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Theorem lspfixed 19060
Description: Show membership in the span of the sum of two vectors, one of which (𝑌) is fixed in advance. (Contributed by NM, 27-May-2015.)
Hypotheses
Ref Expression
lspfixed.v 𝑉 = (Base‘𝑊)
lspfixed.p + = (+g𝑊)
lspfixed.o 0 = (0g𝑊)
lspfixed.n 𝑁 = (LSpan‘𝑊)
lspfixed.w (𝜑𝑊 ∈ LVec)
lspfixed.x (𝜑𝑋𝑉)
lspfixed.y (𝜑𝑌𝑉)
lspfixed.z (𝜑𝑍𝑉)
lspfixed.e (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌}))
lspfixed.f (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑍}))
lspfixed.g (𝜑𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
Assertion
Ref Expression
lspfixed (𝜑 → ∃𝑧 ∈ ((𝑁‘{𝑍}) ∖ { 0 })𝑋 ∈ (𝑁‘{(𝑌 + 𝑧)}))
Distinct variable groups:   𝑧,𝑁   𝑧, 0   𝑧, +   𝑧,𝑊   𝑧,𝑋   𝑧,𝑌   𝑧,𝑍
Allowed substitution hints:   𝜑(𝑧)   𝑉(𝑧)

Proof of Theorem lspfixed
Dummy variables 𝑘 𝑙 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lspfixed.g . . 3 (𝜑𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
2 lspfixed.v . . . 4 𝑉 = (Base‘𝑊)
3 lspfixed.p . . . 4 + = (+g𝑊)
4 eqid 2621 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
5 eqid 2621 . . . 4 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6 eqid 2621 . . . 4 ( ·𝑠𝑊) = ( ·𝑠𝑊)
7 lspfixed.n . . . 4 𝑁 = (LSpan‘𝑊)
8 lspfixed.w . . . . 5 (𝜑𝑊 ∈ LVec)
9 lveclmod 19038 . . . . 5 (𝑊 ∈ LVec → 𝑊 ∈ LMod)
108, 9syl 17 . . . 4 (𝜑𝑊 ∈ LMod)
11 lspfixed.y . . . 4 (𝜑𝑌𝑉)
12 lspfixed.z . . . 4 (𝜑𝑍𝑉)
132, 3, 4, 5, 6, 7, 10, 11, 12lspprel 19026 . . 3 (𝜑 → (𝑋 ∈ (𝑁‘{𝑌, 𝑍}) ↔ ∃𝑘 ∈ (Base‘(Scalar‘𝑊))∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))))
141, 13mpbid 222 . 2 (𝜑 → ∃𝑘 ∈ (Base‘(Scalar‘𝑊))∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍)))
15103ad2ant1 1080 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → 𝑊 ∈ LMod)
16 eqid 2621 . . . . . . . . . 10 (LSubSp‘𝑊) = (LSubSp‘𝑊)
172, 16, 7lspsncl 18909 . . . . . . . . 9 ((𝑊 ∈ LMod ∧ 𝑍𝑉) → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊))
1810, 12, 17syl2anc 692 . . . . . . . 8 (𝜑 → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊))
19183ad2ant1 1080 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊))
2083ad2ant1 1080 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → 𝑊 ∈ LVec)
214lvecdrng 19037 . . . . . . . . 9 (𝑊 ∈ LVec → (Scalar‘𝑊) ∈ DivRing)
2220, 21syl 17 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → (Scalar‘𝑊) ∈ DivRing)
23 simp2l 1085 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → 𝑘 ∈ (Base‘(Scalar‘𝑊)))
24 lspfixed.f . . . . . . . . . 10 (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑍}))
25243ad2ant1 1080 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → ¬ 𝑋 ∈ (𝑁‘{𝑍}))
26 simpl3 1064 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) ∧ 𝑘 = (0g‘(Scalar‘𝑊))) → 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍)))
27 simpr 477 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) ∧ 𝑘 = (0g‘(Scalar‘𝑊))) → 𝑘 = (0g‘(Scalar‘𝑊)))
2827oveq1d 6625 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) ∧ 𝑘 = (0g‘(Scalar‘𝑊))) → (𝑘( ·𝑠𝑊)𝑌) = ((0g‘(Scalar‘𝑊))( ·𝑠𝑊)𝑌))
29 simpl1 1062 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) ∧ 𝑘 = (0g‘(Scalar‘𝑊))) → 𝜑)
3029, 10syl 17 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) ∧ 𝑘 = (0g‘(Scalar‘𝑊))) → 𝑊 ∈ LMod)
3129, 11syl 17 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) ∧ 𝑘 = (0g‘(Scalar‘𝑊))) → 𝑌𝑉)
32 eqid 2621 . . . . . . . . . . . . . . . . 17 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
33 lspfixed.o . . . . . . . . . . . . . . . . 17 0 = (0g𝑊)
342, 4, 6, 32, 33lmod0vs 18828 . . . . . . . . . . . . . . . 16 ((𝑊 ∈ LMod ∧ 𝑌𝑉) → ((0g‘(Scalar‘𝑊))( ·𝑠𝑊)𝑌) = 0 )
3530, 31, 34syl2anc 692 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) ∧ 𝑘 = (0g‘(Scalar‘𝑊))) → ((0g‘(Scalar‘𝑊))( ·𝑠𝑊)𝑌) = 0 )
3628, 35eqtrd 2655 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) ∧ 𝑘 = (0g‘(Scalar‘𝑊))) → (𝑘( ·𝑠𝑊)𝑌) = 0 )
3736oveq1d 6625 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) ∧ 𝑘 = (0g‘(Scalar‘𝑊))) → ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍)) = ( 0 + (𝑙( ·𝑠𝑊)𝑍)))
38 simp2r 1086 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → 𝑙 ∈ (Base‘(Scalar‘𝑊)))
39123ad2ant1 1080 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → 𝑍𝑉)
402, 4, 6, 5lmodvscl 18812 . . . . . . . . . . . . . . . 16 ((𝑊 ∈ LMod ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑍𝑉) → (𝑙( ·𝑠𝑊)𝑍) ∈ 𝑉)
4115, 38, 39, 40syl3anc 1323 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → (𝑙( ·𝑠𝑊)𝑍) ∈ 𝑉)
4241adantr 481 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) ∧ 𝑘 = (0g‘(Scalar‘𝑊))) → (𝑙( ·𝑠𝑊)𝑍) ∈ 𝑉)
432, 3, 33lmod0vlid 18825 . . . . . . . . . . . . . 14 ((𝑊 ∈ LMod ∧ (𝑙( ·𝑠𝑊)𝑍) ∈ 𝑉) → ( 0 + (𝑙( ·𝑠𝑊)𝑍)) = (𝑙( ·𝑠𝑊)𝑍))
4430, 42, 43syl2anc 692 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) ∧ 𝑘 = (0g‘(Scalar‘𝑊))) → ( 0 + (𝑙( ·𝑠𝑊)𝑍)) = (𝑙( ·𝑠𝑊)𝑍))
4526, 37, 443eqtrd 2659 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) ∧ 𝑘 = (0g‘(Scalar‘𝑊))) → 𝑋 = (𝑙( ·𝑠𝑊)𝑍))
4629, 18syl 17 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) ∧ 𝑘 = (0g‘(Scalar‘𝑊))) → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊))
47 simpl2r 1113 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) ∧ 𝑘 = (0g‘(Scalar‘𝑊))) → 𝑙 ∈ (Base‘(Scalar‘𝑊)))
482, 7lspsnid 18925 . . . . . . . . . . . . . . 15 ((𝑊 ∈ LMod ∧ 𝑍𝑉) → 𝑍 ∈ (𝑁‘{𝑍}))
4910, 12, 48syl2anc 692 . . . . . . . . . . . . . 14 (𝜑𝑍 ∈ (𝑁‘{𝑍}))
5029, 49syl 17 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) ∧ 𝑘 = (0g‘(Scalar‘𝑊))) → 𝑍 ∈ (𝑁‘{𝑍}))
514, 6, 5, 16lssvscl 18887 . . . . . . . . . . . . 13 (((𝑊 ∈ LMod ∧ (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊)) ∧ (𝑙 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑍 ∈ (𝑁‘{𝑍}))) → (𝑙( ·𝑠𝑊)𝑍) ∈ (𝑁‘{𝑍}))
5230, 46, 47, 50, 51syl22anc 1324 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) ∧ 𝑘 = (0g‘(Scalar‘𝑊))) → (𝑙( ·𝑠𝑊)𝑍) ∈ (𝑁‘{𝑍}))
5345, 52eqeltrd 2698 . . . . . . . . . . 11 (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) ∧ 𝑘 = (0g‘(Scalar‘𝑊))) → 𝑋 ∈ (𝑁‘{𝑍}))
5453ex 450 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → (𝑘 = (0g‘(Scalar‘𝑊)) → 𝑋 ∈ (𝑁‘{𝑍})))
5554necon3bd 2804 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → (¬ 𝑋 ∈ (𝑁‘{𝑍}) → 𝑘 ≠ (0g‘(Scalar‘𝑊))))
5625, 55mpd 15 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → 𝑘 ≠ (0g‘(Scalar‘𝑊)))
57 eqid 2621 . . . . . . . . 9 (invr‘(Scalar‘𝑊)) = (invr‘(Scalar‘𝑊))
585, 32, 57drnginvrcl 18696 . . . . . . . 8 (((Scalar‘𝑊) ∈ DivRing ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ≠ (0g‘(Scalar‘𝑊))) → ((invr‘(Scalar‘𝑊))‘𝑘) ∈ (Base‘(Scalar‘𝑊)))
5922, 23, 56, 58syl3anc 1323 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → ((invr‘(Scalar‘𝑊))‘𝑘) ∈ (Base‘(Scalar‘𝑊)))
60493ad2ant1 1080 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → 𝑍 ∈ (𝑁‘{𝑍}))
6115, 19, 38, 60, 51syl22anc 1324 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → (𝑙( ·𝑠𝑊)𝑍) ∈ (𝑁‘{𝑍}))
624, 6, 5, 16lssvscl 18887 . . . . . . 7 (((𝑊 ∈ LMod ∧ (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊)) ∧ (((invr‘(Scalar‘𝑊))‘𝑘) ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑙( ·𝑠𝑊)𝑍) ∈ (𝑁‘{𝑍}))) → (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑙( ·𝑠𝑊)𝑍)) ∈ (𝑁‘{𝑍}))
6315, 19, 59, 61, 62syl22anc 1324 . . . . . 6 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑙( ·𝑠𝑊)𝑍)) ∈ (𝑁‘{𝑍}))
645, 32, 57drnginvrn0 18697 . . . . . . . 8 (((Scalar‘𝑊) ∈ DivRing ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ≠ (0g‘(Scalar‘𝑊))) → ((invr‘(Scalar‘𝑊))‘𝑘) ≠ (0g‘(Scalar‘𝑊)))
6522, 23, 56, 64syl3anc 1323 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → ((invr‘(Scalar‘𝑊))‘𝑘) ≠ (0g‘(Scalar‘𝑊)))
66 lspfixed.e . . . . . . . . . 10 (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌}))
67663ad2ant1 1080 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → ¬ 𝑋 ∈ (𝑁‘{𝑌}))
68 simpl3 1064 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) ∧ 𝑙 = (0g‘(Scalar‘𝑊))) → 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍)))
69 oveq1 6617 . . . . . . . . . . . . . . 15 (𝑙 = (0g‘(Scalar‘𝑊)) → (𝑙( ·𝑠𝑊)𝑍) = ((0g‘(Scalar‘𝑊))( ·𝑠𝑊)𝑍))
702, 4, 6, 32, 33lmod0vs 18828 . . . . . . . . . . . . . . . 16 ((𝑊 ∈ LMod ∧ 𝑍𝑉) → ((0g‘(Scalar‘𝑊))( ·𝑠𝑊)𝑍) = 0 )
7115, 39, 70syl2anc 692 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → ((0g‘(Scalar‘𝑊))( ·𝑠𝑊)𝑍) = 0 )
7269, 71sylan9eqr 2677 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) ∧ 𝑙 = (0g‘(Scalar‘𝑊))) → (𝑙( ·𝑠𝑊)𝑍) = 0 )
7372oveq2d 6626 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) ∧ 𝑙 = (0g‘(Scalar‘𝑊))) → ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍)) = ((𝑘( ·𝑠𝑊)𝑌) + 0 ))
74113ad2ant1 1080 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → 𝑌𝑉)
752, 4, 6, 5lmodvscl 18812 . . . . . . . . . . . . . . . 16 ((𝑊 ∈ LMod ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑌𝑉) → (𝑘( ·𝑠𝑊)𝑌) ∈ 𝑉)
7615, 23, 74, 75syl3anc 1323 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → (𝑘( ·𝑠𝑊)𝑌) ∈ 𝑉)
772, 3, 33lmod0vrid 18826 . . . . . . . . . . . . . . 15 ((𝑊 ∈ LMod ∧ (𝑘( ·𝑠𝑊)𝑌) ∈ 𝑉) → ((𝑘( ·𝑠𝑊)𝑌) + 0 ) = (𝑘( ·𝑠𝑊)𝑌))
7815, 76, 77syl2anc 692 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → ((𝑘( ·𝑠𝑊)𝑌) + 0 ) = (𝑘( ·𝑠𝑊)𝑌))
7978adantr 481 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) ∧ 𝑙 = (0g‘(Scalar‘𝑊))) → ((𝑘( ·𝑠𝑊)𝑌) + 0 ) = (𝑘( ·𝑠𝑊)𝑌))
8068, 73, 793eqtrd 2659 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) ∧ 𝑙 = (0g‘(Scalar‘𝑊))) → 𝑋 = (𝑘( ·𝑠𝑊)𝑌))
812, 16, 7lspsncl 18909 . . . . . . . . . . . . . . . 16 ((𝑊 ∈ LMod ∧ 𝑌𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
8210, 11, 81syl2anc 692 . . . . . . . . . . . . . . 15 (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
83823ad2ant1 1080 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
842, 7lspsnid 18925 . . . . . . . . . . . . . . . 16 ((𝑊 ∈ LMod ∧ 𝑌𝑉) → 𝑌 ∈ (𝑁‘{𝑌}))
8510, 11, 84syl2anc 692 . . . . . . . . . . . . . . 15 (𝜑𝑌 ∈ (𝑁‘{𝑌}))
86853ad2ant1 1080 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → 𝑌 ∈ (𝑁‘{𝑌}))
874, 6, 5, 16lssvscl 18887 . . . . . . . . . . . . . 14 (((𝑊 ∈ LMod ∧ (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑌 ∈ (𝑁‘{𝑌}))) → (𝑘( ·𝑠𝑊)𝑌) ∈ (𝑁‘{𝑌}))
8815, 83, 23, 86, 87syl22anc 1324 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → (𝑘( ·𝑠𝑊)𝑌) ∈ (𝑁‘{𝑌}))
8988adantr 481 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) ∧ 𝑙 = (0g‘(Scalar‘𝑊))) → (𝑘( ·𝑠𝑊)𝑌) ∈ (𝑁‘{𝑌}))
9080, 89eqeltrd 2698 . . . . . . . . . . 11 (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) ∧ 𝑙 = (0g‘(Scalar‘𝑊))) → 𝑋 ∈ (𝑁‘{𝑌}))
9190ex 450 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → (𝑙 = (0g‘(Scalar‘𝑊)) → 𝑋 ∈ (𝑁‘{𝑌})))
9291necon3bd 2804 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → (¬ 𝑋 ∈ (𝑁‘{𝑌}) → 𝑙 ≠ (0g‘(Scalar‘𝑊))))
9367, 92mpd 15 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → 𝑙 ≠ (0g‘(Scalar‘𝑊)))
94 simpl1 1062 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) ∧ 𝑍 = 0 ) → 𝜑)
9594, 1syl 17 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) ∧ 𝑍 = 0 ) → 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
96 preq2 4244 . . . . . . . . . . . . . 14 (𝑍 = 0 → {𝑌, 𝑍} = {𝑌, 0 })
9796fveq2d 6157 . . . . . . . . . . . . 13 (𝑍 = 0 → (𝑁‘{𝑌, 𝑍}) = (𝑁‘{𝑌, 0 }))
982, 33, 7, 15, 74lsppr0 19024 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → (𝑁‘{𝑌, 0 }) = (𝑁‘{𝑌}))
9997, 98sylan9eqr 2677 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) ∧ 𝑍 = 0 ) → (𝑁‘{𝑌, 𝑍}) = (𝑁‘{𝑌}))
10095, 99eleqtrd 2700 . . . . . . . . . . 11 (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) ∧ 𝑍 = 0 ) → 𝑋 ∈ (𝑁‘{𝑌}))
101100ex 450 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → (𝑍 = 0𝑋 ∈ (𝑁‘{𝑌})))
102101necon3bd 2804 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → (¬ 𝑋 ∈ (𝑁‘{𝑌}) → 𝑍0 ))
10367, 102mpd 15 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → 𝑍0 )
1042, 6, 4, 5, 32, 33, 20, 38, 39lvecvsn0 19041 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → ((𝑙( ·𝑠𝑊)𝑍) ≠ 0 ↔ (𝑙 ≠ (0g‘(Scalar‘𝑊)) ∧ 𝑍0 )))
10593, 103, 104mpbir2and 956 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → (𝑙( ·𝑠𝑊)𝑍) ≠ 0 )
1062, 6, 4, 5, 32, 33, 20, 59, 41lvecvsn0 19041 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → ((((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑙( ·𝑠𝑊)𝑍)) ≠ 0 ↔ (((invr‘(Scalar‘𝑊))‘𝑘) ≠ (0g‘(Scalar‘𝑊)) ∧ (𝑙( ·𝑠𝑊)𝑍) ≠ 0 )))
10765, 105, 106mpbir2and 956 . . . . . 6 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑙( ·𝑠𝑊)𝑍)) ≠ 0 )
108 eldifsn 4292 . . . . . 6 ((((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑙( ·𝑠𝑊)𝑍)) ∈ ((𝑁‘{𝑍}) ∖ { 0 }) ↔ ((((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑙( ·𝑠𝑊)𝑍)) ∈ (𝑁‘{𝑍}) ∧ (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑙( ·𝑠𝑊)𝑍)) ≠ 0 ))
10963, 107, 108sylanbrc 697 . . . . 5 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑙( ·𝑠𝑊)𝑍)) ∈ ((𝑁‘{𝑍}) ∖ { 0 }))
110 simp3 1061 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍)))
1112, 3lmodvacl 18809 . . . . . . . . 9 ((𝑊 ∈ LMod ∧ (𝑘( ·𝑠𝑊)𝑌) ∈ 𝑉 ∧ (𝑙( ·𝑠𝑊)𝑍) ∈ 𝑉) → ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍)) ∈ 𝑉)
11215, 76, 41, 111syl3anc 1323 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍)) ∈ 𝑉)
1132, 7lspsnid 18925 . . . . . . . 8 ((𝑊 ∈ LMod ∧ ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍)) ∈ 𝑉) → ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍)) ∈ (𝑁‘{((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))}))
11415, 112, 113syl2anc 692 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍)) ∈ (𝑁‘{((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))}))
115110, 114eqeltrd 2698 . . . . . 6 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → 𝑋 ∈ (𝑁‘{((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))}))
1162, 4, 6, 5, 32, 7lspsnvs 19046 . . . . . . . 8 ((𝑊 ∈ LVec ∧ (((invr‘(Scalar‘𝑊))‘𝑘) ∈ (Base‘(Scalar‘𝑊)) ∧ ((invr‘(Scalar‘𝑊))‘𝑘) ≠ (0g‘(Scalar‘𝑊))) ∧ ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍)) ∈ 𝑉) → (𝑁‘{(((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍)))}) = (𝑁‘{((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))}))
11720, 59, 65, 112, 116syl121anc 1328 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → (𝑁‘{(((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍)))}) = (𝑁‘{((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))}))
1182, 3, 4, 6, 5lmodvsdi 18818 . . . . . . . . . . 11 ((𝑊 ∈ LMod ∧ (((invr‘(Scalar‘𝑊))‘𝑘) ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑘( ·𝑠𝑊)𝑌) ∈ 𝑉 ∧ (𝑙( ·𝑠𝑊)𝑍) ∈ 𝑉)) → (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) = ((((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑘( ·𝑠𝑊)𝑌)) + (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑙( ·𝑠𝑊)𝑍))))
11915, 59, 76, 41, 118syl13anc 1325 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) = ((((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑘( ·𝑠𝑊)𝑌)) + (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑙( ·𝑠𝑊)𝑍))))
120 eqid 2621 . . . . . . . . . . . . . . 15 (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊))
121 eqid 2621 . . . . . . . . . . . . . . 15 (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑊))
1225, 32, 120, 121, 57drnginvrl 18698 . . . . . . . . . . . . . 14 (((Scalar‘𝑊) ∈ DivRing ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ≠ (0g‘(Scalar‘𝑊))) → (((invr‘(Scalar‘𝑊))‘𝑘)(.r‘(Scalar‘𝑊))𝑘) = (1r‘(Scalar‘𝑊)))
12322, 23, 56, 122syl3anc 1323 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → (((invr‘(Scalar‘𝑊))‘𝑘)(.r‘(Scalar‘𝑊))𝑘) = (1r‘(Scalar‘𝑊)))
124123oveq1d 6625 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → ((((invr‘(Scalar‘𝑊))‘𝑘)(.r‘(Scalar‘𝑊))𝑘)( ·𝑠𝑊)𝑌) = ((1r‘(Scalar‘𝑊))( ·𝑠𝑊)𝑌))
1252, 4, 6, 5, 120lmodvsass 18820 . . . . . . . . . . . . 13 ((𝑊 ∈ LMod ∧ (((invr‘(Scalar‘𝑊))‘𝑘) ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑌𝑉)) → ((((invr‘(Scalar‘𝑊))‘𝑘)(.r‘(Scalar‘𝑊))𝑘)( ·𝑠𝑊)𝑌) = (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑘( ·𝑠𝑊)𝑌)))
12615, 59, 23, 74, 125syl13anc 1325 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → ((((invr‘(Scalar‘𝑊))‘𝑘)(.r‘(Scalar‘𝑊))𝑘)( ·𝑠𝑊)𝑌) = (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑘( ·𝑠𝑊)𝑌)))
1272, 4, 6, 121lmodvs1 18823 . . . . . . . . . . . . 13 ((𝑊 ∈ LMod ∧ 𝑌𝑉) → ((1r‘(Scalar‘𝑊))( ·𝑠𝑊)𝑌) = 𝑌)
12815, 74, 127syl2anc 692 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → ((1r‘(Scalar‘𝑊))( ·𝑠𝑊)𝑌) = 𝑌)
129124, 126, 1283eqtr3d 2663 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑘( ·𝑠𝑊)𝑌)) = 𝑌)
130129oveq1d 6625 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → ((((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑘( ·𝑠𝑊)𝑌)) + (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑙( ·𝑠𝑊)𝑍))) = (𝑌 + (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑙( ·𝑠𝑊)𝑍))))
131119, 130eqtrd 2655 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) = (𝑌 + (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑙( ·𝑠𝑊)𝑍))))
132131sneqd 4165 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → {(((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍)))} = {(𝑌 + (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑙( ·𝑠𝑊)𝑍)))})
133132fveq2d 6157 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → (𝑁‘{(((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍)))}) = (𝑁‘{(𝑌 + (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑙( ·𝑠𝑊)𝑍)))}))
134117, 133eqtr3d 2657 . . . . . 6 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → (𝑁‘{((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))}) = (𝑁‘{(𝑌 + (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑙( ·𝑠𝑊)𝑍)))}))
135115, 134eleqtrd 2700 . . . . 5 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → 𝑋 ∈ (𝑁‘{(𝑌 + (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑙( ·𝑠𝑊)𝑍)))}))
136 oveq2 6618 . . . . . . . . 9 (𝑧 = (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑙( ·𝑠𝑊)𝑍)) → (𝑌 + 𝑧) = (𝑌 + (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑙( ·𝑠𝑊)𝑍))))
137136sneqd 4165 . . . . . . . 8 (𝑧 = (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑙( ·𝑠𝑊)𝑍)) → {(𝑌 + 𝑧)} = {(𝑌 + (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑙( ·𝑠𝑊)𝑍)))})
138137fveq2d 6157 . . . . . . 7 (𝑧 = (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑙( ·𝑠𝑊)𝑍)) → (𝑁‘{(𝑌 + 𝑧)}) = (𝑁‘{(𝑌 + (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑙( ·𝑠𝑊)𝑍)))}))
139138eleq2d 2684 . . . . . 6 (𝑧 = (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑙( ·𝑠𝑊)𝑍)) → (𝑋 ∈ (𝑁‘{(𝑌 + 𝑧)}) ↔ 𝑋 ∈ (𝑁‘{(𝑌 + (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑙( ·𝑠𝑊)𝑍)))})))
140139rspcev 3298 . . . . 5 (((((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑙( ·𝑠𝑊)𝑍)) ∈ ((𝑁‘{𝑍}) ∖ { 0 }) ∧ 𝑋 ∈ (𝑁‘{(𝑌 + (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑙( ·𝑠𝑊)𝑍)))})) → ∃𝑧 ∈ ((𝑁‘{𝑍}) ∖ { 0 })𝑋 ∈ (𝑁‘{(𝑌 + 𝑧)}))
141109, 135, 140syl2anc 692 . . . 4 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → ∃𝑧 ∈ ((𝑁‘{𝑍}) ∖ { 0 })𝑋 ∈ (𝑁‘{(𝑌 + 𝑧)}))
1421413exp 1261 . . 3 (𝜑 → ((𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) → (𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍)) → ∃𝑧 ∈ ((𝑁‘{𝑍}) ∖ { 0 })𝑋 ∈ (𝑁‘{(𝑌 + 𝑧)}))))
143142rexlimdvv 3031 . 2 (𝜑 → (∃𝑘 ∈ (Base‘(Scalar‘𝑊))∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍)) → ∃𝑧 ∈ ((𝑁‘{𝑍}) ∖ { 0 })𝑋 ∈ (𝑁‘{(𝑌 + 𝑧)})))
14414, 143mpd 15 1 (𝜑 → ∃𝑧 ∈ ((𝑁‘{𝑍}) ∖ { 0 })𝑋 ∈ (𝑁‘{(𝑌 + 𝑧)}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wrex 2908  cdif 3556  {csn 4153  {cpr 4155  cfv 5852  (class class class)co 6610  Basecbs 15792  +gcplusg 15873  .rcmulr 15874  Scalarcsca 15876   ·𝑠 cvsca 15877  0gc0g 16032  1rcur 18433  invrcinvr 18603  DivRingcdr 18679  LModclmod 18795  LSubSpclss 18864  LSpanclspn 18903  LVecclvec 19034
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-tpos 7304  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-er 7694  df-en 7908  df-dom 7909  df-sdom 7910  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-2 11031  df-3 11032  df-ndx 15795  df-slot 15796  df-base 15797  df-sets 15798  df-ress 15799  df-plusg 15886  df-mulr 15887  df-0g 16034  df-mgm 17174  df-sgrp 17216  df-mnd 17227  df-submnd 17268  df-grp 17357  df-minusg 17358  df-sbg 17359  df-subg 17523  df-cntz 17682  df-lsm 17983  df-cmn 18127  df-abl 18128  df-mgp 18422  df-ur 18434  df-ring 18481  df-oppr 18555  df-dvdsr 18573  df-unit 18574  df-invr 18604  df-drng 18681  df-lmod 18797  df-lss 18865  df-lsp 18904  df-lvec 19035
This theorem is referenced by:  lsatfixedN  33811
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