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Theorem lspfixed 21023
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 2728 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
5 eqid 2728 . . . 4 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6 eqid 2728 . . . 4 ( ·𝑠𝑊) = ( ·𝑠𝑊)
7 lspfixed.n . . . 4 𝑁 = (LSpan‘𝑊)
8 lspfixed.w . . . . 5 (𝜑𝑊 ∈ LVec)
9 lveclmod 20998 . . . . 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 20986 . . 3 (𝜑 → (𝑋 ∈ (𝑁‘{𝑌, 𝑍}) ↔ ∃𝑘 ∈ (Base‘(Scalar‘𝑊))∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))))
141, 13mpbid 231 . 2 (𝜑 → ∃𝑘 ∈ (Base‘(Scalar‘𝑊))∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍)))
15103ad2ant1 1130 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → 𝑊 ∈ LMod)
16 eqid 2728 . . . . . . . . . 10 (LSubSp‘𝑊) = (LSubSp‘𝑊)
172, 16, 7lspsncl 20868 . . . . . . . . 9 ((𝑊 ∈ LMod ∧ 𝑍𝑉) → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊))
1810, 12, 17syl2anc 582 . . . . . . . 8 (𝜑 → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊))
19183ad2ant1 1130 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊))
2083ad2ant1 1130 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → 𝑊 ∈ LVec)
214lvecdrng 20997 . . . . . . . . 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 1196 . . . . . . . 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 1130 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → ¬ 𝑋 ∈ (𝑁‘{𝑍}))
26 simpl3 1190 . . . . . . . . . . . . 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 483 . . . . . . . . . . . . . . . 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 7441 . . . . . . . . . . . . . . 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 1188 . . . . . . . . . . . . . . . . 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 2728 . . . . . . . . . . . . . . . . 17 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
33 lspfixed.o . . . . . . . . . . . . . . . . 17 0 = (0g𝑊)
342, 4, 6, 32, 33lmod0vs 20785 . . . . . . . . . . . . . . . 16 ((𝑊 ∈ LMod ∧ 𝑌𝑉) → ((0g‘(Scalar‘𝑊))( ·𝑠𝑊)𝑌) = 0 )
3530, 31, 34syl2anc 582 . . . . . . . . . . . . . . 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 2768 . . . . . . . . . . . . . 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 7441 . . . . . . . . . . . . 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 1197 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → 𝑙 ∈ (Base‘(Scalar‘𝑊)))
39123ad2ant1 1130 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → 𝑍𝑉)
402, 4, 6, 5lmodvscl 20768 . . . . . . . . . . . . . . . 16 ((𝑊 ∈ LMod ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑍𝑉) → (𝑙( ·𝑠𝑊)𝑍) ∈ 𝑉)
4115, 38, 39, 40syl3anc 1368 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → (𝑙( ·𝑠𝑊)𝑍) ∈ 𝑉)
4241adantr 479 . . . . . . . . . . . . . 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 20782 . . . . . . . . . . . . . 14 ((𝑊 ∈ LMod ∧ (𝑙( ·𝑠𝑊)𝑍) ∈ 𝑉) → ( 0 + (𝑙( ·𝑠𝑊)𝑍)) = (𝑙( ·𝑠𝑊)𝑍))
4430, 42, 43syl2anc 582 . . . . . . . . . . . . 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 2772 . . . . . . . . . . . 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 1224 . . . . . . . . . . . . 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 20884 . . . . . . . . . . . . . . 15 ((𝑊 ∈ LMod ∧ 𝑍𝑉) → 𝑍 ∈ (𝑁‘{𝑍}))
4910, 12, 48syl2anc 582 . . . . . . . . . . . . . 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 20846 . . . . . . . . . . . . 13 (((𝑊 ∈ LMod ∧ (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊)) ∧ (𝑙 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑍 ∈ (𝑁‘{𝑍}))) → (𝑙( ·𝑠𝑊)𝑍) ∈ (𝑁‘{𝑍}))
5230, 46, 47, 50, 51syl22anc 837 . . . . . . . . . . . 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 2829 . . . . . . . . . . 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 411 . . . . . . . . . 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 2951 . . . . . . . . 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 2728 . . . . . . . . 9 (invr‘(Scalar‘𝑊)) = (invr‘(Scalar‘𝑊))
585, 32, 57drnginvrcl 20653 . . . . . . . 8 (((Scalar‘𝑊) ∈ DivRing ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ≠ (0g‘(Scalar‘𝑊))) → ((invr‘(Scalar‘𝑊))‘𝑘) ∈ (Base‘(Scalar‘𝑊)))
5922, 23, 56, 58syl3anc 1368 . . . . . . 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 1130 . . . . . . . 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 837 . . . . . . 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 20846 . . . . . . 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 837 . . . . . 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 20654 . . . . . . . 8 (((Scalar‘𝑊) ∈ DivRing ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ≠ (0g‘(Scalar‘𝑊))) → ((invr‘(Scalar‘𝑊))‘𝑘) ≠ (0g‘(Scalar‘𝑊)))
6522, 23, 56, 64syl3anc 1368 . . . . . . 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 1130 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → ¬ 𝑋 ∈ (𝑁‘{𝑌}))
68 simpl3 1190 . . . . . . . . . . . . 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 7433 . . . . . . . . . . . . . . 15 (𝑙 = (0g‘(Scalar‘𝑊)) → (𝑙( ·𝑠𝑊)𝑍) = ((0g‘(Scalar‘𝑊))( ·𝑠𝑊)𝑍))
702, 4, 6, 32, 33lmod0vs 20785 . . . . . . . . . . . . . . . 16 ((𝑊 ∈ LMod ∧ 𝑍𝑉) → ((0g‘(Scalar‘𝑊))( ·𝑠𝑊)𝑍) = 0 )
7115, 39, 70syl2anc 582 . . . . . . . . . . . . . . 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 2790 . . . . . . . . . . . . . 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 7442 . . . . . . . . . . . . 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 1130 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → 𝑌𝑉)
752, 4, 6, 5lmodvscl 20768 . . . . . . . . . . . . . . . 16 ((𝑊 ∈ LMod ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑌𝑉) → (𝑘( ·𝑠𝑊)𝑌) ∈ 𝑉)
7615, 23, 74, 75syl3anc 1368 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → (𝑘( ·𝑠𝑊)𝑌) ∈ 𝑉)
772, 3, 33lmod0vrid 20783 . . . . . . . . . . . . . . 15 ((𝑊 ∈ LMod ∧ (𝑘( ·𝑠𝑊)𝑌) ∈ 𝑉) → ((𝑘( ·𝑠𝑊)𝑌) + 0 ) = (𝑘( ·𝑠𝑊)𝑌))
7815, 76, 77syl2anc 582 . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . 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 2772 . . . . . . . . . . . 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 20868 . . . . . . . . . . . . . . . 16 ((𝑊 ∈ LMod ∧ 𝑌𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
8210, 11, 81syl2anc 582 . . . . . . . . . . . . . . 15 (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
83823ad2ant1 1130 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
842, 7lspsnid 20884 . . . . . . . . . . . . . . . 16 ((𝑊 ∈ LMod ∧ 𝑌𝑉) → 𝑌 ∈ (𝑁‘{𝑌}))
8510, 11, 84syl2anc 582 . . . . . . . . . . . . . . 15 (𝜑𝑌 ∈ (𝑁‘{𝑌}))
86853ad2ant1 1130 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → 𝑌 ∈ (𝑁‘{𝑌}))
874, 6, 5, 16lssvscl 20846 . . . . . . . . . . . . . 14 (((𝑊 ∈ LMod ∧ (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑌 ∈ (𝑁‘{𝑌}))) → (𝑘( ·𝑠𝑊)𝑌) ∈ (𝑁‘{𝑌}))
8815, 83, 23, 86, 87syl22anc 837 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → (𝑘( ·𝑠𝑊)𝑌) ∈ (𝑁‘{𝑌}))
8988adantr 479 . . . . . . . . . . . 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 2829 . . . . . . . . . . 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 411 . . . . . . . . . 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 2951 . . . . . . . . 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 1188 . . . . . . . . . . . . 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 4743 . . . . . . . . . . . . . 14 (𝑍 = 0 → {𝑌, 𝑍} = {𝑌, 0 })
9796fveq2d 6906 . . . . . . . . . . . . 13 (𝑍 = 0 → (𝑁‘{𝑌, 𝑍}) = (𝑁‘{𝑌, 0 }))
982, 33, 7, 15, 74lsppr0 20984 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → (𝑁‘{𝑌, 0 }) = (𝑁‘{𝑌}))
9997, 98sylan9eqr 2790 . . . . . . . . . . . 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 2831 . . . . . . . . . . 11 (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) ∧ 𝑍 = 0 ) → 𝑋 ∈ (𝑁‘{𝑌}))
101100ex 411 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → (𝑍 = 0𝑋 ∈ (𝑁‘{𝑌})))
102101necon3bd 2951 . . . . . . . . 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 21004 . . . . . . . 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 711 . . . . . . 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 21004 . . . . . . 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 711 . . . . . 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 4795 . . . . . 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 581 . . . . 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 1135 . . . . . . 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 20765 . . . . . . . . 9 ((𝑊 ∈ LMod ∧ (𝑘( ·𝑠𝑊)𝑌) ∈ 𝑉 ∧ (𝑙( ·𝑠𝑊)𝑍) ∈ 𝑉) → ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍)) ∈ 𝑉)
11215, 76, 41, 111syl3anc 1368 . . . . . . . 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 20884 . . . . . . . 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 582 . . . . . . 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 2829 . . . . . 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 21009 . . . . . . . 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 1372 . . . . . . 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 20775 . . . . . . . . . . 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 1369 . . . . . . . . . 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 2728 . . . . . . . . . . . . . . 15 (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊))
121 eqid 2728 . . . . . . . . . . . . . . 15 (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑊))
1225, 32, 120, 121, 57drnginvrl 20656 . . . . . . . . . . . . . 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 1368 . . . . . . . . . . . . 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 7441 . . . . . . . . . . . 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 20777 . . . . . . . . . . . . 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 1369 . . . . . . . . . . . 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 20780 . . . . . . . . . . . . 13 ((𝑊 ∈ LMod ∧ 𝑌𝑉) → ((1r‘(Scalar‘𝑊))( ·𝑠𝑊)𝑌) = 𝑌)
12815, 74, 127syl2anc 582 . . . . . . . . . . . 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 2776 . . . . . . . . . . 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 7441 . . . . . . . . . 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 2768 . . . . . . . . 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 4644 . . . . . . . 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 6906 . . . . . . 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 2770 . . . . . 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 2831 . . . . 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 7434 . . . . . . . . 9 (𝑧 = (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑙( ·𝑠𝑊)𝑍)) → (𝑌 + 𝑧) = (𝑌 + (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑙( ·𝑠𝑊)𝑍))))
137136sneqd 4644 . . . . . . . 8 (𝑧 = (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑙( ·𝑠𝑊)𝑍)) → {(𝑌 + 𝑧)} = {(𝑌 + (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑙( ·𝑠𝑊)𝑍)))})
138137fveq2d 6906 . . . . . . 7 (𝑧 = (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑙( ·𝑠𝑊)𝑍)) → (𝑁‘{(𝑌 + 𝑧)}) = (𝑁‘{(𝑌 + (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑙( ·𝑠𝑊)𝑍)))}))
139138eleq2d 2815 . . . . . 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 3611 . . . . 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 582 . . . 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 1116 . . 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 3208 . 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 394  w3a 1084   = wceq 1533  wcel 2098  wne 2937  wrex 3067  cdif 3946  {csn 4632  {cpr 4634  cfv 6553  (class class class)co 7426  Basecbs 17187  +gcplusg 17240  .rcmulr 17241  Scalarcsca 17243   ·𝑠 cvsca 17244  0gc0g 17428  1rcur 20128  invrcinvr 20333  DivRingcdr 20631  LModclmod 20750  LSubSpclss 20822  LSpanclspn 20862  LVecclvec 20994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-tpos 8238  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-er 8731  df-en 8971  df-dom 8972  df-sdom 8973  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12251  df-2 12313  df-3 12314  df-sets 17140  df-slot 17158  df-ndx 17170  df-base 17188  df-ress 17217  df-plusg 17253  df-mulr 17254  df-0g 17430  df-mgm 18607  df-sgrp 18686  df-mnd 18702  df-submnd 18748  df-grp 18900  df-minusg 18901  df-sbg 18902  df-subg 19085  df-cntz 19275  df-lsm 19598  df-cmn 19744  df-abl 19745  df-mgp 20082  df-rng 20100  df-ur 20129  df-ring 20182  df-oppr 20280  df-dvdsr 20303  df-unit 20304  df-invr 20334  df-drng 20633  df-lmod 20752  df-lss 20823  df-lsp 20863  df-lvec 20995
This theorem is referenced by:  lsatfixedN  38513
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