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Theorem lincscmcl 44767
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 2824 . . . . 5 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2824 . . . . 5 (Scalar‘𝑀) = (Scalar‘𝑀)
3 lincscmcl.r . . . . 5 𝑅 = (Base‘(Scalar‘𝑀))
41, 2, 3lcoval 44747 . . . 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 484 . . 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 486 . . . . . . 7 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑀 ∈ LMod)
76ad2antrr 725 . . . . . 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 488 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅) → 𝐶𝑅)
98adantr 484 . . . . . 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 770 . . . . . 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 19651 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝐶𝑅𝐷 ∈ (Base‘𝑀)) → (𝐶 · 𝐷) ∈ (Base‘𝑀))
137, 9, 10, 12syl3anc 1368 . . . . 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 19642 . . . . . . . . . . . . . . . . 17 (𝑀 ∈ LMod → (Scalar‘𝑀) ∈ Ring)
1514ad2antrr 725 . . . . . . . . . . . . . . . 16 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅) → (Scalar‘𝑀) ∈ Ring)
1615adantl 485 . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . 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 8424 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝑅m 𝑉) → 𝑥:𝑉𝑅)
21 ffvelrn 6840 . . . . . . . . . . . . . . . . . . 19 ((𝑥:𝑉𝑅𝑣𝑉) → (𝑥𝑣) ∈ 𝑅)
2221ex 416 . . . . . . . . . . . . . . . . . 18 (𝑥:𝑉𝑅 → (𝑣𝑉 → (𝑥𝑣) ∈ 𝑅))
2320, 22syl 17 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (𝑅m 𝑉) → (𝑣𝑉 → (𝑥𝑣) ∈ 𝑅))
2423adantr 484 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ (𝑅m 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → (𝑣𝑉 → (𝑥𝑣) ∈ 𝑅))
2524ad2antrr 725 . . . . . . . . . . . . . . 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 410 . . . . . . . . . . . . . 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 2824 . . . . . . . . . . . . . . 15 (.r‘(Scalar‘𝑀)) = (.r‘(Scalar‘𝑀))
283, 27ringcl 19314 . . . . . . . . . . . . . 14 (((Scalar‘𝑀) ∈ Ring ∧ 𝐶𝑅 ∧ (𝑥𝑣) ∈ 𝑅) → (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣)) ∈ 𝑅)
2917, 19, 26, 28syl3anc 1368 . . . . . . . . . . . . 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 6870 . . . . . . . . . . . 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 6675 . . . . . . . . . . . . 13 𝑅 ∈ V
32 simpr 488 . . . . . . . . . . . . . . 15 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑉 ∈ 𝒫 (Base‘𝑀))
3332adantr 484 . . . . . . . . . . . . . 14 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅) → 𝑉 ∈ 𝒫 (Base‘𝑀))
3433adantl 485 . . . . . . . . . . . . 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 8415 . . . . . . . . . . . . 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 590 . . . . . . . . . . . 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 260 . . . . . . . . . . 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 1125 . . . . . . . . . . . . 13 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅) → ((Scalar‘𝑀) ∈ Ring ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐶𝑅))
3938adantl 485 . . . . . . . . . . . 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 486 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝑅m 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → 𝑥 ∈ (𝑅m 𝑉))
4140ad2antrr 725 . . . . . . . . . . . 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 770 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝑅m 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → 𝑥 finSupp (0g‘(Scalar‘𝑀)))
4342ad2antrr 725 . . . . . . . . . . . 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 44702 . . . . . . . . . . . 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 1368 . . . . . . . . . . 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 7157 . . . . . . . . . . . . . . 15 (𝐷 = (𝑥( linC ‘𝑀)𝑉) → (𝐶 · 𝐷) = (𝐶 · (𝑥( linC ‘𝑀)𝑉)))
4746adantl 485 . . . . . . . . . . . . . 14 ((𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)) → (𝐶 · 𝐷) = (𝐶 · (𝑥( linC ‘𝑀)𝑉)))
4847adantl 485 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝑅m 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → (𝐶 · 𝐷) = (𝐶 · (𝑥( linC ‘𝑀)𝑉)))
4948ad2antrr 725 . . . . . . . . . . . 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 770 . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . 14 (((𝑥 ∈ (𝑅m 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) → 𝑥 ∈ (𝑅m 𝑉))
5251, 8anim12i 615 . . . . . . . . . . . . 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 2824 . . . . . . . . . . . . . 14 (𝑥( linC ‘𝑀)𝑉) = (𝑥( linC ‘𝑀)𝑉)
54 eqid 2824 . . . . . . . . . . . . . 14 (𝑣𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣))) = (𝑣𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣)))
5511, 27, 53, 3, 54lincscm 44765 . . . . . . . . . . . . 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 1368 . . . . . . . . . . . 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 2859 . . . . . . . . . . 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 5055 . . . . . . . . . . . . 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 7156 . . . . . . . . . . . . . 14 (𝑠 = (𝑣𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣))) → (𝑠( linC ‘𝑀)𝑉) = ((𝑣𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥𝑣)))( linC ‘𝑀)𝑉))
6059eqeq2d 2835 . . . . . . . . . . . . 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 633 . . . . . . . . . . . 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 3609 . . . . . . . . . . 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 416 . . . . . . . . 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 416 . . . . . . . 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 3273 . . . . . . 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 411 . . . . . 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 411 . . . . 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 44747 . . . . . 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 725 . . . . 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 712 . . . 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 416 . . 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 243 . 2 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅) → (𝐷 ∈ (𝑀 LinCo 𝑉) → (𝐶 · 𝐷) ∈ (𝑀 LinCo 𝑉)))
74733impia 1114 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wrex 3134  Vcvv 3480  𝒫 cpw 4522   class class class wbr 5052  cmpt 5132  wf 6339  cfv 6343  (class class class)co 7149  m cmap 8402   finSupp cfsupp 8830  Basecbs 16483  .rcmulr 16566  Scalarcsca 16568   ·𝑠 cvsca 16569  0gc0g 16713  Ringcrg 19297  LModclmod 19634   linC clinc 44739   LinCo clinco 44740
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-se 5502  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-isom 6352  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-supp 7827  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8285  df-map 8404  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-fsupp 8831  df-oi 8971  df-card 9365  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-nn 11635  df-2 11697  df-n0 11895  df-z 11979  df-uz 12241  df-fz 12895  df-fzo 13038  df-seq 13374  df-hash 13696  df-ndx 16486  df-slot 16487  df-base 16489  df-sets 16490  df-plusg 16578  df-0g 16715  df-gsum 16716  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-mhm 17956  df-grp 18106  df-minusg 18107  df-ghm 18356  df-cntz 18447  df-cmn 18908  df-abl 18909  df-mgp 19240  df-ur 19252  df-ring 19299  df-lmod 19636  df-linc 44741  df-lco 44742
This theorem is referenced by:  lincsumscmcl  44768
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