mapdpglem3 - Mathbox for Norm Megill

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Theorem mapdpglem3 41713

Description: Lemma for mapdpg 41744. 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 7361	. . . 4 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌}))) = ((𝐽‘{𝐺}) ⊕ (𝑀‘(𝑁‘{𝑌}))))
4	1, 3	eleqtrd 2833	. . 3 ⊢ (𝜑 → 𝑡 ∈ ((𝐽‘{𝐺}) ⊕ (𝑀‘(𝑁‘{𝑌}))))
5		r19.41v 3162	. . . . . . 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 41630	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ LMod)
10		mapdpglem3.g	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
11		eqid 2731	. . . . . . . . . . 11 ⊢ (Scalar‘𝐶) = (Scalar‘𝐶)
12		eqid 2731	. . . . . . . . . . 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	ellspsn 20934	. . . . . . . . . 10 ⊢ ((𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (𝑤 ∈ (𝐽‘{𝐺}) ↔ ∃𝑔 ∈ (Base‘(Scalar‘𝐶))𝑤 = (𝑔 · 𝐺)))
17	9, 10, 16	syl2anc 584	. . . . . . . . 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 41638	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘(Scalar‘𝐶)) = 𝐵)
22	21	rexeqdv 3293	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑔 ∈ (Base‘(Scalar‘𝐶))𝑤 = (𝑔 · 𝐺) ↔ ∃𝑔 ∈ 𝐵 𝑤 = (𝑔 · 𝐺)))
23	17, 22	bitrd 279	. . . . . . . 8 ⊢ (𝜑 → (𝑤 ∈ (𝐽‘{𝐺}) ↔ ∃𝑔 ∈ 𝐵 𝑤 = (𝑔 · 𝐺)))
24	23	anbi1d 631	. . . . . . 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 290	. . . . . 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 1922	. . . . 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 3057	. . . . 5 ⊢ (∃𝑤 ∈ (𝐽‘{𝐺})∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧) ↔ ∃𝑤(𝑤 ∈ (𝐽‘{𝐺}) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧)))
28	26, 27	bitr4di 289	. . . 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 2731	. . . . . . . 8 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶)
32	31	lsssssubg 20889	. . . . . . 7 ⊢ (𝐶 ∈ LMod → (LSubSp‘𝐶) ⊆ (SubGrp‘𝐶))
33	9, 32	syl 17	. . . . . 6 ⊢ (𝜑 → (LSubSp‘𝐶) ⊆ (SubGrp‘𝐶))
34	13, 31, 15	lspsncl 20908	. . . . . . 7 ⊢ ((𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (𝐽‘{𝐺}) ∈ (LSubSp‘𝐶))
35	9, 10, 34	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝐽‘{𝐺}) ∈ (LSubSp‘𝐶))
36	33, 35	sseldd 3935	. . . . 5 ⊢ (𝜑 → (𝐽‘{𝐺}) ∈ (SubGrp‘𝐶))
37		mapdpglem.m	. . . . . . 7 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
38		eqid 2731	. . . . . . 7 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
39	6, 18, 8	dvhlmod 41148	. . . . . . . 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 20908	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
44	39, 40, 43	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
45	6, 37, 18, 38, 7, 31, 8, 44	mapdcl2 41694	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) ∈ (LSubSp‘𝐶))
46	33, 45	sseldd 3935	. . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) ∈ (SubGrp‘𝐶))
47	29, 30, 36, 46	lsmelvalm 19561	. . . 4 ⊢ (𝜑 → (𝑡 ∈ ((𝐽‘{𝐺}) ⊕ (𝑀‘(𝑁‘{𝑌}))) ↔ ∃𝑤 ∈ (𝐽‘{𝐺})∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧)))
48	28, 47	bitr4d 282	. . 3 ⊢ (𝜑 → (∃𝑤∃𝑔 ∈ 𝐵 (𝑤 = (𝑔 · 𝐺) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧)) ↔ 𝑡 ∈ ((𝐽‘{𝐺}) ⊕ (𝑀‘(𝑁‘{𝑌})))))
49	4, 48	mpbird 257	. 2 ⊢ (𝜑 → ∃𝑤∃𝑔 ∈ 𝐵 (𝑤 = (𝑔 · 𝐺) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧)))
50		ovex 7379	. . . . 5 ⊢ (𝑔 · 𝐺) ∈ V
51		oveq1 7353	. . . . . . 7 ⊢ (𝑤 = (𝑔 · 𝐺) → (𝑤𝑅𝑧) = ((𝑔 · 𝐺)𝑅𝑧))
52	51	eqeq2d 2742	. . . . . 6 ⊢ (𝑤 = (𝑔 · 𝐺) → (𝑡 = (𝑤𝑅𝑧) ↔ 𝑡 = ((𝑔 · 𝐺)𝑅𝑧)))
53	52	rexbidv 3156	. . . . 5 ⊢ (𝑤 = (𝑔 · 𝐺) → (∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧) ↔ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = ((𝑔 · 𝐺)𝑅𝑧)))
54	50, 53	ceqsexv 3487	. . . 4 ⊢ (∃𝑤(𝑤 = (𝑔 · 𝐺) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧)) ↔ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = ((𝑔 · 𝐺)𝑅𝑧))
55	54	rexbii 3079	. . 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 3259	. . 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 277	. 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 234	1 ⊢ (𝜑 → ∃𝑔 ∈ 𝐵 ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = ((𝑔 · 𝐺)𝑅𝑧))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∃wex 1780 ∈ wcel 2111 ∃wrex 3056 ⊆ wss 3902 {csn 4576 ‘cfv 6481 (class class class)co 7346 Basecbs 17117 Scalarcsca 17161 ·_𝑠 cvsca 17162 -_gcsg 18845 SubGrpcsubg 19030 LSSumclsm 19544 LModclmod 20791 LSubSpclss 20862 LSpanclspn 20902 HLchlt 39388 LHypclh 40022 DVecHcdvh 41116 LCDualclcd 41624 mapdcmpd 41662

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-cnex 11059 ax-resscn 11060 ax-1cn 11061 ax-icn 11062 ax-addcl 11063 ax-addrcl 11064 ax-mulcl 11065 ax-mulrcl 11066 ax-mulcom 11067 ax-addass 11068 ax-mulass 11069 ax-distr 11070 ax-i2m1 11071 ax-1ne0 11072 ax-1rid 11073 ax-rnegex 11074 ax-rrecex 11075 ax-cnre 11076 ax-pre-lttri 11077 ax-pre-lttrn 11078 ax-pre-ltadd 11079 ax-pre-mulgt0 11080 ax-riotaBAD 38991

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-uni 4860 df-int 4898 df-iun 4943 df-iin 4944 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-tpos 8156 df-undef 8203 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-pnf 11145 df-mnf 11146 df-xr 11147 df-ltxr 11148 df-le 11149 df-sub 11343 df-neg 11344 df-nn 12123 df-2 12185 df-3 12186 df-4 12187 df-5 12188 df-6 12189 df-n0 12379 df-z 12466 df-uz 12730 df-fz 13405 df-struct 17055 df-sets 17072 df-slot 17090 df-ndx 17102 df-base 17118 df-ress 17139 df-plusg 17171 df-mulr 17172 df-sca 17174 df-vsca 17175 df-0g 17342 df-mre 17485 df-mrc 17486 df-acs 17488 df-proset 18197 df-poset 18216 df-plt 18231 df-lub 18247 df-glb 18248 df-join 18249 df-meet 18250 df-p0 18326 df-p1 18327 df-lat 18335 df-clat 18402 df-mgm 18545 df-sgrp 18624 df-mnd 18640 df-submnd 18689 df-grp 18846 df-minusg 18847 df-sbg 18848 df-subg 19033 df-cntz 19227 df-oppg 19256 df-lsm 19546 df-cmn 19692 df-abl 19693 df-mgp 20057 df-rng 20069 df-ur 20098 df-ring 20151 df-oppr 20253 df-dvdsr 20273 df-unit 20274 df-invr 20304 df-dvr 20317 df-nzr 20426 df-rlreg 20607 df-domn 20608 df-drng 20644 df-lmod 20793 df-lss 20863 df-lsp 20903 df-lvec 21035 df-lsatoms 39014 df-lshyp 39015 df-lcv 39057 df-lfl 39096 df-lkr 39124 df-ldual 39162 df-oposet 39214 df-ol 39216 df-oml 39217 df-covers 39304 df-ats 39305 df-atl 39336 df-cvlat 39360 df-hlat 39389 df-llines 39536 df-lplanes 39537 df-lvols 39538 df-lines 39539 df-psubsp 39541 df-pmap 39542 df-padd 39834 df-lhyp 40026 df-laut 40027 df-ldil 40142 df-ltrn 40143 df-trl 40197 df-tgrp 40781 df-tendo 40793 df-edring 40795 df-dveca 41041 df-disoa 41067 df-dvech 41117 df-dib 41177 df-dic 41211 df-dih 41267 df-doch 41386 df-djh 41433 df-lcdual 41625 df-mapd 41663

This theorem is referenced by: mapdpglem24 41742

W3C validator