hdmapglem7a - Mathbox for Norm Megill

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Theorem hdmapglem7a 42734

Description: Lemma for hdmapg 42737. (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 2765	. . . . . 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 41917	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LMod)
10		eqid 2765	. . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾)
11		eqid 2765	. . . . . . . . 9 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
12		eqid 2765	. . . . . . . . 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 41919	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ {(0_g‘𝑈)}))
15	14	eldifad 3918	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ 𝑉)
16		hdmapglem7.n	. . . . . . . 8 ⊢ 𝑁 = (LSpan‘𝑈)
17	5, 6, 16	lspsncl 21128	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝐸 ∈ 𝑉) → (𝑁‘{𝐸}) ∈ (LSubSp‘𝑈))
18	9, 15, 17	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝐸}) ∈ (LSubSp‘𝑈))
19	15	snssd 4754	. . . . . . . . 9 ⊢ (𝜑 → {𝐸} ⊆ 𝑉)
20	2, 4, 3, 5, 16, 8, 19	dochocsp 42186	. . . . . . . 8 ⊢ (𝜑 → (𝑂‘(𝑁‘{𝐸})) = (𝑂‘{𝐸}))
21	20	fveq2d 6889	. . . . . . 7 ⊢ (𝜑 → (𝑂‘(𝑂‘(𝑁‘{𝐸}))) = (𝑂‘(𝑂‘{𝐸})))
22	2, 4, 3, 5, 16, 8, 15	dochocsn 42188	. . . . . . 7 ⊢ (𝜑 → (𝑂‘(𝑂‘{𝐸})) = (𝑁‘{𝐸}))
23	21, 22	eqtrd 2800	. . . . . 6 ⊢ (𝜑 → (𝑂‘(𝑂‘(𝑁‘{𝐸}))) = (𝑁‘{𝐸}))
24	2, 3, 4, 5, 6, 7, 8, 18, 23	dochexmid 42275	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝐸}) ⊕ (𝑂‘(𝑁‘{𝐸}))) = 𝑉)
25	20	oveq2d 7432	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝐸}) ⊕ (𝑂‘(𝑁‘{𝐸}))) = ((𝑁‘{𝐸}) ⊕ (𝑂‘{𝐸})))
26	24, 25	eqtr3d 2802	. . . 4 ⊢ (𝜑 → 𝑉 = ((𝑁‘{𝐸}) ⊕ (𝑂‘{𝐸})))
27	1, 26	eleqtrd 2867	. . 3 ⊢ (𝜑 → 𝑋 ∈ ((𝑁‘{𝐸}) ⊕ (𝑂‘{𝐸})))
28	6	lsssssubg 21109	. . . . . 6 ⊢ (𝑈 ∈ LMod → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
29	9, 28	syl 18	. . . . 5 ⊢ (𝜑 → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
30	29, 18	sseldd 3939	. . . 4 ⊢ (𝜑 → (𝑁‘{𝐸}) ∈ (SubGrp‘𝑈))
31	2, 4, 5, 6, 3	dochlss 42161	. . . . . 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 3939	. . . 4 ⊢ (𝜑 → (𝑂‘{𝐸}) ∈ (SubGrp‘𝑈))
34		hdmapglem7.p	. . . . 5 ⊢ + = (+_g‘𝑈)
35	34, 7	lsmelval 19743	. . . 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 3296	. . 3 ⊢ (∃𝑎 ∈ (𝑁‘{𝐸})∃𝑢 ∈ (𝑂‘{𝐸})𝑋 = (𝑎 + 𝑢) ↔ ∃𝑢 ∈ (𝑂‘{𝐸})∃𝑎 ∈ (𝑁‘{𝐸})𝑋 = (𝑎 + 𝑢))
39		df-rex 3092	. . . . 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 21154	. . . . . . . . . 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 3197	. . . . . . . 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 3294	. . . . . . 7 ⊢ (∃𝑘 ∈ 𝐵 ∃𝑎(𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ ∃𝑎∃𝑘 ∈ 𝐵 (𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)))
50		ovex 7449	. . . . . . . . 9 ⊢ (𝑘 · 𝐸) ∈ V
51		oveq1 7423	. . . . . . . . . 10 ⊢ (𝑎 = (𝑘 · 𝐸) → (𝑎 + 𝑢) = ((𝑘 · 𝐸) + 𝑢))
52	51	eqeq2d 2776	. . . . . . . . 9 ⊢ (𝑎 = (𝑘 · 𝐸) → (𝑋 = (𝑎 + 𝑢) ↔ 𝑋 = ((𝑘 · 𝐸) + 𝑢)))
53	50, 52	ceqsexv 3505	. . . . . . . 8 ⊢ (∃𝑎(𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ 𝑋 = ((𝑘 · 𝐸) + 𝑢))
54	53	rexbii 3114	. . . . . . 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 3191	. . 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 2146 ∃wrex 3091 ⊆ wss 3906 {csn 4591 ⟨cop 4597 I cid 5557 ↾ cres 5665 ‘cfv 6540 (class class class)co 7416 Basecbs 17287 +_gcplusg 17328 Scalarcsca 17331 ·_𝑠 cvsca 17332 0_gc0g 17510 SubGrpcsubg 19210 LSSumclsm 19728 LModclmod 21011 LSubSpclss 21082 LSpanclspn 21122 HLchlt 40157 LHypclh 40791 LTrncltrn 40908 DVecHcdvh 41885 ocHcoch 42154

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7738 ax-cnex 11167 ax-resscn 11168 ax-1cn 11169 ax-icn 11170 ax-addcl 11171 ax-addrcl 11172 ax-mulcl 11173 ax-mulrcl 11174 ax-mulcom 11175 ax-addass 11176 ax-mulass 11177 ax-distr 11178 ax-i2m1 11179 ax-1ne0 11180 ax-1rid 11181 ax-rnegex 11182 ax-rrecex 11183 ax-cnre 11184 ax-pre-lttri 11185 ax-pre-lttrn 11186 ax-pre-ltadd 11187 ax-pre-mulgt0 11188 ax-riotaBAD 39760

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 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7865 df-1st 7988 df-2nd 7989 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 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11454 df-neg 11455 df-nn 12245 df-2 12314 df-3 12315 df-4 12316 df-5 12317 df-6 12318 df-n0 12516 df-z 12603 df-uz 12875 df-fz 13548 df-struct 17225 df-sets 17242 df-slot 17260 df-ndx 17272 df-base 17288 df-ress 17309 df-plusg 17341 df-mulr 17342 df-sca 17344 df-vsca 17345 df-0g 17512 df-mre 17656 df-mrc 17657 df-acs 17659 df-proset 18368 df-poset 18387 df-plt 18402 df-lub 18418 df-glb 18419 df-join 18420 df-meet 18421 df-p0 18497 df-p1 18498 df-lat 18506 df-clat 18573 df-mgm 18716 df-sgrp 18799 df-mnd 18815 df-submnd 18866 df-grp 19027 df-minusg 19028 df-sbg 19029 df-subg 19213 df-cntz 19411 df-oppg 19440 df-lsm 19730 df-cmn 19876 df-abl 19877 df-mgp 20241 df-rng 20255 df-ur 20288 df-ring 20341 df-oppr 20445 df-dvdsr 20465 df-unit 20466 df-invr 20496 df-dvr 20509 df-drng 20859 df-lmod 21013 df-lss 21083 df-lsp 21123 df-lvec 21254 df-lsatoms 39783 df-lcv 39826 df-oposet 39983 df-ol 39985 df-oml 39986 df-covers 40073 df-ats 40074 df-atl 40105 df-cvlat 40129 df-hlat 40158 df-llines 40305 df-lplanes 40306 df-lvols 40307 df-lines 40308 df-psubsp 40310 df-pmap 40311 df-padd 40603 df-lhyp 40795 df-laut 40796 df-ldil 40911 df-ltrn 40912 df-trl 40966 df-tgrp 41550 df-tendo 41562 df-edring 41564 df-dveca 41810 df-disoa 41836 df-dvech 41886 df-dib 41946 df-dic 41980 df-dih 42036 df-doch 42155 df-djh 42202

This theorem is used by: hdmapglem7 42736

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