lsatfixedN - Mathbox for Norm Megill

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Theorem lsatfixedN 37471

Description: Show equality with the span of the sum of two vectors, one of which (𝑋) is fixed in advance. Compare lspfixed 20589. (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 37453	. . . 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 231	. 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 2742	. . . . . . 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 3011	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}))
22	3, 5, 4, 11, 16, 13, 21	lspsnne1 20578	. . . . 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 3011	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑌}))
26	3, 5, 4, 11, 16, 15, 25	lspsnne1 20578	. . . . 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 3982	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → (𝑁‘{𝑤}) ⊆ (𝑁‘{𝑋, 𝑌}))
30		eqid 2736	. . . . . . 7 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
31		lveclmod 20567	. . . . . . . . 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 20439	. . . . . . . 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 3922	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → 𝑤 ∈ 𝑉)
37	3, 30, 4, 33, 35, 36	lspsnel5 20456	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → (𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ↔ (𝑁‘{𝑤}) ⊆ (𝑁‘{𝑋, 𝑌})))
38	29, 37	mpbird 256	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))
39	3, 10, 5, 4, 11, 13, 15, 22, 26, 38	lspfixed 20589	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → ∃𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })𝑤 ∈ (𝑁‘{(𝑋 + 𝑧)}))
40		simpl1 1191	. . . . . . . 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 1192	. . . . . . 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 4769	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑌} ⊆ 𝑉)
46	3, 4	lspssv 20444	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LMod ∧ {𝑌} ⊆ 𝑉) → (𝑁‘{𝑌}) ⊆ 𝑉)
47	32, 45, 46	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁‘{𝑌}) ⊆ 𝑉)
48	47	ssdifssd 4102	. . . . . . . . . 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 3944	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) ∧ 𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })) → 𝑧 ∈ 𝑉)
51	3, 10	lmodvacl 20336	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → (𝑋 + 𝑧) ∈ 𝑉)
52	43, 44, 50, 51	syl3anc 1371	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) ∧ 𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })) → (𝑋 + 𝑧) ∈ 𝑉)
53	3, 5, 4, 41, 42, 52	lspsncmp 20577	. . . . . 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 20438	. . . . . . . 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 3922	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) ∧ 𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })) → 𝑤 ∈ 𝑉)
57	3, 30, 4, 43, 55, 56	lspsnel5 20456	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) ∧ 𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })) → (𝑤 ∈ (𝑁‘{(𝑋 + 𝑧)}) ↔ (𝑁‘{𝑤}) ⊆ (𝑁‘{(𝑋 + 𝑧)})))
58		simpl3 1193	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) ∧ 𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })) → 𝑄 = (𝑁‘{𝑤}))
59	58	eqeq1d 2738	. . . . . 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 311	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) ∧ 𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })) → (𝑄 = (𝑁‘{(𝑋 + 𝑧)}) ↔ 𝑤 ∈ (𝑁‘{(𝑋 + 𝑧)})))
61	60	rexbidva 3173	. . . 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 256	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → ∃𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })𝑄 = (𝑁‘{(𝑋 + 𝑧)}))
63	62	rexlimdv3a 3156	. 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 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∃wrex 3073 ∖ cdif 3907 ⊆ wss 3910 {csn 4586 {cpr 4588 ‘cfv 6496 (class class class)co 7357 Basecbs 17083 +_gcplusg 17133 0_gc0g 17321 LModclmod 20322 LSubSpclss 20392 LSpanclspn 20432 LVecclvec 20563 LSAtomsclsa 37436

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-tpos 8157 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-0g 17323 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-grp 18751 df-minusg 18752 df-sbg 18753 df-subg 18925 df-cntz 19097 df-lsm 19418 df-cmn 19564 df-abl 19565 df-mgp 19897 df-ur 19914 df-ring 19966 df-oppr 20049 df-dvdsr 20070 df-unit 20071 df-invr 20101 df-drng 20187 df-lmod 20324 df-lss 20393 df-lsp 20433 df-lvec 20564 df-lsatoms 37438

This theorem is referenced by: hdmaprnlem3eN 40321

W3C validator