lsatfixedN - Mathbox for Norm Megill

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Theorem lsatfixedN 36635

Description: Show equality with the span of the sum of two vectors, one of which (𝑋) is fixed in advance. Compare lspfixed 20012. (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 36617	. . . 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 235	. 2 ⊢ (𝜑 → ∃𝑤 ∈ (𝑉 ∖ { 0 })𝑄 = (𝑁‘{𝑤}))
10		lsatfixed.p	. . . . 5 ⊢ + = (+_g‘𝑊)
11	2	3ad2ant1 1134	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → 𝑊 ∈ LVec)
12		lsatfixed.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
13	12	3ad2ant1 1134	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → 𝑋 ∈ 𝑉)
14		lsatfixed.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
15	14	3ad2ant1 1134	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → 𝑌 ∈ 𝑉)
16		simp2 1138	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → 𝑤 ∈ (𝑉 ∖ { 0 }))
17		simp3 1139	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → 𝑄 = (𝑁‘{𝑤}))
18	17	eqcomd 2744	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → (𝑁‘{𝑤}) = 𝑄)
19		lsatfixed.e	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ≠ (𝑁‘{𝑋}))
20	19	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → 𝑄 ≠ (𝑁‘{𝑋}))
21	18, 20	eqnetrd 3001	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}))
22	3, 5, 4, 11, 16, 13, 21	lspsnne1 20001	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → ¬ 𝑤 ∈ (𝑁‘{𝑋}))
23		lsatfixed.f	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ≠ (𝑁‘{𝑌}))
24	23	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → 𝑄 ≠ (𝑁‘{𝑌}))
25	18, 24	eqnetrd 3001	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑌}))
26	3, 5, 4, 11, 16, 15, 25	lspsnne1 20001	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → ¬ 𝑤 ∈ (𝑁‘{𝑌}))
27		lsatfixed.g	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ⊆ (𝑁‘{𝑋, 𝑌}))
28	27	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → 𝑄 ⊆ (𝑁‘{𝑋, 𝑌}))
29	18, 28	eqsstrd 3913	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → (𝑁‘{𝑤}) ⊆ (𝑁‘{𝑋, 𝑌}))
30		eqid 2738	. . . . . . 7 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
31		lveclmod 19990	. . . . . . . . 9 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
32	2, 31	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ LMod)
33	32	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → 𝑊 ∈ LMod)
34	3, 30, 4, 32, 12, 14	lspprcl 19862	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑊))
35	34	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑊))
36	16	eldifad 3853	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → 𝑤 ∈ 𝑉)
37	3, 30, 4, 33, 35, 36	lspsnel5 19879	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → (𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ↔ (𝑁‘{𝑤}) ⊆ (𝑁‘{𝑋, 𝑌})))
38	29, 37	mpbird 260	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))
39	3, 10, 5, 4, 11, 13, 15, 22, 26, 38	lspfixed 20012	. . . 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 4694	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑌} ⊆ 𝑉)
46	3, 4	lspssv 19867	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LMod ∧ {𝑌} ⊆ 𝑉) → (𝑁‘{𝑌}) ⊆ 𝑉)
47	32, 45, 46	syl2anc 587	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁‘{𝑌}) ⊆ 𝑉)
48	47	ssdifssd 4031	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑁‘{𝑌}) ∖ { 0 }) ⊆ 𝑉)
49	48	3ad2ant1 1134	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → ((𝑁‘{𝑌}) ∖ { 0 }) ⊆ 𝑉)
50	49	sselda 3875	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) ∧ 𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })) → 𝑧 ∈ 𝑉)
51	3, 10	lmodvacl 19760	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → (𝑋 + 𝑧) ∈ 𝑉)
52	43, 44, 50, 51	syl3anc 1372	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) ∧ 𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })) → (𝑋 + 𝑧) ∈ 𝑉)
53	3, 5, 4, 41, 42, 52	lspsncmp 20000	. . . . . 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 19861	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ (𝑋 + 𝑧) ∈ 𝑉) → (𝑁‘{(𝑋 + 𝑧)}) ∈ (LSubSp‘𝑊))
55	43, 52, 54	syl2anc 587	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) ∧ 𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })) → (𝑁‘{(𝑋 + 𝑧)}) ∈ (LSubSp‘𝑊))
56	42	eldifad 3853	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) ∧ 𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })) → 𝑤 ∈ 𝑉)
57	3, 30, 4, 43, 55, 56	lspsnel5 19879	. . . . . 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 2740	. . . . . 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 315	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) ∧ 𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })) → (𝑄 = (𝑁‘{(𝑋 + 𝑧)}) ↔ 𝑤 ∈ (𝑁‘{(𝑋 + 𝑧)})))
61	60	rexbidva 3205	. . . 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 260	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → ∃𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })𝑄 = (𝑁‘{(𝑋 + 𝑧)}))
63	62	rexlimdv3a 3195	. 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 209 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2113 ≠ wne 2934 ∃wrex 3054 ∖ cdif 3838 ⊆ wss 3841 {csn 4513 {cpr 4515 ‘cfv 6333 (class class class)co 7164 Basecbs 16579 +_gcplusg 16661 0_gc0g 16809 LModclmod 19746 LSubSpclss 19815 LSpanclspn 19855 LVecclvec 19986 LSAtomsclsa 36600

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-int 4834 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-1st 7707 df-2nd 7708 df-tpos 7914 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-er 8313 df-en 8549 df-dom 8550 df-sdom 8551 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-nn 11710 df-2 11772 df-3 11773 df-ndx 16582 df-slot 16583 df-base 16585 df-sets 16586 df-ress 16587 df-plusg 16674 df-mulr 16675 df-0g 16811 df-mgm 17961 df-sgrp 18010 df-mnd 18021 df-submnd 18066 df-grp 18215 df-minusg 18216 df-sbg 18217 df-subg 18387 df-cntz 18558 df-lsm 18872 df-cmn 19019 df-abl 19020 df-mgp 19352 df-ur 19364 df-ring 19411 df-oppr 19488 df-dvdsr 19506 df-unit 19507 df-invr 19537 df-drng 19616 df-lmod 19748 df-lss 19816 df-lsp 19856 df-lvec 19987 df-lsatoms 36602

This theorem is referenced by: hdmaprnlem3eN 39484

W3C validator