mapdpglem3 - Mathbox for Norm Megill

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Theorem mapdpglem3 42121

Description: Lemma for mapdpg 42152. 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 7382	. . . 4 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌}))) = ((𝐽‘{𝐺}) ⊕ (𝑀‘(𝑁‘{𝑌}))))
4	1, 3	eleqtrd 2838	. . 3 ⊢ (𝜑 → 𝑡 ∈ ((𝐽‘{𝐺}) ⊕ (𝑀‘(𝑁‘{𝑌}))))
5		r19.41v 3167	. . . . . . 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 42038	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ LMod)
10		mapdpglem3.g	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
11		eqid 2736	. . . . . . . . . . 11 ⊢ (Scalar‘𝐶) = (Scalar‘𝐶)
12		eqid 2736	. . . . . . . . . . 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 20998	. . . . . . . . . 10 ⊢ ((𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (𝑤 ∈ (𝐽‘{𝐺}) ↔ ∃𝑔 ∈ (Base‘(Scalar‘𝐶))𝑤 = (𝑔 · 𝐺)))
17	9, 10, 16	syl2anc 585	. . . . . . . . 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 42046	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘(Scalar‘𝐶)) = 𝐵)
22	21	rexeqdv 3296	. . . . . . . . 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 632	. . . . . . 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 1923	. . . . 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 3062	. . . . 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 2736	. . . . . . . 8 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶)
32	31	lsssssubg 20953	. . . . . . 7 ⊢ (𝐶 ∈ LMod → (LSubSp‘𝐶) ⊆ (SubGrp‘𝐶))
33	9, 32	syl 17	. . . . . 6 ⊢ (𝜑 → (LSubSp‘𝐶) ⊆ (SubGrp‘𝐶))
34	13, 31, 15	lspsncl 20972	. . . . . . 7 ⊢ ((𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (𝐽‘{𝐺}) ∈ (LSubSp‘𝐶))
35	9, 10, 34	syl2anc 585	. . . . . 6 ⊢ (𝜑 → (𝐽‘{𝐺}) ∈ (LSubSp‘𝐶))
36	33, 35	sseldd 3922	. . . . 5 ⊢ (𝜑 → (𝐽‘{𝐺}) ∈ (SubGrp‘𝐶))
37		mapdpglem.m	. . . . . . 7 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
38		eqid 2736	. . . . . . 7 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
39	6, 18, 8	dvhlmod 41556	. . . . . . . 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 20972	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
44	39, 40, 43	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
45	6, 37, 18, 38, 7, 31, 8, 44	mapdcl2 42102	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) ∈ (LSubSp‘𝐶))
46	33, 45	sseldd 3922	. . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) ∈ (SubGrp‘𝐶))
47	29, 30, 36, 46	lsmelvalm 19626	. . . 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 7400	. . . . 5 ⊢ (𝑔 · 𝐺) ∈ V
51		oveq1 7374	. . . . . . 7 ⊢ (𝑤 = (𝑔 · 𝐺) → (𝑤𝑅𝑧) = ((𝑔 · 𝐺)𝑅𝑧))
52	51	eqeq2d 2747	. . . . . 6 ⊢ (𝑤 = (𝑔 · 𝐺) → (𝑡 = (𝑤𝑅𝑧) ↔ 𝑡 = ((𝑔 · 𝐺)𝑅𝑧)))
53	52	rexbidv 3161	. . . . 5 ⊢ (𝑤 = (𝑔 · 𝐺) → (∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧) ↔ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = ((𝑔 · 𝐺)𝑅𝑧)))
54	50, 53	ceqsexv 3478	. . . 4 ⊢ (∃𝑤(𝑤 = (𝑔 · 𝐺) ∧ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = (𝑤𝑅𝑧)) ↔ ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = ((𝑔 · 𝐺)𝑅𝑧))
55	54	rexbii 3084	. . 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 3264	. . 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 1542 ∃wex 1781 ∈ wcel 2114 ∃wrex 3061 ⊆ wss 3889 {csn 4567 ‘cfv 6498 (class class class)co 7367 Basecbs 17179 Scalarcsca 17223 ·_𝑠 cvsca 17224 -_gcsg 18911 SubGrpcsubg 19096 LSSumclsm 19609 LModclmod 20855 LSubSpclss 20926 LSpanclspn 20966 HLchlt 39796 LHypclh 40430 DVecHcdvh 41524 LCDualclcd 42032 mapdcmpd 42070

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-riotaBAD 39399

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-tpos 8176 df-undef 8223 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-n0 12438 df-z 12525 df-uz 12789 df-fz 13462 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-sca 17236 df-vsca 17237 df-0g 17404 df-mre 17548 df-mrc 17549 df-acs 17551 df-proset 18260 df-poset 18279 df-plt 18294 df-lub 18310 df-glb 18311 df-join 18312 df-meet 18313 df-p0 18389 df-p1 18390 df-lat 18398 df-clat 18465 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-grp 18912 df-minusg 18913 df-sbg 18914 df-subg 19099 df-cntz 19292 df-oppg 19321 df-lsm 19611 df-cmn 19757 df-abl 19758 df-mgp 20122 df-rng 20134 df-ur 20163 df-ring 20216 df-oppr 20317 df-dvdsr 20337 df-unit 20338 df-invr 20368 df-dvr 20381 df-nzr 20490 df-rlreg 20671 df-domn 20672 df-drng 20708 df-lmod 20857 df-lss 20927 df-lsp 20967 df-lvec 21098 df-lsatoms 39422 df-lshyp 39423 df-lcv 39465 df-lfl 39504 df-lkr 39532 df-ldual 39570 df-oposet 39622 df-ol 39624 df-oml 39625 df-covers 39712 df-ats 39713 df-atl 39744 df-cvlat 39768 df-hlat 39797 df-llines 39944 df-lplanes 39945 df-lvols 39946 df-lines 39947 df-psubsp 39949 df-pmap 39950 df-padd 40242 df-lhyp 40434 df-laut 40435 df-ldil 40550 df-ltrn 40551 df-trl 40605 df-tgrp 41189 df-tendo 41201 df-edring 41203 df-dveca 41449 df-disoa 41475 df-dvech 41525 df-dib 41585 df-dic 41619 df-dih 41675 df-doch 41794 df-djh 41841 df-lcdual 42033 df-mapd 42071

This theorem is referenced by: mapdpglem24 42150

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