lsatfixedN - Mathbox for Norm Megill

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Theorem lsatfixedN 40325

Description: Show equality with the span of the sum of two vectors, one of which (𝑋) is fixed in advance. Compare lspfixed 21448. (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 40307	. . . 4 ⊢ (𝑊 ∈ LVec → (𝑄 ∈ 𝐴 ↔ ∃𝑤 ∈ (𝑉 ∖ { 0 })𝑄 = (𝑁‘{𝑤})))
8	2, 7	syl 18	. . 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 1151	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → 𝑊 ∈ LVec)
12		lsatfixed.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
13	12	3ad2ant1 1151	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → 𝑋 ∈ 𝑉)
14		lsatfixed.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
15	14	3ad2ant1 1151	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → 𝑌 ∈ 𝑉)
16		simp2 1155	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → 𝑤 ∈ (𝑉 ∖ { 0 }))
17		simp3 1156	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → 𝑄 = (𝑁‘{𝑤}))
18	17	eqcomd 2769	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → (𝑁‘{𝑤}) = 𝑄)
19		lsatfixed.e	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ≠ (𝑁‘{𝑋}))
20	19	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → 𝑄 ≠ (𝑁‘{𝑋}))
21	18, 20	eqnetrd 3026	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}))
22	3, 5, 4, 11, 16, 13, 21	lspsnne1 21437	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → ¬ 𝑤 ∈ (𝑁‘{𝑋}))
23		lsatfixed.f	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ≠ (𝑁‘{𝑌}))
24	23	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → 𝑄 ≠ (𝑁‘{𝑌}))
25	18, 24	eqnetrd 3026	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑌}))
26	3, 5, 4, 11, 16, 15, 25	lspsnne1 21437	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → ¬ 𝑤 ∈ (𝑁‘{𝑌}))
27		lsatfixed.g	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ⊆ (𝑁‘{𝑋, 𝑌}))
28	27	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → 𝑄 ⊆ (𝑁‘{𝑋, 𝑌}))
29	18, 28	eqsstrd 3947	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → (𝑁‘{𝑤}) ⊆ (𝑁‘{𝑋, 𝑌}))
30		eqid 2763	. . . . . . 7 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
31		lveclmod 21423	. . . . . . . . 9 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
32	2, 31	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ LMod)
33	32	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → 𝑊 ∈ LMod)
34	3, 30, 4, 32, 12, 14	lspprcl 21295	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑊))
35	34	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑊))
36	16	eldifad 4054	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → 𝑤 ∈ 𝑉)
37	3, 30, 4, 33, 35, 36	ellspsn5b 21312	. . . . . 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 21448	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → ∃𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })𝑤 ∈ (𝑁‘{(𝑋 + 𝑧)}))
40		simpl1 1210	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) ∧ 𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })) → 𝜑)
41	40, 2	syl 18	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) ∧ 𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })) → 𝑊 ∈ LVec)
42		simpl2 1211	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) ∧ 𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })) → 𝑤 ∈ (𝑉 ∖ { 0 }))
43	40, 32	syl 18	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) ∧ 𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })) → 𝑊 ∈ LMod)
44	40, 12	syl 18	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) ∧ 𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })) → 𝑋 ∈ 𝑉)
45	14	snssd 4747	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑌} ⊆ 𝑉)
46	3, 4	lspssv 21300	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LMod ∧ {𝑌} ⊆ 𝑉) → (𝑁‘{𝑌}) ⊆ 𝑉)
47	32, 45, 46	syl2anc 596	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁‘{𝑌}) ⊆ 𝑉)
48	47	ssdifssd 4086	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑁‘{𝑌}) ∖ { 0 }) ⊆ 𝑉)
49	48	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) → ((𝑁‘{𝑌}) ∖ { 0 }) ⊆ 𝑉)
50	49	sselda 3913	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) ∧ 𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })) → 𝑧 ∈ 𝑉)
51	3, 10	lmodvacl 21192	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → (𝑋 + 𝑧) ∈ 𝑉)
52	43, 44, 50, 51	syl3anc 1399	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) ∧ 𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })) → (𝑋 + 𝑧) ∈ 𝑉)
53	3, 5, 4, 41, 42, 52	lspsncmp 21436	. . . . . 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 21294	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ (𝑋 + 𝑧) ∈ 𝑉) → (𝑁‘{(𝑋 + 𝑧)}) ∈ (LSubSp‘𝑊))
55	43, 52, 54	syl2anc 596	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) ∧ 𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })) → (𝑁‘{(𝑋 + 𝑧)}) ∈ (LSubSp‘𝑊))
56	42	eldifad 4054	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) ∧ 𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })) → 𝑤 ∈ 𝑉)
57	3, 30, 4, 43, 55, 56	ellspsn5b 21312	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) ∧ 𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })) → (𝑤 ∈ (𝑁‘{(𝑋 + 𝑧)}) ↔ (𝑁‘{𝑤}) ⊆ (𝑁‘{(𝑋 + 𝑧)})))
58		simpl3 1212	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝑉 ∖ { 0 }) ∧ 𝑄 = (𝑁‘{𝑤})) ∧ 𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })) → 𝑄 = (𝑁‘{𝑤}))
59	58	eqeq1d 2765	. . . . . 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 3188	. . . 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 3171	. 2 ⊢ (𝜑 → (∃𝑤 ∈ (𝑉 ∖ { 0 })𝑄 = (𝑁‘{𝑤}) → ∃𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })𝑄 = (𝑁‘{(𝑋 + 𝑧)})))
64	9, 63	mpd 16	1 ⊢ (𝜑 → ∃𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })𝑄 = (𝑁‘{(𝑋 + 𝑧)}))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 ∃wrex 3090 ⊆ wss 3900 ∖ cdif 4048 {csn 4588 {cpr 4590 ‘cfv 6564 (class class class)co 7449 Basecbs 17428 +_gcplusg 17469 0_gc0g 17651 LModclmod 21177 LSubSpclss 21248 LSpanclspn 21288 LVecclvec 21419 LSAtomsclsa 40290

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5236 ax-sep 5253 ax-nul 5264 ax-pow 5332 ax-pr 5396 ax-un 7781 ax-cnex 11293 ax-resscn 11294 ax-1cn 11295 ax-icn 11296 ax-addcl 11297 ax-addrcl 11298 ax-mulcl 11299 ax-mulrcl 11300 ax-mulcom 11301 ax-addass 11302 ax-mulass 11303 ax-distr 11304 ax-i2m1 11305 ax-1ne0 11306 ax-1rid 11307 ax-rnegex 11308 ax-rrecex 11309 ax-cnre 11310 ax-pre-lttri 11311 ax-pre-lttrn 11312 ax-pre-ltadd 11313 ax-pre-mulgt0 11314

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-br 5107 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5551 df-eprel 5556 df-po 5564 df-so 5565 df-fr 5609 df-we 5611 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-pred 6328 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6518 df-fun 6566 df-fn 6567 df-f 6568 df-f1 6569 df-fo 6570 df-f1o 6571 df-fv 6572 df-riota 7406 df-ov 7452 df-oprab 7453 df-mpo 7454 df-om 7910 df-1st 8033 df-2nd 8034 df-tpos 8279 df-frecs 8335 df-wrecs 8366 df-recs 8415 df-rdg 8454 df-er 8753 df-en 9010 df-dom 9011 df-sdom 9012 df-pnf 11382 df-mnf 11383 df-xr 11384 df-ltxr 11385 df-le 11386 df-sub 11580 df-neg 11581 df-nn 12374 df-2 12443 df-3 12444 df-sets 17383 df-slot 17401 df-ndx 17413 df-base 17429 df-ress 17450 df-plusg 17482 df-mulr 17483 df-0g 17653 df-mgm 18857 df-sgrp 18949 df-mnd 18965 df-submnd 19020 df-grp 19188 df-minusg 19189 df-sbg 19190 df-subg 19374 df-cntz 19573 df-lsm 19892 df-cmn 20038 df-abl 20039 df-mgp 20403 df-rng 20417 df-ur 20450 df-ring 20503 df-oppr 20609 df-dvdsr 20629 df-unit 20630 df-invr 20660 df-drng 21024 df-lmod 21179 df-lss 21249 df-lsp 21289 df-lvec 21420 df-lsatoms 40292

This theorem is used by: hdmaprnlem3eN 43174

W3C validator