mapdpglem3 - Mathbox for Norm Megill

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Theorem mapdpglem3 41278

Description: Lemma for mapdpg 41309. Baer p. 45, line 3: "infer ... the existence of a number g in G and of an element z in (Fy)* such that t = gx'-z." (We scope $d 𝑔𝑤𝑧𝜑 locally to avoid clashes with later substitutions into 𝜑.) (Contributed by NM, 18-Mar-2015.)

**Hypotheses**
Ref	Expression
mapdpglem.h	⊢ 𝐻 = (LHyp‘𝐾)
mapdpglem.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
mapdpglem.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
mapdpglem.v	⊢ 𝑉 = (Base‘𝑈)
mapdpglem.s	⊢ − = (-_g‘𝑈)
mapdpglem.n	⊢ 𝑁 = (LSpan‘𝑈)
mapdpglem.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
mapdpglem.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
mapdpglem.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
mapdpglem.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)
mapdpglem1.p	⊢ ⊕ = (LSSum‘𝐶)
mapdpglem2.j	⊢ 𝐽 = (LSpan‘𝐶)
mapdpglem3.f	⊢ 𝐹 = (Base‘𝐶)
mapdpglem3.te	⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌}))))
mapdpglem3.a	⊢ 𝐴 = (Scalar‘𝑈)
mapdpglem3.b	⊢ 𝐵 = (Base‘𝐴)
mapdpglem3.t	⊢ · = ( ·_𝑠 ‘𝐶)
mapdpglem3.r	⊢ 𝑅 = (-_g‘𝐶)
mapdpglem3.g	⊢ (𝜑 → 𝐺 ∈ 𝐹)
mapdpglem3.e	⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))

**Assertion**
Ref	Expression
mapdpglem3	⊢ (𝜑 → ∃𝑔 ∈ 𝐵 ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = ((𝑔 · 𝐺)𝑅𝑧))

Distinct variable groups: 𝑡, − 𝑡,𝐶 𝑡,𝐽 𝑡,𝑀 𝑡,𝑁 𝑡,𝑋 𝑡,𝑌 𝐵,𝑔 𝑧,𝑔,𝐶 𝑔,𝐹 𝑔,𝐺,𝑧 𝑔,𝐽,𝑧 𝑔,𝑀,𝑧 𝑔,𝑁,𝑧 𝑅,𝑔,𝑧 · ,𝑔,𝑧 𝑔,𝑌,𝑧,𝑡 𝜑,𝑔,𝑧 Allowed substitution hints: 𝜑(𝑡) 𝐴(𝑧,𝑡,𝑔) 𝐵(𝑧,𝑡) ⊕ (𝑧,𝑡,𝑔) 𝑅(𝑡) · (𝑡) 𝑈(𝑧,𝑡,𝑔) 𝐹(𝑧,𝑡) 𝐺(𝑡) 𝐻(𝑧,𝑡,𝑔) 𝐾(𝑧,𝑡,𝑔) − (𝑧,𝑔) 𝑉(𝑧,𝑡,𝑔) 𝑊(𝑧,𝑡,𝑔) 𝑋(𝑧,𝑔)

Proof of Theorem mapdpglem3 Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mapdpglem3.te	. . . 4 ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌}))))
2		mapdpglem3.e	. . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))
3	2	oveq1d 7434	. . . 4 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌}))) = ((𝐽‘{𝐺}) ⊕ (𝑀‘(𝑁‘{𝑌}))))
4	1, 3	eleqtrd 2827	. . 3 ⊢ (𝜑 → 𝑡 ∈ ((𝐽‘{𝐺}) ⊕ (𝑀‘(𝑁‘{𝑌}))))
5		r19.41v 3178	. . . . . . 7 ⊢ (∃𝑔 ∈ 𝐵 (𝑤 = (𝑔 · 𝐺) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧)) ↔ (∃𝑔 ∈ 𝐵 𝑤 = (𝑔 · 𝐺) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧)))
6		mapdpglem.h	. . . . . . . . . . 11 ⊢ 𝐻 = (LHyp‘𝐾)
7		mapdpglem.c	. . . . . . . . . . 11 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
8		mapdpglem.k	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
9	6, 7, 8	lcdlmod 41195	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ LMod)
10		mapdpglem3.g	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
11		eqid 2725	. . . . . . . . . . 11 ⊢ (Scalar‘𝐶) = (Scalar‘𝐶)
12		eqid 2725	. . . . . . . . . . 11 ⊢ (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶))
13		mapdpglem3.f	. . . . . . . . . . 11 ⊢ 𝐹 = (Base‘𝐶)
14		mapdpglem3.t	. . . . . . . . . . 11 ⊢ · = ( ·_𝑠 ‘𝐶)
15		mapdpglem2.j	. . . . . . . . . . 11 ⊢ 𝐽 = (LSpan‘𝐶)
16	11, 12, 13, 14, 15	lspsnel 20899	. . . . . . . . . 10 ⊢ ((𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (𝑤 ∈ (𝐽‘{𝐺}) ↔ ∃𝑔 ∈ (Base‘(Scalar‘𝐶))𝑤 = (𝑔 · 𝐺)))
17	9, 10, 16	syl2anc 582	. . . . . . . . 9 ⊢ (𝜑 → (𝑤 ∈ (𝐽‘{𝐺}) ↔ ∃𝑔 ∈ (Base‘(Scalar‘𝐶))𝑤 = (𝑔 · 𝐺)))
18		mapdpglem.u	. . . . . . . . . . 11 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
19		mapdpglem3.a	. . . . . . . . . . 11 ⊢ 𝐴 = (Scalar‘𝑈)
20		mapdpglem3.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝐴)
21	6, 18, 19, 20, 7, 11, 12, 8	lcdsbase 41203	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘(Scalar‘𝐶)) = 𝐵)
22	21	rexeqdv 3315	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑔 ∈ (Base‘(Scalar‘𝐶))𝑤 = (𝑔 · 𝐺) ↔ ∃𝑔 ∈ 𝐵 𝑤 = (𝑔 · 𝐺)))
23	17, 22	bitrd 278	. . . . . . . 8 ⊢ (𝜑 → (𝑤 ∈ (𝐽‘{𝐺}) ↔ ∃𝑔 ∈ 𝐵 𝑤 = (𝑔 · 𝐺)))
24	23	anbi1d 629	. . . . . . 7 ⊢ (𝜑 → ((𝑤 ∈ (𝐽‘{𝐺}) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧)) ↔ (∃𝑔 ∈ 𝐵 𝑤 = (𝑔 · 𝐺) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧))))
25	5, 24	bitr4id 289	. . . . . 6 ⊢ (𝜑 → (∃𝑔 ∈ 𝐵 (𝑤 = (𝑔 · 𝐺) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧)) ↔ (𝑤 ∈ (𝐽‘{𝐺}) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧))))
26	25	exbidv 1916	. . . . 5 ⊢ (𝜑 → (∃𝑤∃𝑔 ∈ 𝐵 (𝑤 = (𝑔 · 𝐺) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧)) ↔ ∃𝑤(𝑤 ∈ (𝐽‘{𝐺}) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧))))
27		df-rex 3060	. . . . 5 ⊢ (∃𝑤 ∈ (𝐽‘{𝐺})∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧) ↔ ∃𝑤(𝑤 ∈ (𝐽‘{𝐺}) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧)))
28	26, 27	bitr4di 288	. . . 4 ⊢ (𝜑 → (∃𝑤∃𝑔 ∈ 𝐵 (𝑤 = (𝑔 · 𝐺) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧)) ↔ ∃𝑤 ∈ (𝐽‘{𝐺})∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧)))
29		mapdpglem3.r	. . . . 5 ⊢ 𝑅 = (-_g‘𝐶)
30		mapdpglem1.p	. . . . 5 ⊢ ⊕ = (LSSum‘𝐶)
31		eqid 2725	. . . . . . . 8 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶)
32	31	lsssssubg 20854	. . . . . . 7 ⊢ (𝐶 ∈ LMod → (LSubSp‘𝐶) ⊆ (SubGrp‘𝐶))
33	9, 32	syl 17	. . . . . 6 ⊢ (𝜑 → (LSubSp‘𝐶) ⊆ (SubGrp‘𝐶))
34	13, 31, 15	lspsncl 20873	. . . . . . 7 ⊢ ((𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (𝐽‘{𝐺}) ∈ (LSubSp‘𝐶))
35	9, 10, 34	syl2anc 582	. . . . . 6 ⊢ (𝜑 → (𝐽‘{𝐺}) ∈ (LSubSp‘𝐶))
36	33, 35	sseldd 3977	. . . . 5 ⊢ (𝜑 → (𝐽‘{𝐺}) ∈ (SubGrp‘𝐶))
37		mapdpglem.m	. . . . . . 7 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
38		eqid 2725	. . . . . . 7 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
39	6, 18, 8	dvhlmod 40713	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod)
40		mapdpglem.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
41		mapdpglem.v	. . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑈)
42		mapdpglem.n	. . . . . . . . 9 ⊢ 𝑁 = (LSpan‘𝑈)
43	41, 38, 42	lspsncl 20873	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
44	39, 40, 43	syl2anc 582	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
45	6, 37, 18, 38, 7, 31, 8, 44	mapdcl2 41259	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) ∈ (LSubSp‘𝐶))
46	33, 45	sseldd 3977	. . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) ∈ (SubGrp‘𝐶))
47	29, 30, 36, 46	lsmelvalm 19618	. . . 4 ⊢ (𝜑 → (𝑡 ∈ ((𝐽‘{𝐺}) ⊕ (𝑀‘(𝑁‘{𝑌}))) ↔ ∃𝑤 ∈ (𝐽‘{𝐺})∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧)))
48	28, 47	bitr4d 281	. . 3 ⊢ (𝜑 → (∃𝑤∃𝑔 ∈ 𝐵 (𝑤 = (𝑔 · 𝐺) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧)) ↔ 𝑡 ∈ ((𝐽‘{𝐺}) ⊕ (𝑀‘(𝑁‘{𝑌})))))
49	4, 48	mpbird 256	. 2 ⊢ (𝜑 → ∃𝑤∃𝑔 ∈ 𝐵 (𝑤 = (𝑔 · 𝐺) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧)))
50		ovex 7452	. . . . 5 ⊢ (𝑔 · 𝐺) ∈ V
51		oveq1 7426	. . . . . . 7 ⊢ (𝑤 = (𝑔 · 𝐺) → (𝑤𝑅𝑧) = ((𝑔 · 𝐺)𝑅𝑧))
52	51	eqeq2d 2736	. . . . . 6 ⊢ (𝑤 = (𝑔 · 𝐺) → (𝑡 = (𝑤𝑅𝑧) ↔ 𝑡 = ((𝑔 · 𝐺)𝑅𝑧)))
53	52	rexbidv 3168	. . . . 5 ⊢ (𝑤 = (𝑔 · 𝐺) → (∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧) ↔ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = ((𝑔 · 𝐺)𝑅𝑧)))
54	50, 53	ceqsexv 3514	. . . 4 ⊢ (∃𝑤(𝑤 = (𝑔 · 𝐺) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧)) ↔ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = ((𝑔 · 𝐺)𝑅𝑧))
55	54	rexbii 3083	. . 3 ⊢ (∃𝑔 ∈ 𝐵 ∃𝑤(𝑤 = (𝑔 · 𝐺) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧)) ↔ ∃𝑔 ∈ 𝐵 ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = ((𝑔 · 𝐺)𝑅𝑧))
56		rexcom4 3275	. . 3 ⊢ (∃𝑔 ∈ 𝐵 ∃𝑤(𝑤 = (𝑔 · 𝐺) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧)) ↔ ∃𝑤∃𝑔 ∈ 𝐵 (𝑤 = (𝑔 · 𝐺) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧)))
57	55, 56	bitr3i 276	. 2 ⊢ (∃𝑔 ∈ 𝐵 ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = ((𝑔 · 𝐺)𝑅𝑧) ↔ ∃𝑤∃𝑔 ∈ 𝐵 (𝑤 = (𝑔 · 𝐺) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧)))
58	49, 57	sylibr 233	1 ⊢ (𝜑 → ∃𝑔 ∈ 𝐵 ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = ((𝑔 · 𝐺)𝑅𝑧))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ∃wrex 3059 ⊆ wss 3944 {csn 4630 ‘cfv 6549 (class class class)co 7419 Basecbs 17183 Scalarcsca 17239 ·_𝑠 cvsca 17240 -_gcsg 18900 SubGrpcsubg 19083 LSSumclsm 19601 LModclmod 20755 LSubSpclss 20827 LSpanclspn 20867 HLchlt 38952 LHypclh 39587 DVecHcdvh 40681 LCDualclcd 41189 mapdcmpd 41227

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-riotaBAD 38555

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 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 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-n0 12506 df-z 12592 df-uz 12856 df-fz 13520 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-sca 17252 df-vsca 17253 df-0g 17426 df-mre 17569 df-mrc 17570 df-acs 17572 df-proset 18290 df-poset 18308 df-plt 18325 df-lub 18341 df-glb 18342 df-join 18343 df-meet 18344 df-p0 18420 df-p1 18421 df-lat 18427 df-clat 18494 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18744 df-grp 18901 df-minusg 18902 df-sbg 18903 df-subg 19086 df-cntz 19280 df-oppg 19309 df-lsm 19603 df-cmn 19749 df-abl 19750 df-mgp 20087 df-rng 20105 df-ur 20134 df-ring 20187 df-oppr 20285 df-dvdsr 20308 df-unit 20309 df-invr 20339 df-dvr 20352 df-drng 20638 df-lmod 20757 df-lss 20828 df-lsp 20868 df-lvec 21000 df-lsatoms 38578 df-lshyp 38579 df-lcv 38621 df-lfl 38660 df-lkr 38688 df-ldual 38726 df-oposet 38778 df-ol 38780 df-oml 38781 df-covers 38868 df-ats 38869 df-atl 38900 df-cvlat 38924 df-hlat 38953 df-llines 39101 df-lplanes 39102 df-lvols 39103 df-lines 39104 df-psubsp 39106 df-pmap 39107 df-padd 39399 df-lhyp 39591 df-laut 39592 df-ldil 39707 df-ltrn 39708 df-trl 39762 df-tgrp 40346 df-tendo 40358 df-edring 40360 df-dveca 40606 df-disoa 40632 df-dvech 40682 df-dib 40742 df-dic 40776 df-dih 40832 df-doch 40951 df-djh 40998 df-lcdual 41190 df-mapd 41228

This theorem is referenced by: mapdpglem24 41307

W3C validator