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

Theorem lincscm 42013
 Description: A linear combinations multiplied with a scalar is a linear combination, see also the proof in [Lang] p. 129. (Contributed by AV, 9-Apr-2019.) (Revised by AV, 28-Jul-2019.)
Hypotheses
Ref Expression
lincscm.s = ( ·𝑠𝑀)
lincscm.t · = (.r‘(Scalar‘𝑀))
lincscm.x 𝑋 = (𝐴( linC ‘𝑀)𝑉)
lincscm.r 𝑅 = (Base‘(Scalar‘𝑀))
lincscm.f 𝐹 = (𝑥𝑉 ↦ (𝑆 · (𝐴𝑥)))
Assertion
Ref Expression
lincscm (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → (𝑆 𝑋) = (𝐹( linC ‘𝑀)𝑉))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑀   𝑥,𝑅   𝑥,𝑆   𝑥,𝑉   𝑥, ·
Allowed substitution hints:   (𝑥)   𝐹(𝑥)   𝑋(𝑥)

Proof of Theorem lincscm
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 eqid 2610 . . 3 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2610 . . 3 (Scalar‘𝑀) = (Scalar‘𝑀)
3 lincscm.r . . 3 𝑅 = (Base‘(Scalar‘𝑀))
4 eqid 2610 . . 3 (0g𝑀) = (0g𝑀)
5 eqid 2610 . . 3 (+g𝑀) = (+g𝑀)
6 lincscm.s . . 3 = ( ·𝑠𝑀)
7 simp1l 1078 . . 3 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → 𝑀 ∈ LMod)
8 simpr 476 . . . 4 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑉 ∈ 𝒫 (Base‘𝑀))
983ad2ant1 1075 . . 3 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → 𝑉 ∈ 𝒫 (Base‘𝑀))
10 simpr 476 . . . 4 ((𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) → 𝑆𝑅)
11103ad2ant2 1076 . . 3 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → 𝑆𝑅)
127adantr 480 . . . 4 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) ∧ 𝑣𝑉) → 𝑀 ∈ LMod)
13 elmapi 7765 . . . . . . . 8 (𝐴 ∈ (𝑅𝑚 𝑉) → 𝐴:𝑉𝑅)
14 ffvelrn 6265 . . . . . . . . 9 ((𝐴:𝑉𝑅𝑣𝑉) → (𝐴𝑣) ∈ 𝑅)
1514ex 449 . . . . . . . 8 (𝐴:𝑉𝑅 → (𝑣𝑉 → (𝐴𝑣) ∈ 𝑅))
1613, 15syl 17 . . . . . . 7 (𝐴 ∈ (𝑅𝑚 𝑉) → (𝑣𝑉 → (𝐴𝑣) ∈ 𝑅))
1716adantr 480 . . . . . 6 ((𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) → (𝑣𝑉 → (𝐴𝑣) ∈ 𝑅))
18173ad2ant2 1076 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → (𝑣𝑉 → (𝐴𝑣) ∈ 𝑅))
1918imp 444 . . . 4 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) ∧ 𝑣𝑉) → (𝐴𝑣) ∈ 𝑅)
20 elelpwi 4119 . . . . . . . 8 ((𝑣𝑉𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑣 ∈ (Base‘𝑀))
2120expcom 450 . . . . . . 7 (𝑉 ∈ 𝒫 (Base‘𝑀) → (𝑣𝑉𝑣 ∈ (Base‘𝑀)))
2221adantl 481 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑣𝑉𝑣 ∈ (Base‘𝑀)))
23223ad2ant1 1075 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → (𝑣𝑉𝑣 ∈ (Base‘𝑀)))
2423imp 444 . . . 4 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) ∧ 𝑣𝑉) → 𝑣 ∈ (Base‘𝑀))
25 eqid 2610 . . . . 5 ( ·𝑠𝑀) = ( ·𝑠𝑀)
261, 2, 25, 3lmodvscl 18703 . . . 4 ((𝑀 ∈ LMod ∧ (𝐴𝑣) ∈ 𝑅𝑣 ∈ (Base‘𝑀)) → ((𝐴𝑣)( ·𝑠𝑀)𝑣) ∈ (Base‘𝑀))
2712, 19, 24, 26syl3anc 1318 . . 3 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) ∧ 𝑣𝑉) → ((𝐴𝑣)( ·𝑠𝑀)𝑣) ∈ (Base‘𝑀))
282, 3scmfsupp 41953 . . . 4 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) finSupp (0g𝑀))
29283adant2r 1313 . . 3 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) finSupp (0g𝑀))
301, 2, 3, 4, 5, 6, 7, 9, 11, 27, 29gsumvsmul 18750 . 2 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → (𝑀 Σg (𝑣𝑉 ↦ (𝑆 ((𝐴𝑣)( ·𝑠𝑀)𝑣)))) = (𝑆 (𝑀 Σg (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)))))
312lmodring 18694 . . . . . . . . . 10 (𝑀 ∈ LMod → (Scalar‘𝑀) ∈ Ring)
3231adantr 480 . . . . . . . . 9 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (Scalar‘𝑀) ∈ Ring)
33323ad2ant1 1075 . . . . . . . 8 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → (Scalar‘𝑀) ∈ Ring)
3433adantr 480 . . . . . . 7 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) ∧ 𝑥𝑉) → (Scalar‘𝑀) ∈ Ring)
353eleq2i 2680 . . . . . . . . . . 11 (𝑆𝑅𝑆 ∈ (Base‘(Scalar‘𝑀)))
3635biimpi 205 . . . . . . . . . 10 (𝑆𝑅𝑆 ∈ (Base‘(Scalar‘𝑀)))
3736adantl 481 . . . . . . . . 9 ((𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) → 𝑆 ∈ (Base‘(Scalar‘𝑀)))
38373ad2ant2 1076 . . . . . . . 8 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → 𝑆 ∈ (Base‘(Scalar‘𝑀)))
3938adantr 480 . . . . . . 7 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) ∧ 𝑥𝑉) → 𝑆 ∈ (Base‘(Scalar‘𝑀)))
40 ffvelrn 6265 . . . . . . . . . . . . 13 ((𝐴:𝑉𝑅𝑥𝑉) → (𝐴𝑥) ∈ 𝑅)
4140, 3syl6eleq 2698 . . . . . . . . . . . 12 ((𝐴:𝑉𝑅𝑥𝑉) → (𝐴𝑥) ∈ (Base‘(Scalar‘𝑀)))
4241ex 449 . . . . . . . . . . 11 (𝐴:𝑉𝑅 → (𝑥𝑉 → (𝐴𝑥) ∈ (Base‘(Scalar‘𝑀))))
4313, 42syl 17 . . . . . . . . . 10 (𝐴 ∈ (𝑅𝑚 𝑉) → (𝑥𝑉 → (𝐴𝑥) ∈ (Base‘(Scalar‘𝑀))))
4443adantr 480 . . . . . . . . 9 ((𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) → (𝑥𝑉 → (𝐴𝑥) ∈ (Base‘(Scalar‘𝑀))))
45443ad2ant2 1076 . . . . . . . 8 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → (𝑥𝑉 → (𝐴𝑥) ∈ (Base‘(Scalar‘𝑀))))
4645imp 444 . . . . . . 7 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) ∧ 𝑥𝑉) → (𝐴𝑥) ∈ (Base‘(Scalar‘𝑀)))
47 eqid 2610 . . . . . . . 8 (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
48 lincscm.t . . . . . . . 8 · = (.r‘(Scalar‘𝑀))
4947, 48ringcl 18384 . . . . . . 7 (((Scalar‘𝑀) ∈ Ring ∧ 𝑆 ∈ (Base‘(Scalar‘𝑀)) ∧ (𝐴𝑥) ∈ (Base‘(Scalar‘𝑀))) → (𝑆 · (𝐴𝑥)) ∈ (Base‘(Scalar‘𝑀)))
5034, 39, 46, 49syl3anc 1318 . . . . . 6 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) ∧ 𝑥𝑉) → (𝑆 · (𝐴𝑥)) ∈ (Base‘(Scalar‘𝑀)))
51 lincscm.f . . . . . 6 𝐹 = (𝑥𝑉 ↦ (𝑆 · (𝐴𝑥)))
5250, 51fmptd 6292 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → 𝐹:𝑉⟶(Base‘(Scalar‘𝑀)))
53 fvex 6113 . . . . . 6 (Base‘(Scalar‘𝑀)) ∈ V
54 elmapg 7757 . . . . . 6 (((Base‘(Scalar‘𝑀)) ∈ V ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ↔ 𝐹:𝑉⟶(Base‘(Scalar‘𝑀))))
5553, 9, 54sylancr 694 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → (𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ↔ 𝐹:𝑉⟶(Base‘(Scalar‘𝑀))))
5652, 55mpbird 246 . . . 4 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → 𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉))
57 lincval 41992 . . . 4 ((𝑀 ∈ LMod ∧ 𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝐹( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑣𝑉 ↦ ((𝐹𝑣)( ·𝑠𝑀)𝑣))))
587, 56, 9, 57syl3anc 1318 . . 3 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → (𝐹( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑣𝑉 ↦ ((𝐹𝑣)( ·𝑠𝑀)𝑣))))
59 simpr 476 . . . . . . . 8 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) ∧ 𝑣𝑉) → 𝑣𝑉)
60 ovex 6577 . . . . . . . 8 (𝑆 · (𝐴𝑣)) ∈ V
61 fveq2 6103 . . . . . . . . . 10 (𝑥 = 𝑣 → (𝐴𝑥) = (𝐴𝑣))
6261oveq2d 6565 . . . . . . . . 9 (𝑥 = 𝑣 → (𝑆 · (𝐴𝑥)) = (𝑆 · (𝐴𝑣)))
6362, 51fvmptg 6189 . . . . . . . 8 ((𝑣𝑉 ∧ (𝑆 · (𝐴𝑣)) ∈ V) → (𝐹𝑣) = (𝑆 · (𝐴𝑣)))
6459, 60, 63sylancl 693 . . . . . . 7 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) ∧ 𝑣𝑉) → (𝐹𝑣) = (𝑆 · (𝐴𝑣)))
6564oveq1d 6564 . . . . . 6 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) ∧ 𝑣𝑉) → ((𝐹𝑣)( ·𝑠𝑀)𝑣) = ((𝑆 · (𝐴𝑣))( ·𝑠𝑀)𝑣))
6611adantr 480 . . . . . . . 8 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) ∧ 𝑣𝑉) → 𝑆𝑅)
671, 2, 25, 3, 48lmodvsass 18711 . . . . . . . 8 ((𝑀 ∈ LMod ∧ (𝑆𝑅 ∧ (𝐴𝑣) ∈ 𝑅𝑣 ∈ (Base‘𝑀))) → ((𝑆 · (𝐴𝑣))( ·𝑠𝑀)𝑣) = (𝑆( ·𝑠𝑀)((𝐴𝑣)( ·𝑠𝑀)𝑣)))
6812, 66, 19, 24, 67syl13anc 1320 . . . . . . 7 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) ∧ 𝑣𝑉) → ((𝑆 · (𝐴𝑣))( ·𝑠𝑀)𝑣) = (𝑆( ·𝑠𝑀)((𝐴𝑣)( ·𝑠𝑀)𝑣)))
696eqcomi 2619 . . . . . . . . 9 ( ·𝑠𝑀) =
7069a1i 11 . . . . . . . 8 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) ∧ 𝑣𝑉) → ( ·𝑠𝑀) = )
7170oveqd 6566 . . . . . . 7 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) ∧ 𝑣𝑉) → (𝑆( ·𝑠𝑀)((𝐴𝑣)( ·𝑠𝑀)𝑣)) = (𝑆 ((𝐴𝑣)( ·𝑠𝑀)𝑣)))
7268, 71eqtrd 2644 . . . . . 6 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) ∧ 𝑣𝑉) → ((𝑆 · (𝐴𝑣))( ·𝑠𝑀)𝑣) = (𝑆 ((𝐴𝑣)( ·𝑠𝑀)𝑣)))
7365, 72eqtrd 2644 . . . . 5 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) ∧ 𝑣𝑉) → ((𝐹𝑣)( ·𝑠𝑀)𝑣) = (𝑆 ((𝐴𝑣)( ·𝑠𝑀)𝑣)))
7473mpteq2dva 4672 . . . 4 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → (𝑣𝑉 ↦ ((𝐹𝑣)( ·𝑠𝑀)𝑣)) = (𝑣𝑉 ↦ (𝑆 ((𝐴𝑣)( ·𝑠𝑀)𝑣))))
7574oveq2d 6565 . . 3 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → (𝑀 Σg (𝑣𝑉 ↦ ((𝐹𝑣)( ·𝑠𝑀)𝑣))) = (𝑀 Σg (𝑣𝑉 ↦ (𝑆 ((𝐴𝑣)( ·𝑠𝑀)𝑣)))))
7658, 75eqtrd 2644 . 2 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → (𝐹( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑣𝑉 ↦ (𝑆 ((𝐴𝑣)( ·𝑠𝑀)𝑣)))))
77 lincscm.x . . . . 5 𝑋 = (𝐴( linC ‘𝑀)𝑉)
7877a1i 11 . . . 4 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → 𝑋 = (𝐴( linC ‘𝑀)𝑉))
793oveq1i 6559 . . . . . . . . 9 (𝑅𝑚 𝑉) = ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)
8079eleq2i 2680 . . . . . . . 8 (𝐴 ∈ (𝑅𝑚 𝑉) ↔ 𝐴 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉))
8180biimpi 205 . . . . . . 7 (𝐴 ∈ (𝑅𝑚 𝑉) → 𝐴 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉))
8281adantr 480 . . . . . 6 ((𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) → 𝐴 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉))
83823ad2ant2 1076 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → 𝐴 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉))
84 lincval 41992 . . . . 5 ((𝑀 ∈ LMod ∧ 𝐴 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝐴( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))))
857, 83, 9, 84syl3anc 1318 . . . 4 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → (𝐴( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))))
8678, 85eqtrd 2644 . . 3 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → 𝑋 = (𝑀 Σg (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))))
8786oveq2d 6565 . 2 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → (𝑆 𝑋) = (𝑆 (𝑀 Σg (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)))))
8830, 76, 873eqtr4rd 2655 1 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → (𝑆 𝑋) = (𝐹( linC ‘𝑀)𝑉))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  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  +gcplusg 15768  .rcmulr 15769  Scalarcsca 15771   ·𝑠 cvsca 15772  0gc0g 15923   Σg cgsu 15924  Ringcrg 18370  LModclmod 18686   linC clinc 41987 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 This theorem is referenced by:  lincscmcl  42015
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