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Theorem lincscmcl 42783
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 2806 . . . . 5 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2806 . . . . 5 (Scalar‘𝑀) = (Scalar‘𝑀)
3 lincscmcl.r . . . . 5 𝑅 = (Base‘(Scalar‘𝑀))
41, 2, 3lcoval 42763 . . . 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 468 . . 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 470 . . . . . . 7 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑀 ∈ LMod)
76ad2antrr 708 . . . . . 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 473 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅) → 𝐶𝑅)
98adantr 468 . . . . . 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 778 . . . . . 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 19080 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝐶𝑅𝐷 ∈ (Base‘𝑀)) → (𝐶 · 𝐷) ∈ (Base‘𝑀))
137, 9, 10, 12syl3anc 1483 . . . . 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 19071 . . . . . . . . . . . . . . . . 17 (𝑀 ∈ LMod → (Scalar‘𝑀) ∈ Ring)
1514ad2antrr 708 . . . . . . . . . . . . . . . 16 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅) → (Scalar‘𝑀) ∈ Ring)
1615adantl 469 . . . . . . . . . . . . . . 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 468 . . . . . . . . . . . . . 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 469 . . . . . . . . . . . . . . 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 468 . . . . . . . . . . . . . 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 8110 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝑅𝑚 𝑉) → 𝑥:𝑉𝑅)
21 ffvelrn 6575 . . . . . . . . . . . . . . . . . . 19 ((𝑥:𝑉𝑅𝑣𝑉) → (𝑥𝑣) ∈ 𝑅)
2221ex 399 . . . . . . . . . . . . . . . . . 18 (𝑥:𝑉𝑅 → (𝑣𝑉 → (𝑥𝑣) ∈ 𝑅))
2320, 22syl 17 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (𝑅𝑚 𝑉) → (𝑣𝑉 → (𝑥𝑣) ∈ 𝑅))
2423adantr 468 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ (𝑅𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → (𝑣𝑉 → (𝑥𝑣) ∈ 𝑅))
2524ad2antrr 708 . . . . . . . . . . . . . . 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 395 . . . . . . . . . . . . . 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 2806 . . . . . . . . . . . . . . 15 (.r‘(Scalar‘𝑀)) = (.r‘(Scalar‘𝑀))
283, 27ringcl 18759 . . . . . . . . . . . . . 14 (((Scalar‘𝑀) ∈ Ring ∧ 𝐶𝑅 ∧ (𝑥𝑣) ∈ 𝑅) → (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣)) ∈ 𝑅)
2917, 19, 26, 28syl3anc 1483 . . . . . . . . . . . . 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)
3029fmpttd 6603 . . . . . . . . . . . 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)
313fvexi 6418 . . . . . . . . . . . . 13 𝑅 ∈ V
32 simpr 473 . . . . . . . . . . . . . . 15 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑉 ∈ 𝒫 (Base‘𝑀))
3332adantr 468 . . . . . . . . . . . . . 14 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅) → 𝑉 ∈ 𝒫 (Base‘𝑀))
3433adantl 469 . . . . . . . . . . . . 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))
35 elmapg 8101 . . . . . . . . . . . . 13 ((𝑅 ∈ V ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ((𝑣𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣))) ∈ (𝑅𝑚 𝑉) ↔ (𝑣𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣))):𝑉𝑅))
3631, 34, 35sylancr 577 . . . . . . . . . . . 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))
3730, 36mpbird 248 . . . . . . . . . . 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))
3815, 33, 83jca 1151 . . . . . . . . . . . . 13 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅) → ((Scalar‘𝑀) ∈ Ring ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐶𝑅))
3938adantl 469 . . . . . . . . . . . 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))
40 simpl 470 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝑅𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → 𝑥 ∈ (𝑅𝑚 𝑉))
4140ad2antrr 708 . . . . . . . . . . . 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))
42 simprl 778 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝑅𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → 𝑥 finSupp (0g‘(Scalar‘𝑀)))
4342ad2antrr 708 . . . . . . . . . . . 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)))
443rmfsupp 42717 . . . . . . . . . . . 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)))
4539, 41, 43, 44syl3anc 1483 . . . . . . . . . . 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)))
46 oveq2 6878 . . . . . . . . . . . . . . 15 (𝐷 = (𝑥( linC ‘𝑀)𝑉) → (𝐶 · 𝐷) = (𝐶 · (𝑥( linC ‘𝑀)𝑉)))
4746adantl 469 . . . . . . . . . . . . . 14 ((𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)) → (𝐶 · 𝐷) = (𝐶 · (𝑥( linC ‘𝑀)𝑉)))
4847adantl 469 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝑅𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → (𝐶 · 𝐷) = (𝐶 · (𝑥( linC ‘𝑀)𝑉)))
4948ad2antrr 708 . . . . . . . . . . . 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)))
50 simprl 778 . . . . . . . . . . . . 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)))
5140adantr 468 . . . . . . . . . . . . . 14 (((𝑥 ∈ (𝑅𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) → 𝑥 ∈ (𝑅𝑚 𝑉))
5251, 8anim12i 602 . . . . . . . . . . . . 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))
53 eqid 2806 . . . . . . . . . . . . . 14 (𝑥( linC ‘𝑀)𝑉) = (𝑥( linC ‘𝑀)𝑉)
54 eqid 2806 . . . . . . . . . . . . . 14 (𝑣𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣))) = (𝑣𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣)))
5511, 27, 53, 3, 54lincscm 42781 . . . . . . . . . . . . 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))
5650, 52, 43, 55syl3anc 1483 . . . . . . . . . . . 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))
5749, 56eqtrd 2840 . . . . . . . . . . 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))
58 breq1 4847 . . . . . . . . . . . . 13 (𝑠 = (𝑣𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣))) → (𝑠 finSupp (0g‘(Scalar‘𝑀)) ↔ (𝑣𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣))) finSupp (0g‘(Scalar‘𝑀))))
59 oveq1 6877 . . . . . . . . . . . . . 14 (𝑠 = (𝑣𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣))) → (𝑠( linC ‘𝑀)𝑉) = ((𝑣𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣)))( linC ‘𝑀)𝑉))
6059eqeq2d 2816 . . . . . . . . . . . . 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)))
6158, 60anbi12d 618 . . . . . . . . . . . 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))))
6261rspcev 3502 . . . . . . . . . . 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)))
6337, 45, 57, 62syl12anc 856 . . . . . . . . . 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)))
6463ex 399 . . . . . . . . 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))))
6564ex 399 . . . . . . . 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)))))
6665rexlimiva 3216 . . . . . . 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)))))
6766impcom 396 . . . . . 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))))
6867impcom 396 . . . . 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)))
691, 2, 3lcoval 42763 . . . . . 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)))))
7069ad2antrr 708 . . . . 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)))))
7113, 68, 70mpbir2and 695 . . . 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))
7271ex 399 . . 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)))
735, 72sylbid 231 . 2 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅) → (𝐷 ∈ (𝑀 LinCo 𝑉) → (𝐶 · 𝐷) ∈ (𝑀 LinCo 𝑉)))
74733impia 1138 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 197  wa 384  w3a 1100   = wceq 1637  wcel 2156  wrex 3097  Vcvv 3391  𝒫 cpw 4351   class class class wbr 4844  cmpt 4923  wf 6093  cfv 6097  (class class class)co 6870  𝑚 cmap 8088   finSupp cfsupp 8510  Basecbs 16064  .rcmulr 16150  Scalarcsca 16152   ·𝑠 cvsca 16153  0gc0g 16301  Ringcrg 18745  LModclmod 19063   linC clinc 42755   LinCo clinco 42756
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175  ax-cnex 10273  ax-resscn 10274  ax-1cn 10275  ax-icn 10276  ax-addcl 10277  ax-addrcl 10278  ax-mulcl 10279  ax-mulrcl 10280  ax-mulcom 10281  ax-addass 10282  ax-mulass 10283  ax-distr 10284  ax-i2m1 10285  ax-1ne0 10286  ax-1rid 10287  ax-rnegex 10288  ax-rrecex 10289  ax-cnre 10290  ax-pre-lttri 10291  ax-pre-lttrn 10292  ax-pre-ltadd 10293  ax-pre-mulgt0 10294
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-nel 3082  df-ral 3101  df-rex 3102  df-reu 3103  df-rmo 3104  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-int 4670  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-se 5271  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-f1 6102  df-fo 6103  df-f1o 6104  df-fv 6105  df-isom 6106  df-riota 6831  df-ov 6873  df-oprab 6874  df-mpt2 6875  df-om 7292  df-1st 7394  df-2nd 7395  df-supp 7526  df-wrecs 7638  df-recs 7700  df-rdg 7738  df-1o 7792  df-oadd 7796  df-er 7975  df-map 8090  df-en 8189  df-dom 8190  df-sdom 8191  df-fin 8192  df-fsupp 8511  df-oi 8650  df-card 9044  df-pnf 10357  df-mnf 10358  df-xr 10359  df-ltxr 10360  df-le 10361  df-sub 10549  df-neg 10550  df-nn 11302  df-2 11360  df-n0 11556  df-z 11640  df-uz 11901  df-fz 12546  df-fzo 12686  df-seq 13021  df-hash 13334  df-ndx 16067  df-slot 16068  df-base 16070  df-sets 16071  df-plusg 16162  df-0g 16303  df-gsum 16304  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-mhm 17536  df-grp 17626  df-minusg 17627  df-ghm 17856  df-cntz 17947  df-cmn 18392  df-abl 18393  df-mgp 18688  df-ur 18700  df-ring 18747  df-lmod 19065  df-linc 42757  df-lco 42758
This theorem is referenced by:  lincsumscmcl  42784
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