lsatfixedN - Mathbox for Norm Megill

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Theorem lsatfixedN 39009

Description: Show equality with the span of the sum of two vectors, one of which (𝑋) is fixed in advance. Compare lspfixed 21045. (Contributed by NM, 29-May-2015.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
lsatfixed.v	⊢ 𝑉 = (Base‘𝑊)
lsatfixed.p	⊢ + = (+_g‘𝑊)
lsatfixed.o	⊢ 0 = (0_g‘𝑊)
lsatfixed.n	⊢ 𝑁 = (LSpan‘𝑊)
lsatfixed.a	⊢ 𝐴 = (LSAtoms‘𝑊)
lsatfixed.w	⊢ (𝜑 → 𝑊 ∈ LVec)
lsatfixed.q	⊢ (𝜑 → 𝑄 ∈ 𝐴)
lsatfixed.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
lsatfixed.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)
lsatfixed.e	⊢ (𝜑 → 𝑄 ≠ (𝑁‘{𝑋}))
lsatfixed.f	⊢ (𝜑 → 𝑄 ≠ (𝑁‘{𝑌}))
lsatfixed.g	⊢ (𝜑 → 𝑄 ⊆ (𝑁‘{𝑋, 𝑌}))

**Assertion**
Ref	Expression
lsatfixedN	⊢ (𝜑 → ∃𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })𝑄 = (𝑁‘{(𝑋 + 𝑧)}))

Distinct variable groups: 𝑧,𝑁 𝑧, 0 𝑧, + 𝜑,𝑧 𝑧,𝑄 𝑧,𝑉 𝑧,𝑊 𝑧,𝑋 𝑧,𝑌 Allowed substitution hint: 𝐴(𝑧)

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

Step	Hyp	Ref	Expression
1		lsatfixed.q	. . 3 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
2		lsatfixed.w	. . . 4 ⊢ (𝜑 → 𝑊 ∈ LVec)
3		lsatfixed.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑊)
4		lsatfixed.n	. . . . 5 ⊢ 𝑁 = (LSpan‘𝑊)
5		lsatfixed.o	. . . . 5 ⊢ 0 = (0_g‘𝑊)
6		lsatfixed.a	. . . . 5 ⊢ 𝐴 = (LSAtoms‘𝑊)
7	3, 4, 5, 6	islsat 38991	. . . 4 ⊢ (𝑊 ∈ LVec → (𝑄 ∈ 𝐴 ↔ ∃𝑤 ∈ (𝑉 ∖ { 0 })𝑄 = (𝑁‘{𝑤})))
8	2, 7	syl 17	. . 3 ⊢ (𝜑 → (𝑄 ∈ 𝐴 ↔ ∃𝑤 ∈ (𝑉 ∖ { 0 })𝑄 = (𝑁‘{𝑤})))
9	1, 8	mpbid 232	. 2 ⊢ (𝜑 → ∃𝑤 ∈ (𝑉 ∖ { 0 })𝑄 = (𝑁‘{𝑤}))
10		lsatfixed.p	. . . . 5 ⊢ + = (+_g‘𝑊)
11	2	3ad2ant1 1133	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → 𝑊 ∈ LVec)
12		lsatfixed.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
13	12	3ad2ant1 1133	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → 𝑋 ∈ 𝑉)
14		lsatfixed.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
15	14	3ad2ant1 1133	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → 𝑌 ∈ 𝑉)
16		simp2 1137	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → 𝑤 ∈ (𝑉 ∖ { 0 }))
17		simp3 1138	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → 𝑄 = (𝑁‘{𝑤}))
18	17	eqcomd 2736	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → (𝑁‘{𝑤}) = 𝑄)
19		lsatfixed.e	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ≠ (𝑁‘{𝑋}))
20	19	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → 𝑄 ≠ (𝑁‘{𝑋}))
21	18, 20	eqnetrd 2993	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}))
22	3, 5, 4, 11, 16, 13, 21	lspsnne1 21034	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → ¬ 𝑤 ∈ (𝑁‘{𝑋}))
23		lsatfixed.f	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ≠ (𝑁‘{𝑌}))
24	23	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → 𝑄 ≠ (𝑁‘{𝑌}))
25	18, 24	eqnetrd 2993	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑌}))
26	3, 5, 4, 11, 16, 15, 25	lspsnne1 21034	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → ¬ 𝑤 ∈ (𝑁‘{𝑌}))
27		lsatfixed.g	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ⊆ (𝑁‘{𝑋, 𝑌}))
28	27	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → 𝑄 ⊆ (𝑁‘{𝑋, 𝑌}))
29	18, 28	eqsstrd 3984	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → (𝑁‘{𝑤}) ⊆ (𝑁‘{𝑋, 𝑌}))
30		eqid 2730	. . . . . . 7 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
31		lveclmod 21020	. . . . . . . . 9 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
32	2, 31	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ LMod)
33	32	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → 𝑊 ∈ LMod)
34	3, 30, 4, 32, 12, 14	lspprcl 20891	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑊))
35	34	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑊))
36	16	eldifad 3929	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → 𝑤 ∈ 𝑉)
37	3, 30, 4, 33, 35, 36	ellspsn5b 20908	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → (𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ↔ (𝑁‘{𝑤}) ⊆ (𝑁‘{𝑋, 𝑌})))
38	29, 37	mpbird 257	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))
39	3, 10, 5, 4, 11, 13, 15, 22, 26, 38	lspfixed 21045	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → ∃𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })𝑤 ∈ (𝑁‘{(𝑋 + 𝑧)}))
40		simpl1 1192	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) ∧ 𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })) → 𝜑)
41	40, 2	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) ∧ 𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })) → 𝑊 ∈ LVec)
42		simpl2 1193	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) ∧ 𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })) → 𝑤 ∈ (𝑉 ∖ { 0 }))
43	40, 32	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) ∧ 𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })) → 𝑊 ∈ LMod)
44	40, 12	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) ∧ 𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })) → 𝑋 ∈ 𝑉)
45	14	snssd 4776	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑌} ⊆ 𝑉)
46	3, 4	lspssv 20896	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LMod ∧ {𝑌} ⊆ 𝑉) → (𝑁‘{𝑌}) ⊆ 𝑉)
47	32, 45, 46	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁‘{𝑌}) ⊆ 𝑉)
48	47	ssdifssd 4113	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑁‘{𝑌}) ∖ { 0 }) ⊆ 𝑉)
49	48	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → ((𝑁‘{𝑌}) ∖ { 0 }) ⊆ 𝑉)
50	49	sselda 3949	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) ∧ 𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })) → 𝑧 ∈ 𝑉)
51	3, 10	lmodvacl 20788	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → (𝑋 + 𝑧) ∈ 𝑉)
52	43, 44, 50, 51	syl3anc 1373	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) ∧ 𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })) → (𝑋 + 𝑧) ∈ 𝑉)
53	3, 5, 4, 41, 42, 52	lspsncmp 21033	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) ∧ 𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })) → ((𝑁‘{𝑤}) ⊆ (𝑁‘{(𝑋 + 𝑧)}) ↔ (𝑁‘{𝑤}) = (𝑁‘{(𝑋 + 𝑧)})))
54	3, 30, 4	lspsncl 20890	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ (𝑋 + 𝑧) ∈ 𝑉) → (𝑁‘{(𝑋 + 𝑧)}) ∈ (LSubSp‘𝑊))
55	43, 52, 54	syl2anc 584	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) ∧ 𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })) → (𝑁‘{(𝑋 + 𝑧)}) ∈ (LSubSp‘𝑊))
56	42	eldifad 3929	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) ∧ 𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })) → 𝑤 ∈ 𝑉)
57	3, 30, 4, 43, 55, 56	ellspsn5b 20908	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) ∧ 𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })) → (𝑤 ∈ (𝑁‘{(𝑋 + 𝑧)}) ↔ (𝑁‘{𝑤}) ⊆ (𝑁‘{(𝑋 + 𝑧)})))
58		simpl3 1194	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) ∧ 𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })) → 𝑄 = (𝑁‘{𝑤}))
59	58	eqeq1d 2732	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) ∧ 𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })) → (𝑄 = (𝑁‘{(𝑋 + 𝑧)}) ↔ (𝑁‘{𝑤}) = (𝑁‘{(𝑋 + 𝑧)})))
60	53, 57, 59	3bitr4rd 312	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) ∧ 𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })) → (𝑄 = (𝑁‘{(𝑋 + 𝑧)}) ↔ 𝑤 ∈ (𝑁‘{(𝑋 + 𝑧)})))
61	60	rexbidva 3156	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → (∃𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })𝑄 = (𝑁‘{(𝑋 + 𝑧)}) ↔ ∃𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })𝑤 ∈ (𝑁‘{(𝑋 + 𝑧)})))
62	39, 61	mpbird 257	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → ∃𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })𝑄 = (𝑁‘{(𝑋 + 𝑧)}))
63	62	rexlimdv3a 3139	. 2 ⊢ (𝜑 → (∃𝑤 ∈ (𝑉 ∖ { 0 })𝑄 = (𝑁‘{𝑤}) → ∃𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })𝑄 = (𝑁‘{(𝑋 + 𝑧)})))
64	9, 63	mpd 15	1 ⊢ (𝜑 → ∃𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })𝑄 = (𝑁‘{(𝑋 + 𝑧)}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 ∃wrex 3054 ∖ cdif 3914 ⊆ wss 3917 {csn 4592 {cpr 4594 ‘cfv 6514 (class class class)co 7390 Basecbs 17186 +_gcplusg 17227 0_gc0g 17409 LModclmod 20773 LSubSpclss 20844 LSpanclspn 20884 LVecclvec 21016 LSAtomsclsa 38974

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-tpos 8208 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-2 12256 df-3 12257 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-0g 17411 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-submnd 18718 df-grp 18875 df-minusg 18876 df-sbg 18877 df-subg 19062 df-cntz 19256 df-lsm 19573 df-cmn 19719 df-abl 19720 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-drng 20647 df-lmod 20775 df-lss 20845 df-lsp 20885 df-lvec 21017 df-lsatoms 38976

This theorem is referenced by: hdmaprnlem3eN 41859

W3C validator