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Theorem lspfixed 21148
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 2735 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
5 eqid 2735 . . . 4 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6 eqid 2735 . . . 4 ( ·𝑠𝑊) = ( ·𝑠𝑊)
7 lspfixed.n . . . 4 𝑁 = (LSpan‘𝑊)
8 lspfixed.w . . . . 5 (𝜑𝑊 ∈ LVec)
9 lveclmod 21123 . . . . 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 21111 . . 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 1132 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → 𝑊 ∈ LMod)
16 eqid 2735 . . . . . . . . . 10 (LSubSp‘𝑊) = (LSubSp‘𝑊)
172, 16, 7lspsncl 20993 . . . . . . . . 9 ((𝑊 ∈ LMod ∧ 𝑍𝑉) → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊))
1810, 12, 17syl2anc 584 . . . . . . . 8 (𝜑 → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊))
19183ad2ant1 1132 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊))
2083ad2ant1 1132 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → 𝑊 ∈ LVec)
214lvecdrng 21122 . . . . . . . . 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 1198 . . . . . . . 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 1132 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → ¬ 𝑋 ∈ (𝑁‘{𝑍}))
26 simpl3 1192 . . . . . . . . . . . . 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 1190 . . . . . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . . . . 17 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
33 lspfixed.o . . . . . . . . . . . . . . . . 17 0 = (0g𝑊)
342, 4, 6, 32, 33lmod0vs 20910 . . . . . . . . . . . . . . . 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 2775 . . . . . . . . . . . . . 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 1199 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → 𝑙 ∈ (Base‘(Scalar‘𝑊)))
39123ad2ant1 1132 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → 𝑍𝑉)
402, 4, 6, 5lmodvscl 20893 . . . . . . . . . . . . . . . 16 ((𝑊 ∈ LMod ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑍𝑉) → (𝑙( ·𝑠𝑊)𝑍) ∈ 𝑉)
4115, 38, 39, 40syl3anc 1370 . . . . . . . . . . . . . . 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 20907 . . . . . . . . . . . . . 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 2779 . . . . . . . . . . . 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 1226 . . . . . . . . . . . . 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 21009 . . . . . . . . . . . . . . 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 20971 . . . . . . . . . . . . 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 2839 . . . . . . . . . . 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 2952 . . . . . . . . 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 2735 . . . . . . . . 9 (invr‘(Scalar‘𝑊)) = (invr‘(Scalar‘𝑊))
585, 32, 57drnginvrcl 20770 . . . . . . . 8 (((Scalar‘𝑊) ∈ DivRing ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ≠ (0g‘(Scalar‘𝑊))) → ((invr‘(Scalar‘𝑊))‘𝑘) ∈ (Base‘(Scalar‘𝑊)))
5922, 23, 56, 58syl3anc 1370 . . . . . . 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 1132 . . . . . . . 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 20971 . . . . . . 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 20771 . . . . . . . 8 (((Scalar‘𝑊) ∈ DivRing ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ≠ (0g‘(Scalar‘𝑊))) → ((invr‘(Scalar‘𝑊))‘𝑘) ≠ (0g‘(Scalar‘𝑊)))
6522, 23, 56, 64syl3anc 1370 . . . . . . 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 1132 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → ¬ 𝑋 ∈ (𝑁‘{𝑌}))
68 simpl3 1192 . . . . . . . . . . . . 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 20910 . . . . . . . . . . . . . . . 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 2797 . . . . . . . . . . . . . 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 1132 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → 𝑌𝑉)
752, 4, 6, 5lmodvscl 20893 . . . . . . . . . . . . . . . 16 ((𝑊 ∈ LMod ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑌𝑉) → (𝑘( ·𝑠𝑊)𝑌) ∈ 𝑉)
7615, 23, 74, 75syl3anc 1370 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → (𝑘( ·𝑠𝑊)𝑌) ∈ 𝑉)
772, 3, 33lmod0vrid 20908 . . . . . . . . . . . . . . 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 2779 . . . . . . . . . . . 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 20993 . . . . . . . . . . . . . . . 16 ((𝑊 ∈ LMod ∧ 𝑌𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
8210, 11, 81syl2anc 584 . . . . . . . . . . . . . . 15 (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
83823ad2ant1 1132 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
842, 7lspsnid 21009 . . . . . . . . . . . . . . . 16 ((𝑊 ∈ LMod ∧ 𝑌𝑉) → 𝑌 ∈ (𝑁‘{𝑌}))
8510, 11, 84syl2anc 584 . . . . . . . . . . . . . . 15 (𝜑𝑌 ∈ (𝑁‘{𝑌}))
86853ad2ant1 1132 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → 𝑌 ∈ (𝑁‘{𝑌}))
874, 6, 5, 16lssvscl 20971 . . . . . . . . . . . . . 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 2839 . . . . . . . . . . 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 2952 . . . . . . . . 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 1190 . . . . . . . . . . . . 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 4739 . . . . . . . . . . . . . 14 (𝑍 = 0 → {𝑌, 𝑍} = {𝑌, 0 })
9796fveq2d 6911 . . . . . . . . . . . . 13 (𝑍 = 0 → (𝑁‘{𝑌, 𝑍}) = (𝑁‘{𝑌, 0 }))
982, 33, 7, 15, 74lsppr0 21109 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍))) → (𝑁‘{𝑌, 0 }) = (𝑁‘{𝑌}))
9997, 98sylan9eqr 2797 . . . . . . . . . . . 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 2841 . . . . . . . . . . 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 2952 . . . . . . . . 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 21129 . . . . . . . 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 21129 . . . . . . 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 4791 . . . . . 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 1137 . . . . . . 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 20890 . . . . . . . . 9 ((𝑊 ∈ LMod ∧ (𝑘( ·𝑠𝑊)𝑌) ∈ 𝑉 ∧ (𝑙( ·𝑠𝑊)𝑍) ∈ 𝑉) → ((𝑘( ·𝑠𝑊)𝑌) + (𝑙( ·𝑠𝑊)𝑍)) ∈ 𝑉)
11215, 76, 41, 111syl3anc 1370 . . . . . . . 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 21009 . . . . . . . 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 2839 . . . . . 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 21134 . . . . . . . 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 1374 . . . . . . 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 20900 . . . . . . . . . . 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 1371 . . . . . . . . . 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 2735 . . . . . . . . . . . . . . 15 (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊))
121 eqid 2735 . . . . . . . . . . . . . . 15 (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑊))
1225, 32, 120, 121, 57drnginvrl 20773 . . . . . . . . . . . . . 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 1370 . . . . . . . . . . . . 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 20902 . . . . . . . . . . . . 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 1371 . . . . . . . . . . . 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 20905 . . . . . . . . . . . . 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 2783 . . . . . . . . . . 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 2775 . . . . . . . . 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 4643 . . . . . . . 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 6911 . . . . . . 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 2777 . . . . . 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 2841 . . . . 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 4643 . . . . . . . 8 (𝑧 = (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑙( ·𝑠𝑊)𝑍)) → {(𝑌 + 𝑧)} = {(𝑌 + (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑙( ·𝑠𝑊)𝑍)))})
138137fveq2d 6911 . . . . . . 7 (𝑧 = (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑙( ·𝑠𝑊)𝑍)) → (𝑁‘{(𝑌 + 𝑧)}) = (𝑁‘{(𝑌 + (((invr‘(Scalar‘𝑊))‘𝑘)( ·𝑠𝑊)(𝑙( ·𝑠𝑊)𝑍)))}))
139138eleq2d 2825 . . . . . 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 1118 . . 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 3210 . 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 1086   = wceq 1537  wcel 2106  wne 2938  wrex 3068  cdif 3960  {csn 4631  {cpr 4633  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  .rcmulr 17299  Scalarcsca 17301   ·𝑠 cvsca 17302  0gc0g 17486  1rcur 20199  invrcinvr 20404  DivRingcdr 20746  LModclmod 20875  LSubSpclss 20947  LSpanclspn 20987  LVecclvec 21119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-tpos 8250  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-submnd 18810  df-grp 18967  df-minusg 18968  df-sbg 18969  df-subg 19154  df-cntz 19348  df-lsm 19669  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-oppr 20351  df-dvdsr 20374  df-unit 20375  df-invr 20405  df-drng 20748  df-lmod 20877  df-lss 20948  df-lsp 20988  df-lvec 21120
This theorem is referenced by:  lsatfixedN  38991
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