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Theorem lincscmcl 46025
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 2737 . . . . 5 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2737 . . . . 5 (Scalar‘𝑀) = (Scalar‘𝑀)
3 lincscmcl.r . . . . 5 𝑅 = (Base‘(Scalar‘𝑀))
41, 2, 3lcoval 46005 . . . 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 723 . . . . . 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 768 . . . . . 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 20212 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝐶𝑅𝐷 ∈ (Base‘𝑀)) → (𝐶 · 𝐷) ∈ (Base‘𝑀))
137, 9, 10, 12syl3anc 1370 . . . . 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 20203 . . . . . . . . . . . . . . . . 17 (𝑀 ∈ LMod → (Scalar‘𝑀) ∈ Ring)
1514ad2antrr 723 . . . . . . . . . . . . . . . 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 8685 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝑅m 𝑉) → 𝑥:𝑉𝑅)
21 ffvelcdm 6998 . . . . . . . . . . . . . . . . . . 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 723 . . . . . . . . . . . . . . 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 2737 . . . . . . . . . . . . . . 15 (.r‘(Scalar‘𝑀)) = (.r‘(Scalar‘𝑀))
283, 27ringcl 19868 . . . . . . . . . . . . . 14 (((Scalar‘𝑀) ∈ Ring ∧ 𝐶𝑅 ∧ (𝑥𝑣) ∈ 𝑅) → (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣)) ∈ 𝑅)
2917, 19, 26, 28syl3anc 1370 . . . . . . . . . . . . 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 7028 . . . . . . . . . . . 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 6825 . . . . . . . . . . . . 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 8676 . . . . . . . . . . . . 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 1127 . . . . . . . . . . . . 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 723 . . . . . . . . . . . 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 768 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝑅m 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → 𝑥 finSupp (0g‘(Scalar‘𝑀)))
4342ad2antrr 723 . . . . . . . . . . . 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 45962 . . . . . . . . . . . 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 1370 . . . . . . . . . . 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 7323 . . . . . . . . . . . . . . 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 723 . . . . . . . . . . . 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 768 . . . . . . . . . . . . 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 2737 . . . . . . . . . . . . . 14 (𝑥( linC ‘𝑀)𝑉) = (𝑥( linC ‘𝑀)𝑉)
54 eqid 2737 . . . . . . . . . . . . . 14 (𝑣𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣))) = (𝑣𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣)))
5511, 27, 53, 3, 54lincscm 46023 . . . . . . . . . . . . 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 1370 . . . . . . . . . . . 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 2777 . . . . . . . . . . 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 5090 . . . . . . . . . . . . 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 7322 . . . . . . . . . . . . . 14 (𝑠 = (𝑣𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣))) → (𝑠( linC ‘𝑀)𝑉) = ((𝑣𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣)))( linC ‘𝑀)𝑉))
6059eqeq2d 2748 . . . . . . . . . . . . 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 3570 . . . . . . . . . . 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 834 . . . . . . . . . 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 3141 . . . . . . 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 46005 . . . . . 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 723 . . . . 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 710 . . . 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 1116 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 1086   = wceq 1540  wcel 2105  wrex 3071  Vcvv 3441  𝒫 cpw 4545   class class class wbr 5087  cmpt 5170  wf 6461  cfv 6465  (class class class)co 7315  m cmap 8663   finSupp cfsupp 9198  Basecbs 16982  .rcmulr 17033  Scalarcsca 17035   ·𝑠 cvsca 17036  0gc0g 17220  Ringcrg 19851  LModclmod 20195   linC clinc 45997   LinCo clinco 45998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-rep 5224  ax-sep 5238  ax-nul 5245  ax-pow 5303  ax-pr 5367  ax-un 7628  ax-cnex 11000  ax-resscn 11001  ax-1cn 11002  ax-icn 11003  ax-addcl 11004  ax-addrcl 11005  ax-mulcl 11006  ax-mulrcl 11007  ax-mulcom 11008  ax-addass 11009  ax-mulass 11010  ax-distr 11011  ax-i2m1 11012  ax-1ne0 11013  ax-1rid 11014  ax-rnegex 11015  ax-rrecex 11016  ax-cnre 11017  ax-pre-lttri 11018  ax-pre-lttrn 11019  ax-pre-ltadd 11020  ax-pre-mulgt0 11021
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3350  df-reu 3351  df-rab 3405  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4268  df-if 4472  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4851  df-int 4893  df-iun 4939  df-br 5088  df-opab 5150  df-mpt 5171  df-tr 5205  df-id 5507  df-eprel 5513  df-po 5521  df-so 5522  df-fr 5562  df-se 5563  df-we 5564  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-res 5619  df-ima 5620  df-pred 6224  df-ord 6291  df-on 6292  df-lim 6293  df-suc 6294  df-iota 6417  df-fun 6467  df-fn 6468  df-f 6469  df-f1 6470  df-fo 6471  df-f1o 6472  df-fv 6473  df-isom 6474  df-riota 7272  df-ov 7318  df-oprab 7319  df-mpo 7320  df-om 7758  df-1st 7876  df-2nd 7877  df-supp 8025  df-frecs 8144  df-wrecs 8175  df-recs 8249  df-rdg 8288  df-1o 8344  df-er 8546  df-map 8665  df-en 8782  df-dom 8783  df-sdom 8784  df-fin 8785  df-fsupp 9199  df-oi 9339  df-card 9768  df-pnf 11084  df-mnf 11085  df-xr 11086  df-ltxr 11087  df-le 11088  df-sub 11280  df-neg 11281  df-nn 12047  df-2 12109  df-n0 12307  df-z 12393  df-uz 12656  df-fz 13313  df-fzo 13456  df-seq 13795  df-hash 14118  df-sets 16935  df-slot 16953  df-ndx 16965  df-base 16983  df-plusg 17045  df-0g 17222  df-gsum 17223  df-mgm 18396  df-sgrp 18445  df-mnd 18456  df-mhm 18500  df-grp 18649  df-minusg 18650  df-ghm 18901  df-cntz 18992  df-cmn 19456  df-abl 19457  df-mgp 19789  df-ur 19806  df-ring 19853  df-lmod 20197  df-linc 45999  df-lco 46000
This theorem is referenced by:  lincsumscmcl  46026
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