . . . 6 ⊢ (𝜑 → (𝑂‘(𝑂‘(𝑁‘{𝐸}))) = (𝑁‘{𝐸}))
24	2, 3, 4, 5, 6, 7, 8, 18, 23	dochexmid 42341	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝐸}) ⊕ (𝑂‘(𝑁‘{𝐸}))) = 𝑉)
25	20	oveq2d 7429	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝐸}) ⊕ (𝑂‘(𝑁‘{𝐸}))) = ((𝑁‘{𝐸}) ⊕ (𝑂‘{𝐸})))
26	24, 25	eqtr3d 2797	. . . 4 ⊢ (𝜑 → 𝑉 = ((𝑁‘{𝐸}) ⊕ (𝑂‘{𝐸})))
27	1, 26	eleqtrd 2862	. . 3 ⊢ (𝜑 → 𝑋 ∈ ((𝑁‘{𝐸}) ⊕ (𝑂‘{𝐸})))
28	6	lsssssubg 21142	. . . . . 6 ⊢ (𝑈 ∈ LMod → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
29	9, 28	syl 18	. . . . 5 ⊢ (𝜑 → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
30	29, 18	sseldd 3932	. . . 4 ⊢ (𝜑 → (𝑁‘{𝐸}) ∈ (SubGrp‘𝑈))
31	2, 4, 5, 6, 3	dochlss 42227	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝐸} ⊆ 𝑉) → (𝑂‘{𝐸}) ∈ (LSubSp‘𝑈))
32	8, 19, 31	syl2anc 596	. . . . 5 ⊢ (𝜑 → (𝑂‘{𝐸}) ∈ (LSubSp‘𝑈))
33	29, 32	sseldd 3932	. . . 4 ⊢ (𝜑 → (𝑂‘{𝐸}) ∈ (SubGrp‘𝑈))
34		hdmapglem7.p	. . . . 5 ⊢ + = (+_g‘𝑈)
35	34, 7	lsmelval 19776	. . . 4 ⊢ (((𝑁‘{𝐸}) ∈ (SubGrp‘𝑈) ∧ (𝑂‘{𝐸}) ∈ (SubGrp‘𝑈)) → (𝑋 ∈ ((𝑁‘{𝐸}) ⊕ (𝑂‘{𝐸})) ↔ ∃𝑎 ∈ (𝑁‘{𝐸})∃𝑢 ∈ (𝑂‘{𝐸})𝑋 = (𝑎 + 𝑢)))
36	30, 33, 35	syl2anc 596	. . 3 ⊢ (𝜑 → (𝑋 ∈ ((𝑁‘{𝐸}) ⊕ (𝑂‘{𝐸})) ↔ ∃𝑎 ∈ (𝑁‘{𝐸})∃𝑢 ∈ (𝑂‘{𝐸})𝑋 = (𝑎 + 𝑢)))
37	27, 36	mpbid 235	. 2 ⊢ (𝜑 → ∃𝑎 ∈ (𝑁‘{𝐸})∃𝑢 ∈ (𝑂‘{𝐸})𝑋 = (𝑎 + 𝑢))
38		rexcom 3291	. . 3 ⊢ (∃𝑎 ∈ (𝑁‘{𝐸})∃𝑢 ∈ (𝑂‘{𝐸})𝑋 = (𝑎 + 𝑢) ↔ ∃𝑢 ∈ (𝑂‘{𝐸})∃𝑎 ∈ (𝑁‘{𝐸})𝑋 = (𝑎 + 𝑢))
39		df-rex 3087	. . . . 5 ⊢ (∃𝑎 ∈ (𝑁‘{𝐸})𝑋 = (𝑎 + 𝑢) ↔ ∃𝑎(𝑎 ∈ (𝑁‘{𝐸}) ∧ 𝑋 = (𝑎 + 𝑢)))
40		hdmapglem7.r	. . . . . . . . . . 11 ⊢ 𝑅 = (Scalar‘𝑈)
41		hdmapglem7.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝑅)
42		hdmapglem7.q	. . . . . . . . . . 11 ⊢ · = ( ·_𝑠 ‘𝑈)
43	40, 41, 5, 42, 16	ellspsn 21187	. . . . . . . . . 10 ⊢ ((𝑈 ∈ LMod ∧ 𝐸 ∈ 𝑉) → (𝑎 ∈ (𝑁‘{𝐸}) ↔ ∃𝑘 ∈ 𝐵 𝑎 = (𝑘 · 𝐸)))
44	9, 15, 43	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → (𝑎 ∈ (𝑁‘{𝐸}) ↔ ∃𝑘 ∈ 𝐵 𝑎 = (𝑘 · 𝐸)))
45	44	anbi1d 643	. . . . . . . 8 ⊢ (𝜑 → ((𝑎 ∈ (𝑁‘{𝐸}) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ (∃𝑘 ∈ 𝐵 𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢))))
46		r19.41v 3192	. . . . . . . 8 ⊢ (∃𝑘 ∈ 𝐵 (𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ (∃𝑘 ∈ 𝐵 𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)))
47	45, 46	bitr4di 292	. . . . . . 7 ⊢ (𝜑 → ((𝑎 ∈ (𝑁‘{𝐸}) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ ∃𝑘 ∈ 𝐵 (𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢))))
48	47	exbidv 1954	. . . . . 6 ⊢ (𝜑 → (∃𝑎(𝑎 ∈ (𝑁‘{𝐸}) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ ∃𝑎∃𝑘 ∈ 𝐵 (𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢))))
49		rexcom4 3289	. . . . . . 7 ⊢ (∃𝑘 ∈ 𝐵 ∃𝑎(𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ ∃𝑎∃𝑘 ∈ 𝐵 (𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)))
50		ovex 7446	. . . . . . . . 9 ⊢ (𝑘 · 𝐸) ∈ V
51		oveq1 7420	. . . . . . . . . 10 ⊢ (𝑎 = (𝑘 · 𝐸) → (𝑎 + 𝑢) = ((𝑘 · 𝐸) + 𝑢))
52	51	eqeq2d 2771	. . . . . . . . 9 ⊢ (𝑎 = (𝑘 · 𝐸) → (𝑋 = (𝑎 + 𝑢) ↔ 𝑋 = ((𝑘 · 𝐸) + 𝑢)))
53	50, 52	ceqsexv 3498	. . . . . . . 8 ⊢ (∃𝑎(𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ 𝑋 = ((𝑘 · 𝐸) + 𝑢))
54	53	rexbii 3109	. . . . . . 7 ⊢ (∃𝑘 ∈ 𝐵 ∃𝑎(𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ ∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢))
55	49, 54	bitr3i 280	. . . . . 6 ⊢ (∃𝑎∃𝑘 ∈ 𝐵 (𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ ∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢))
56	48, 55	bitrdi 290	. . . . 5 ⊢ (𝜑 → (∃𝑎(𝑎 ∈ (𝑁‘{𝐸}) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ ∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢)))
57	39, 56	bitrid 286	. . . 4 ⊢ (𝜑 → (∃𝑎 ∈ (𝑁‘{𝐸})𝑋 = (𝑎 + 𝑢) ↔ ∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢)))
58	57	rexbidv 3186	. . 3 ⊢ (𝜑 → (∃𝑢 ∈ (𝑂‘{𝐸})∃𝑎 ∈ (𝑁‘{𝐸})𝑋 = (𝑎 + 𝑢) ↔ ∃𝑢 ∈ (𝑂‘{𝐸})∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢)))
59	38, 58	bitrid 286	. 2 ⊢ (𝜑 → (∃𝑎 ∈ (𝑁‘{𝐸})∃𝑢 ∈ (𝑂‘{𝐸})𝑋 = (𝑎 + 𝑢) ↔ ∃𝑢 ∈ (𝑂‘{𝐸})∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢)))
60	37, 59	mpbid 235	1 ⊢ (𝜑 → ∃𝑢 ∈ (𝑂‘{𝐸})∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∃wex 1812 ∈ wcel 2145 ∃wrex 3086 ⊆ wss 3899 {csn 4584 ⟨cop 4590 I cid 5549 ↾ cres 5657 ‘cfv 6533 (class class class)co 7413 Basecbs 17301 +_gcplusg 17342 Scalarcsca 17345 ·_𝑠 cvsca 17346 0_gc0g 17524 SubGrpcsubg 19243 LSSumclsm 19761 LModclmod 21044 LSubSpclss 21115 LSpanclspn 21155 HLchlt 40223 LHypclh 40857 LTrncltrn 40974 DVecHcdvh 41951 ocHcoch 42220

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-riotaBAD 39826

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7863 df-1st 7986 df-2nd 7987 df-tpos 8224 df-undef 8271 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-er 8696 df-map 8828 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-nn 12258 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-n0 12529 df-z 12616 df-uz 12888 df-fz 13562 df-struct 17239 df-sets 17256 df-slot 17274 df-ndx 17286 df-base 17302 df-ress 17323 df-plusg 17355 df-mulr 17356 df-sca 17358 df-vsca 17359 df-0g 17526 df-mre 17670 df-mrc 17671 df-acs 17673 df-proset 18382 df-poset 18401 df-plt 18416 df-lub 18432 df-glb 18433 df-join 18434 df-meet 18435 df-p0 18511 df-p1 18512 df-lat 18520 df-clat 18587 df-mgm 18730 df-sgrp 18821 df-mnd 18837 df-submnd 18892 df-grp 19060 df-minusg 19061 df-sbg 19062 df-subg 19246 df-cntz 19444 df-oppg 19473 df-lsm 19763 df-cmn 19909 df-abl 19910 df-mgp 20274 df-rng 20288 df-ur 20321 df-ring 20374 df-oppr 20478 df-dvdsr 20498 df-unit 20499 df-invr 20529 df-dvr 20542 df-drng 20892 df-lmod 21046 df-lss 21116 df-lsp 21156 df-lvec 21287 df-lsatoms 39849 df-lcv 39892 df-oposet 40049 df-ol 40051 df-oml 40052 df-covers 40139 df-ats 40140 df-atl 40171 df-cvlat 40195 df-hlat 40224 df-llines 40371 df-lplanes 40372 df-lvols 40373 df-lines 40374 df-psubsp 40376 df-pmap 40377 df-padd 40669 df-lhyp 40861 df-laut 40862 df-ldil 40977 df-ltrn 40978 df-trl 41032 df-tgrp 41616 df-tendo 41628 df-edring 41630 df-dveca 41876 df-disoa 41902 df-dvech 41952 df-dib 42012 df-dic 42046 df-dih 42102 df-doch 42221 df-djh 42268

This theorem is used by: hdmapglem7 42802

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