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Theorem lspfixed 21130
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.) (Revised by AV, 12-Jul-2022.)
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.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 2737 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
5 eqid 2737 . . . 4 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6 eqid 2737 . . . 4 ( ·𝑠𝑊) = ( ·𝑠𝑊)
7 lspfixed.n . . . 4 𝑁 = (LSpan‘𝑊)
8 lspfixed.w . . . . 5 (𝜑𝑊 ∈ LVec)
9 lveclmod 21105 . . . . 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 21093 . . 3 (𝜑 → (𝑋 ∈ (𝑁‘{𝑌, 𝑍}) ↔ ∃𝑘 ∈ (Base‘(Scalar‘𝑊))∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))))
141, 13mpbid 232 . 2 (𝜑 → ∃𝑘 ∈ (Base‘(Scalar‘𝑊))∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍)))
15103ad2ant1 1134 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → 𝑊 ∈ LMod)
16 eqid 2737 . . . . . . . . . 10 (LSubSp‘𝑊) = (LSubSp‘𝑊)
172, 16, 7lspsncl 20975 . . . . . . . . 9 ((𝑊 ∈ LMod ∧ 𝑍𝑉) → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊))
1810, 12, 17syl2anc 584 . . . . . . . 8 (𝜑 → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊))
19183ad2ant1 1134 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊))
2083ad2ant1 1134 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → 𝑊 ∈ LVec)
214lvecdrng 21104 . . . . . . . . 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 1200 . . . . . . . 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 1134 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → ¬ 𝑋 ∈ (𝑁‘{𝑍}))
26 simpl3 1194 . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . 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 7446 . . . . . . . . . . . . . . 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 1192 . . . . . . . . . . . . . . . . 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 2737 . . . . . . . . . . . . . . . . 17 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
33 lspfixed.o . . . . . . . . . . . . . . . . 17 0 = (0g𝑊)
342, 4, 6, 32, 33lmod0vs 20893 . . . . . . . . . . . . . . . 16 ((𝑊 ∈ LMod ∧ 𝑌𝑉) → ((0g‘(Scalar‘𝑊))( ·𝑠𝑊)𝑌) = 0 )
3530, 31, 34syl2anc 584 . . . . . . . . . . . . . . 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 2777 . . . . . . . . . . . . . 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 7446 . . . . . . . . . . . . 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 1201 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → 𝑙 ∈ (Base‘(Scalar‘𝑊)))
39123ad2ant1 1134 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → 𝑍𝑉)
402, 4, 6, 5lmodvscl 20876 . . . . . . . . . . . . . . . 16 ((𝑊 ∈ LMod ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑍𝑉) → (𝑙( ·𝑠𝑊)𝑍) ∈ 𝑉)
4115, 38, 39, 40syl3anc 1373 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → (𝑙( ·𝑠𝑊)𝑍) ∈ 𝑉)
4241adantr 480 . . . . . . . . . . . . . 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 20890 . . . . . . . . . . . . . 14 ((𝑊 ∈ LMod ∧ (𝑙( ·𝑠𝑊)𝑍) ∈ 𝑉) → ( 0 + (𝑙( ·𝑠𝑊)𝑍)) = (𝑙( ·𝑠𝑊)𝑍))
4430, 42, 43syl2anc 584 . . . . . . . . . . . . 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 2781 . . . . . . . . . . . 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 1228 . . . . . . . . . . . . 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 20991 . . . . . . . . . . . . . . 15 ((𝑊 ∈ LMod ∧ 𝑍𝑉) → 𝑍 ∈ (𝑁‘{𝑍}))
4910, 12, 48syl2anc 584 . . . . . . . . . . . . . 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 20953 . . . . . . . . . . . . 13 (((𝑊 ∈ LMod ∧ (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊)) ∧ (𝑙 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑍 ∈ (𝑁‘{𝑍}))) → (𝑙( ·𝑠𝑊)𝑍) ∈ (𝑁‘{𝑍}))
5230, 46, 47, 50, 51syl22anc 839 . . . . . . . . . . . 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 2841 . . . . . . . . . . 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 412 . . . . . . . . . 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 2954 . . . . . . . . 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 2737 . . . . . . . . 9 (invr‘(Scalar‘𝑊)) = (invr‘(Scalar‘𝑊))
585, 32, 57drnginvrcl 20753 . . . . . . . 8 (((Scalar‘𝑊) ∈ DivRing ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ≠ (0g‘(Scalar‘𝑊))) → ((invr‘(Scalar‘𝑊))‘𝑘) ∈ (Base‘(Scalar‘𝑊)))
5922, 23, 56, 58syl3anc 1373 . . . . . . 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 1134 . . . . . . . 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 839 . . . . . . 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 20953 . . . . . . 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 839 . . . . . 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 20754 . . . . . . . 8 (((Scalar‘𝑊) ∈ DivRing ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ≠ (0g‘(Scalar‘𝑊))) → ((invr‘(Scalar‘𝑊))‘𝑘) ≠ (0g‘(Scalar‘𝑊)))
6522, 23, 56, 64syl3anc 1373 . . . . . . 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 1134 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → ¬ 𝑋 ∈ (𝑁‘{𝑌}))
68 simpl3 1194 . . . . . . . . . . . . 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 7438 . . . . . . . . . . . . . . 15 (𝑙 = (0g‘(Scalar‘𝑊)) → (𝑙( ·𝑠𝑊)𝑍) = ((0g‘(Scalar‘𝑊))( ·𝑠𝑊)𝑍))
702, 4, 6, 32, 33lmod0vs 20893 . . . . . . . . . . . . . . . 16 ((𝑊 ∈ LMod ∧ 𝑍𝑉) → ((0g‘(Scalar‘𝑊))( ·𝑠𝑊)𝑍) = 0 )
7115, 39, 70syl2anc 584 . . . . . . . . . . . . . . 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 2799 . . . . . . . . . . . . . 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 7447 . . . . . . . . . . . . 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 1134 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → 𝑌𝑉)
752, 4, 6, 5lmodvscl 20876 . . . . . . . . . . . . . . . 16 ((𝑊 ∈ LMod ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑌𝑉) → (𝑘( ·𝑠𝑊)𝑌) ∈ 𝑉)
7615, 23, 74, 75syl3anc 1373 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → (𝑘( ·𝑠𝑊)𝑌) ∈ 𝑉)
772, 3, 33lmod0vrid 20891 . . . . . . . . . . . . . . 15 ((𝑊 ∈ LMod ∧ (𝑘( ·𝑠𝑊)𝑌) ∈ 𝑉) → ((𝑘( ·𝑠𝑊)𝑌) + 0 ) = (𝑘( ·𝑠𝑊)𝑌))
7815, 76, 77syl2anc 584 . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . 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 2781 . . . . . . . . . . . 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 20975 . . . . . . . . . . . . . . . 16 ((𝑊 ∈ LMod ∧ 𝑌𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
8210, 11, 81syl2anc 584 . . . . . . . . . . . . . . 15 (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
83823ad2ant1 1134 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
842, 7lspsnid 20991 . . . . . . . . . . . . . . . 16 ((𝑊 ∈ LMod ∧ 𝑌𝑉) → 𝑌 ∈ (𝑁‘{𝑌}))
8510, 11, 84syl2anc 584 . . . . . . . . . . . . . . 15 (𝜑𝑌 ∈ (𝑁‘{𝑌}))
86853ad2ant1 1134 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → 𝑌 ∈ (𝑁‘{𝑌}))
874, 6, 5, 16lssvscl 20953 . . . . . . . . . . . . . 14 (((𝑊 ∈ LMod ∧ (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑌 ∈ (𝑁‘{𝑌}))) → (𝑘( ·𝑠𝑊)𝑌) ∈ (𝑁‘{𝑌}))
8815, 83, 23, 86, 87syl22anc 839 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → (𝑘( ·𝑠𝑊)𝑌) ∈ (𝑁‘{𝑌}))
8988adantr 480 . . . . . . . . . . . 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 2841 . . . . . . . . . . 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 412 . . . . . . . . . 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 2954 . . . . . . . . 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 1192 . . . . . . . . . . . . 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 4734 . . . . . . . . . . . . . 14 (𝑍 = 0 → {𝑌, 𝑍} = {𝑌, 0 })
9796fveq2d 6910 . . . . . . . . . . . . 13 (𝑍 = 0 → (𝑁‘{𝑌, 𝑍}) = (𝑁‘{𝑌, 0 }))
982, 33, 7, 15, 74lsppr0 21091 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → (𝑁‘{𝑌, 0 }) = (𝑁‘{𝑌}))
9997, 98sylan9eqr 2799 . . . . . . . . . . . 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 2843 . . . . . . . . . . 11 (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) ∧ 𝑍 = 0 ) → 𝑋 ∈ (𝑁‘{𝑌}))
101100ex 412 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → (𝑍 = 0𝑋 ∈ (𝑁‘{𝑌})))
102101necon3bd 2954 . . . . . . . . 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 21111 . . . . . . . 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 713 . . . . . . 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 21111 . . . . . . 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 713 . . . . . 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 4786 . . . . . 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 583 . . . . 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 1139 . . . . . . 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 20873 . . . . . . . . 9 ((𝑊 ∈ LMod ∧ (𝑘( ·𝑠𝑊)𝑌) ∈ 𝑉 ∧ (𝑙( ·𝑠𝑊)𝑍) ∈ 𝑉) → ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍)) ∈ 𝑉)
11215, 76, 41, 111syl3anc 1373 . . . . . . . 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 20991 . . . . . . . 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 584 . . . . . . 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 2841 . . . . . 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 21116 . . . . . . . 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 1377 . . . . . . 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 20883 . . . . . . . . . . 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 1374 . . . . . . . . . 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 2737 . . . . . . . . . . . . . . 15 (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊))
121 eqid 2737 . . . . . . . . . . . . . . 15 (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑊))
1225, 32, 120, 121, 57drnginvrl 20756 . . . . . . . . . . . . . 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 1373 . . . . . . . . . . . . 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 7446 . . . . . . . . . . . 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 20885 . . . . . . . . . . . . 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 1374 . . . . . . . . . . . 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 20888 . . . . . . . . . . . . 13 ((𝑊 ∈ LMod ∧ 𝑌𝑉) → ((1r‘(Scalar‘𝑊))( ·𝑠𝑊)𝑌) = 𝑌)
12815, 74, 127syl2anc 584 . . . . . . . . . . . 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 2785 . . . . . . . . . . 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 7446 . . . . . . . . . 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 2777 . . . . . . . . 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 4638 . . . . . . . 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 6910 . . . . . . 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 2779 . . . . . 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 2843 . . . . 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 7439 . . . . . . . . 9 (𝑧 = (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑙( ·𝑠𝑊)𝑍)) → (𝑌 + 𝑧) = (𝑌 + (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑙( ·𝑠𝑊)𝑍))))
137136sneqd 4638 . . . . . . . 8 (𝑧 = (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑙( ·𝑠𝑊)𝑍)) → {(𝑌 + 𝑧)} = {(𝑌 + (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑙( ·𝑠𝑊)𝑍)))})
138137fveq2d 6910 . . . . . . 7 (𝑧 = (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑙( ·𝑠𝑊)𝑍)) → (𝑁‘{(𝑌 + 𝑧)}) = (𝑁‘{(𝑌 + (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑙( ·𝑠𝑊)𝑍)))}))
139138eleq2d 2827 . . . . . 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 3622 . . . . 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 584 . . . 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 1120 . . 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 3212 . 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 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wrex 3070  cdif 3948  {csn 4626  {cpr 4628  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  .rcmulr 17298  Scalarcsca 17300   ·𝑠 cvsca 17301  0gc0g 17484  1rcur 20178  invrcinvr 20387  DivRingcdr 20729  LModclmod 20858  LSubSpclss 20929  LSpanclspn 20969  LVecclvec 21101
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-tpos 8251  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-subg 19141  df-cntz 19335  df-lsm 19654  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-oppr 20334  df-dvdsr 20357  df-unit 20358  df-invr 20388  df-drng 20731  df-lmod 20860  df-lss 20930  df-lsp 20970  df-lvec 21102
This theorem is referenced by:  lsatfixedN  39010
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