Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lincscmcl Structured version   Visualization version   GIF version

Theorem lincscmcl 42015
 Description: The multiplication of a linear combination with a scalar is a linear combination, see also the proof in [Lang] p. 129. (Contributed by AV, 11-Apr-2019.) (Proof shortened by AV, 28-Jul-2019.)
Hypotheses
Ref Expression
lincscmcl.s · = ( ·𝑠𝑀)
lincscmcl.r 𝑅 = (Base‘(Scalar‘𝑀))
Assertion
Ref Expression
lincscmcl (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅𝐷 ∈ (𝑀 LinCo 𝑉)) → (𝐶 · 𝐷) ∈ (𝑀 LinCo 𝑉))

Proof of Theorem lincscmcl
Dummy variables 𝑠 𝑣 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2610 . . . . 5 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2610 . . . . 5 (Scalar‘𝑀) = (Scalar‘𝑀)
3 lincscmcl.r . . . . 5 𝑅 = (Base‘(Scalar‘𝑀))
41, 2, 3lcoval 41995 . . . 4 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝐷 ∈ (𝑀 LinCo 𝑉) ↔ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ (𝑅𝑚 𝑉)(𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))))
54adantr 480 . . 3 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅) → (𝐷 ∈ (𝑀 LinCo 𝑉) ↔ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ (𝑅𝑚 𝑉)(𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))))
6 simpl 472 . . . . . . 7 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑀 ∈ LMod)
76ad2antrr 758 . . . . . 6 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅) ∧ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ (𝑅𝑚 𝑉)(𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) → 𝑀 ∈ LMod)
8 simpr 476 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅) → 𝐶𝑅)
98adantr 480 . . . . . 6 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅) ∧ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ (𝑅𝑚 𝑉)(𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) → 𝐶𝑅)
10 simprl 790 . . . . . 6 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅) ∧ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ (𝑅𝑚 𝑉)(𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) → 𝐷 ∈ (Base‘𝑀))
11 lincscmcl.s . . . . . . 7 · = ( ·𝑠𝑀)
121, 2, 11, 3lmodvscl 18703 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝐶𝑅𝐷 ∈ (Base‘𝑀)) → (𝐶 · 𝐷) ∈ (Base‘𝑀))
137, 9, 10, 12syl3anc 1318 . . . . 5 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅) ∧ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ (𝑅𝑚 𝑉)(𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) → (𝐶 · 𝐷) ∈ (Base‘𝑀))
142lmodring 18694 . . . . . . . . . . . . . . . . 17 (𝑀 ∈ LMod → (Scalar‘𝑀) ∈ Ring)
1514ad2antrr 758 . . . . . . . . . . . . . . . 16 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅) → (Scalar‘𝑀) ∈ Ring)
1615adantl 481 . . . . . . . . . . . . . . 15 ((((𝑥 ∈ (𝑅𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅)) → (Scalar‘𝑀) ∈ Ring)
1716adantr 480 . . . . . . . . . . . . . 14 (((((𝑥 ∈ (𝑅𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅)) ∧ 𝑣𝑉) → (Scalar‘𝑀) ∈ Ring)
188adantl 481 . . . . . . . . . . . . . . 15 ((((𝑥 ∈ (𝑅𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅)) → 𝐶𝑅)
1918adantr 480 . . . . . . . . . . . . . 14 (((((𝑥 ∈ (𝑅𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅)) ∧ 𝑣𝑉) → 𝐶𝑅)
20 elmapi 7765 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝑅𝑚 𝑉) → 𝑥:𝑉𝑅)
21 ffvelrn 6265 . . . . . . . . . . . . . . . . . . 19 ((𝑥:𝑉𝑅𝑣𝑉) → (𝑥𝑣) ∈ 𝑅)
2221ex 449 . . . . . . . . . . . . . . . . . 18 (𝑥:𝑉𝑅 → (𝑣𝑉 → (𝑥𝑣) ∈ 𝑅))
2320, 22syl 17 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (𝑅𝑚 𝑉) → (𝑣𝑉 → (𝑥𝑣) ∈ 𝑅))
2423adantr 480 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ (𝑅𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → (𝑣𝑉 → (𝑥𝑣) ∈ 𝑅))
2524ad2antrr 758 . . . . . . . . . . . . . . 15 ((((𝑥 ∈ (𝑅𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅)) → (𝑣𝑉 → (𝑥𝑣) ∈ 𝑅))
2625imp 444 . . . . . . . . . . . . . 14 (((((𝑥 ∈ (𝑅𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅)) ∧ 𝑣𝑉) → (𝑥𝑣) ∈ 𝑅)
27 eqid 2610 . . . . . . . . . . . . . . 15 (.r‘(Scalar‘𝑀)) = (.r‘(Scalar‘𝑀))
283, 27ringcl 18384 . . . . . . . . . . . . . 14 (((Scalar‘𝑀) ∈ Ring ∧ 𝐶𝑅 ∧ (𝑥𝑣) ∈ 𝑅) → (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣)) ∈ 𝑅)
2917, 19, 26, 28syl3anc 1318 . . . . . . . . . . . . 13 (((((𝑥 ∈ (𝑅𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅)) ∧ 𝑣𝑉) → (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣)) ∈ 𝑅)
30 eqid 2610 . . . . . . . . . . . . 13 (𝑣𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣))) = (𝑣𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣)))
3129, 30fmptd 6292 . . . . . . . . . . . 12 ((((𝑥 ∈ (𝑅𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅)) → (𝑣𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣))):𝑉𝑅)
32 fvex 6113 . . . . . . . . . . . . . 14 (Base‘(Scalar‘𝑀)) ∈ V
333, 32eqeltri 2684 . . . . . . . . . . . . 13 𝑅 ∈ V
34 simpr 476 . . . . . . . . . . . . . . 15 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑉 ∈ 𝒫 (Base‘𝑀))
3534adantr 480 . . . . . . . . . . . . . 14 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅) → 𝑉 ∈ 𝒫 (Base‘𝑀))
3635adantl 481 . . . . . . . . . . . . 13 ((((𝑥 ∈ (𝑅𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅)) → 𝑉 ∈ 𝒫 (Base‘𝑀))
37 elmapg 7757 . . . . . . . . . . . . 13 ((𝑅 ∈ V ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ((𝑣𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣))) ∈ (𝑅𝑚 𝑉) ↔ (𝑣𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣))):𝑉𝑅))
3833, 36, 37sylancr 694 . . . . . . . . . . . 12 ((((𝑥 ∈ (𝑅𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅)) → ((𝑣𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣))) ∈ (𝑅𝑚 𝑉) ↔ (𝑣𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣))):𝑉𝑅))
3931, 38mpbird 246 . . . . . . . . . . 11 ((((𝑥 ∈ (𝑅𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅)) → (𝑣𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣))) ∈ (𝑅𝑚 𝑉))
4015, 35, 83jca 1235 . . . . . . . . . . . . 13 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅) → ((Scalar‘𝑀) ∈ Ring ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐶𝑅))
4140adantl 481 . . . . . . . . . . . 12 ((((𝑥 ∈ (𝑅𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅)) → ((Scalar‘𝑀) ∈ Ring ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐶𝑅))
42 simpl 472 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝑅𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → 𝑥 ∈ (𝑅𝑚 𝑉))
4342ad2antrr 758 . . . . . . . . . . . 12 ((((𝑥 ∈ (𝑅𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅)) → 𝑥 ∈ (𝑅𝑚 𝑉))
44 simprl 790 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝑅𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → 𝑥 finSupp (0g‘(Scalar‘𝑀)))
4544ad2antrr 758 . . . . . . . . . . . 12 ((((𝑥 ∈ (𝑅𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅)) → 𝑥 finSupp (0g‘(Scalar‘𝑀)))
463rmfsupp 41949 . . . . . . . . . . . 12 ((((Scalar‘𝑀) ∈ Ring ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐶𝑅) ∧ 𝑥 ∈ (𝑅𝑚 𝑉) ∧ 𝑥 finSupp (0g‘(Scalar‘𝑀))) → (𝑣𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣))) finSupp (0g‘(Scalar‘𝑀)))
4741, 43, 45, 46syl3anc 1318 . . . . . . . . . . 11 ((((𝑥 ∈ (𝑅𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅)) → (𝑣𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣))) finSupp (0g‘(Scalar‘𝑀)))
48 oveq2 6557 . . . . . . . . . . . . . . 15 (𝐷 = (𝑥( linC ‘𝑀)𝑉) → (𝐶 · 𝐷) = (𝐶 · (𝑥( linC ‘𝑀)𝑉)))
4948adantl 481 . . . . . . . . . . . . . 14 ((𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)) → (𝐶 · 𝐷) = (𝐶 · (𝑥( linC ‘𝑀)𝑉)))
5049adantl 481 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝑅𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → (𝐶 · 𝐷) = (𝐶 · (𝑥( linC ‘𝑀)𝑉)))
5150ad2antrr 758 . . . . . . . . . . . 12 ((((𝑥 ∈ (𝑅𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅)) → (𝐶 · 𝐷) = (𝐶 · (𝑥( linC ‘𝑀)𝑉)))
52 simprl 790 . . . . . . . . . . . . 13 ((((𝑥 ∈ (𝑅𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅)) → (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)))
5342adantr 480 . . . . . . . . . . . . . 14 (((𝑥 ∈ (𝑅𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) → 𝑥 ∈ (𝑅𝑚 𝑉))
5453, 8anim12i 588 . . . . . . . . . . . . 13 ((((𝑥 ∈ (𝑅𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅)) → (𝑥 ∈ (𝑅𝑚 𝑉) ∧ 𝐶𝑅))
55 eqid 2610 . . . . . . . . . . . . . 14 (𝑥( linC ‘𝑀)𝑉) = (𝑥( linC ‘𝑀)𝑉)
5611, 27, 55, 3, 30lincscm 42013 . . . . . . . . . . . . 13 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑥 ∈ (𝑅𝑚 𝑉) ∧ 𝐶𝑅) ∧ 𝑥 finSupp (0g‘(Scalar‘𝑀))) → (𝐶 · (𝑥( linC ‘𝑀)𝑉)) = ((𝑣𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣)))( linC ‘𝑀)𝑉))
5752, 54, 45, 56syl3anc 1318 . . . . . . . . . . . 12 ((((𝑥 ∈ (𝑅𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅)) → (𝐶 · (𝑥( linC ‘𝑀)𝑉)) = ((𝑣𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣)))( linC ‘𝑀)𝑉))
5851, 57eqtrd 2644 . . . . . . . . . . 11 ((((𝑥 ∈ (𝑅𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅)) → (𝐶 · 𝐷) = ((𝑣𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣)))( linC ‘𝑀)𝑉))
59 breq1 4586 . . . . . . . . . . . . 13 (𝑠 = (𝑣𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣))) → (𝑠 finSupp (0g‘(Scalar‘𝑀)) ↔ (𝑣𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣))) finSupp (0g‘(Scalar‘𝑀))))
60 oveq1 6556 . . . . . . . . . . . . . 14 (𝑠 = (𝑣𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣))) → (𝑠( linC ‘𝑀)𝑉) = ((𝑣𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣)))( linC ‘𝑀)𝑉))
6160eqeq2d 2620 . . . . . . . . . . . . 13 (𝑠 = (𝑣𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣))) → ((𝐶 · 𝐷) = (𝑠( linC ‘𝑀)𝑉) ↔ (𝐶 · 𝐷) = ((𝑣𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣)))( linC ‘𝑀)𝑉)))
6259, 61anbi12d 743 . . . . . . . . . . . 12 (𝑠 = (𝑣𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣))) → ((𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝐶 · 𝐷) = (𝑠( linC ‘𝑀)𝑉)) ↔ ((𝑣𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣))) finSupp (0g‘(Scalar‘𝑀)) ∧ (𝐶 · 𝐷) = ((𝑣𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣)))( linC ‘𝑀)𝑉))))
6362rspcev 3282 . . . . . . . . . . 11 (((𝑣𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣))) ∈ (𝑅𝑚 𝑉) ∧ ((𝑣𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣))) finSupp (0g‘(Scalar‘𝑀)) ∧ (𝐶 · 𝐷) = ((𝑣𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣)))( linC ‘𝑀)𝑉))) → ∃𝑠 ∈ (𝑅𝑚 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝐶 · 𝐷) = (𝑠( linC ‘𝑀)𝑉)))
6439, 47, 58, 63syl12anc 1316 . . . . . . . . . 10 ((((𝑥 ∈ (𝑅𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅)) → ∃𝑠 ∈ (𝑅𝑚 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝐶 · 𝐷) = (𝑠( linC ‘𝑀)𝑉)))
6564ex 449 . . . . . . . . 9 (((𝑥 ∈ (𝑅𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) → (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅) → ∃𝑠 ∈ (𝑅𝑚 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝐶 · 𝐷) = (𝑠( linC ‘𝑀)𝑉))))
6665ex 449 . . . . . . . 8 ((𝑥 ∈ (𝑅𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → (𝐷 ∈ (Base‘𝑀) → (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅) → ∃𝑠 ∈ (𝑅𝑚 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝐶 · 𝐷) = (𝑠( linC ‘𝑀)𝑉)))))
6766rexlimiva 3010 . . . . . . 7 (∃𝑥 ∈ (𝑅𝑚 𝑉)(𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)) → (𝐷 ∈ (Base‘𝑀) → (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅) → ∃𝑠 ∈ (𝑅𝑚 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝐶 · 𝐷) = (𝑠( linC ‘𝑀)𝑉)))))
6867impcom 445 . . . . . 6 ((𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ (𝑅𝑚 𝑉)(𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅) → ∃𝑠 ∈ (𝑅𝑚 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝐶 · 𝐷) = (𝑠( linC ‘𝑀)𝑉))))
6968impcom 445 . . . . 5 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅) ∧ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ (𝑅𝑚 𝑉)(𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) → ∃𝑠 ∈ (𝑅𝑚 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝐶 · 𝐷) = (𝑠( linC ‘𝑀)𝑉)))
701, 2, 3lcoval 41995 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ((𝐶 · 𝐷) ∈ (𝑀 LinCo 𝑉) ↔ ((𝐶 · 𝐷) ∈ (Base‘𝑀) ∧ ∃𝑠 ∈ (𝑅𝑚 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝐶 · 𝐷) = (𝑠( linC ‘𝑀)𝑉)))))
7170ad2antrr 758 . . . . 5 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅) ∧ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ (𝑅𝑚 𝑉)(𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) → ((𝐶 · 𝐷) ∈ (𝑀 LinCo 𝑉) ↔ ((𝐶 · 𝐷) ∈ (Base‘𝑀) ∧ ∃𝑠 ∈ (𝑅𝑚 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝐶 · 𝐷) = (𝑠( linC ‘𝑀)𝑉)))))
7213, 69, 71mpbir2and 959 . . . 4 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅) ∧ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ (𝑅𝑚 𝑉)(𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) → (𝐶 · 𝐷) ∈ (𝑀 LinCo 𝑉))
7372ex 449 . . 3 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅) → ((𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ (𝑅𝑚 𝑉)(𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → (𝐶 · 𝐷) ∈ (𝑀 LinCo 𝑉)))
745, 73sylbid 229 . 2 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅) → (𝐷 ∈ (𝑀 LinCo 𝑉) → (𝐶 · 𝐷) ∈ (𝑀 LinCo 𝑉)))
75743impia 1253 1 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅𝐷 ∈ (𝑀 LinCo 𝑉)) → (𝐶 · 𝐷) ∈ (𝑀 LinCo 𝑉))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173  𝒫 cpw 4108   class class class wbr 4583   ↦ cmpt 4643  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744   finSupp cfsupp 8158  Basecbs 15695  .rcmulr 15769  Scalarcsca 15771   ·𝑠 cvsca 15772  0gc0g 15923  Ringcrg 18370  LModclmod 18686   linC clinc 41987   LinCo clinco 41988 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664  df-hash 12980  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-plusg 15781  df-0g 15925  df-gsum 15926  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-grp 17248  df-minusg 17249  df-ghm 17481  df-cntz 17573  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-ring 18372  df-lmod 18688  df-linc 41989  df-lco 41990 This theorem is referenced by:  lincsumscmcl  42016
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