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Theorem lincscmcl 46503
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 2736 . . . . 5 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2736 . . . . 5 (Scalar‘𝑀) = (Scalar‘𝑀)
3 lincscmcl.r . . . . 5 𝑅 = (Base‘(Scalar‘𝑀))
41, 2, 3lcoval 46483 . . . 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 481 . . 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 483 . . . . . . 7 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑀 ∈ LMod)
76ad2antrr 724 . . . . . 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 485 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅) → 𝐶𝑅)
98adantr 481 . . . . . 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 769 . . . . . 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 20339 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝐶𝑅𝐷 ∈ (Base‘𝑀)) → (𝐶 · 𝐷) ∈ (Base‘𝑀))
137, 9, 10, 12syl3anc 1371 . . . . 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 20330 . . . . . . . . . . . . . . . . 17 (𝑀 ∈ LMod → (Scalar‘𝑀) ∈ Ring)
1514ad2antrr 724 . . . . . . . . . . . . . . . 16 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅) → (Scalar‘𝑀) ∈ Ring)
1615adantl 482 . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . 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 8787 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝑅m 𝑉) → 𝑥:𝑉𝑅)
21 ffvelcdm 7032 . . . . . . . . . . . . . . . . . . 19 ((𝑥:𝑉𝑅𝑣𝑉) → (𝑥𝑣) ∈ 𝑅)
2221ex 413 . . . . . . . . . . . . . . . . . 18 (𝑥:𝑉𝑅 → (𝑣𝑉 → (𝑥𝑣) ∈ 𝑅))
2320, 22syl 17 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (𝑅m 𝑉) → (𝑣𝑉 → (𝑥𝑣) ∈ 𝑅))
2423adantr 481 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ (𝑅m 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → (𝑣𝑉 → (𝑥𝑣) ∈ 𝑅))
2524ad2antrr 724 . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . . 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 2736 . . . . . . . . . . . . . . 15 (.r‘(Scalar‘𝑀)) = (.r‘(Scalar‘𝑀))
283, 27ringcl 19981 . . . . . . . . . . . . . 14 (((Scalar‘𝑀) ∈ Ring ∧ 𝐶𝑅 ∧ (𝑥𝑣) ∈ 𝑅) → (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣)) ∈ 𝑅)
2917, 19, 26, 28syl3anc 1371 . . . . . . . . . . . . 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 7063 . . . . . . . . . . . 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 6856 . . . . . . . . . . . . 13 𝑅 ∈ V
32 simpr 485 . . . . . . . . . . . . . . 15 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑉 ∈ 𝒫 (Base‘𝑀))
3332adantr 481 . . . . . . . . . . . . . 14 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅) → 𝑉 ∈ 𝒫 (Base‘𝑀))
3433adantl 482 . . . . . . . . . . . . 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 8778 . . . . . . . . . . . . 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 587 . . . . . . . . . . . 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 256 . . . . . . . . . . 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 1128 . . . . . . . . . . . . 13 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅) → ((Scalar‘𝑀) ∈ Ring ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐶𝑅))
3938adantl 482 . . . . . . . . . . . 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 483 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝑅m 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → 𝑥 ∈ (𝑅m 𝑉))
4140ad2antrr 724 . . . . . . . . . . . 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 769 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝑅m 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → 𝑥 finSupp (0g‘(Scalar‘𝑀)))
4342ad2antrr 724 . . . . . . . . . . . 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 46440 . . . . . . . . . . . 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 1371 . . . . . . . . . . 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 7365 . . . . . . . . . . . . . . 15 (𝐷 = (𝑥( linC ‘𝑀)𝑉) → (𝐶 · 𝐷) = (𝐶 · (𝑥( linC ‘𝑀)𝑉)))
4746adantl 482 . . . . . . . . . . . . . 14 ((𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)) → (𝐶 · 𝐷) = (𝐶 · (𝑥( linC ‘𝑀)𝑉)))
4847adantl 482 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝑅m 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → (𝐶 · 𝐷) = (𝐶 · (𝑥( linC ‘𝑀)𝑉)))
4948ad2antrr 724 . . . . . . . . . . . 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 769 . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . 14 (((𝑥 ∈ (𝑅m 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) → 𝑥 ∈ (𝑅m 𝑉))
5251, 8anim12i 613 . . . . . . . . . . . . 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 2736 . . . . . . . . . . . . . 14 (𝑥( linC ‘𝑀)𝑉) = (𝑥( linC ‘𝑀)𝑉)
54 eqid 2736 . . . . . . . . . . . . . 14 (𝑣𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣))) = (𝑣𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣)))
5511, 27, 53, 3, 54lincscm 46501 . . . . . . . . . . . . 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 1371 . . . . . . . . . . . 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 2776 . . . . . . . . . . 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 5108 . . . . . . . . . . . . 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 7364 . . . . . . . . . . . . . 14 (𝑠 = (𝑣𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣))) → (𝑠( linC ‘𝑀)𝑉) = ((𝑣𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣)))( linC ‘𝑀)𝑉))
6059eqeq2d 2747 . . . . . . . . . . . . 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 631 . . . . . . . . . . . 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 3581 . . . . . . . . . . 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 835 . . . . . . . . . 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 413 . . . . . . . . 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 413 . . . . . . . 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 3144 . . . . . . 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 408 . . . . . 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 408 . . . . 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 46483 . . . . . 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 724 . . . . 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 711 . . . 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 413 . . 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 239 . 2 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅) → (𝐷 ∈ (𝑀 LinCo 𝑉) → (𝐶 · 𝐷) ∈ (𝑀 LinCo 𝑉)))
74733impia 1117 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 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wrex 3073  Vcvv 3445  𝒫 cpw 4560   class class class wbr 5105  cmpt 5188  wf 6492  cfv 6496  (class class class)co 7357  m cmap 8765   finSupp cfsupp 9305  Basecbs 17083  .rcmulr 17134  Scalarcsca 17136   ·𝑠 cvsca 17137  0gc0g 17321  Ringcrg 19964  LModclmod 20322   linC clinc 46475   LinCo clinco 46476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-plusg 17146  df-0g 17323  df-gsum 17324  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-grp 18751  df-minusg 18752  df-ghm 19006  df-cntz 19097  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-lmod 20324  df-linc 46477  df-lco 46478
This theorem is referenced by:  lincsumscmcl  46504
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