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Theorem lincsum 46584
Description: The sum of two linear combinations is a linear combination, see also the proof in [Lang] p. 129. (Contributed by AV, 4-Apr-2019.) (Revised by AV, 28-Jul-2019.)
Hypotheses
Ref Expression
lincsum.p + = (+gβ€˜π‘€)
lincsum.x 𝑋 = (𝐴( linC β€˜π‘€)𝑉)
lincsum.y π‘Œ = (𝐡( linC β€˜π‘€)𝑉)
lincsum.s 𝑆 = (Scalarβ€˜π‘€)
lincsum.r 𝑅 = (Baseβ€˜π‘†)
lincsum.b ✚ = (+gβ€˜π‘†)
Assertion
Ref Expression
lincsum (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0gβ€˜π‘†) ∧ 𝐡 finSupp (0gβ€˜π‘†))) β†’ (𝑋 + π‘Œ) = ((𝐴 ∘f ✚ 𝐡)( linC β€˜π‘€)𝑉))

Proof of Theorem lincsum
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . . 3 (Baseβ€˜π‘€) = (Baseβ€˜π‘€)
2 eqid 2737 . . 3 (0gβ€˜π‘€) = (0gβ€˜π‘€)
3 lincsum.p . . 3 + = (+gβ€˜π‘€)
4 lmodcmn 20386 . . . . 5 (𝑀 ∈ LMod β†’ 𝑀 ∈ CMnd)
54adantr 482 . . . 4 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ 𝑀 ∈ CMnd)
653ad2ant1 1134 . . 3 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0gβ€˜π‘†) ∧ 𝐡 finSupp (0gβ€˜π‘†))) β†’ 𝑀 ∈ CMnd)
7 simpr 486 . . . 4 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))
873ad2ant1 1134 . . 3 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0gβ€˜π‘†) ∧ 𝐡 finSupp (0gβ€˜π‘†))) β†’ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))
9 simpl 484 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ 𝑀 ∈ LMod)
1093ad2ant1 1134 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0gβ€˜π‘†) ∧ 𝐡 finSupp (0gβ€˜π‘†))) β†’ 𝑀 ∈ LMod)
1110adantr 482 . . . 4 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0gβ€˜π‘†) ∧ 𝐡 finSupp (0gβ€˜π‘†))) ∧ π‘₯ ∈ 𝑉) β†’ 𝑀 ∈ LMod)
12 elmapi 8794 . . . . . . . 8 (𝐴 ∈ (𝑅 ↑m 𝑉) β†’ 𝐴:π‘‰βŸΆπ‘…)
13 ffvelcdm 7037 . . . . . . . . 9 ((𝐴:π‘‰βŸΆπ‘… ∧ π‘₯ ∈ 𝑉) β†’ (π΄β€˜π‘₯) ∈ 𝑅)
1413ex 414 . . . . . . . 8 (𝐴:π‘‰βŸΆπ‘… β†’ (π‘₯ ∈ 𝑉 β†’ (π΄β€˜π‘₯) ∈ 𝑅))
1512, 14syl 17 . . . . . . 7 (𝐴 ∈ (𝑅 ↑m 𝑉) β†’ (π‘₯ ∈ 𝑉 β†’ (π΄β€˜π‘₯) ∈ 𝑅))
1615adantr 482 . . . . . 6 ((𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉)) β†’ (π‘₯ ∈ 𝑉 β†’ (π΄β€˜π‘₯) ∈ 𝑅))
17163ad2ant2 1135 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0gβ€˜π‘†) ∧ 𝐡 finSupp (0gβ€˜π‘†))) β†’ (π‘₯ ∈ 𝑉 β†’ (π΄β€˜π‘₯) ∈ 𝑅))
1817imp 408 . . . 4 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0gβ€˜π‘†) ∧ 𝐡 finSupp (0gβ€˜π‘†))) ∧ π‘₯ ∈ 𝑉) β†’ (π΄β€˜π‘₯) ∈ 𝑅)
19 elelpwi 4575 . . . . . . . 8 ((π‘₯ ∈ 𝑉 ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ π‘₯ ∈ (Baseβ€˜π‘€))
2019expcom 415 . . . . . . 7 (𝑉 ∈ 𝒫 (Baseβ€˜π‘€) β†’ (π‘₯ ∈ 𝑉 β†’ π‘₯ ∈ (Baseβ€˜π‘€)))
2120adantl 483 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ (π‘₯ ∈ 𝑉 β†’ π‘₯ ∈ (Baseβ€˜π‘€)))
22213ad2ant1 1134 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0gβ€˜π‘†) ∧ 𝐡 finSupp (0gβ€˜π‘†))) β†’ (π‘₯ ∈ 𝑉 β†’ π‘₯ ∈ (Baseβ€˜π‘€)))
2322imp 408 . . . 4 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0gβ€˜π‘†) ∧ 𝐡 finSupp (0gβ€˜π‘†))) ∧ π‘₯ ∈ 𝑉) β†’ π‘₯ ∈ (Baseβ€˜π‘€))
24 lincsum.s . . . . 5 𝑆 = (Scalarβ€˜π‘€)
25 eqid 2737 . . . . 5 ( ·𝑠 β€˜π‘€) = ( ·𝑠 β€˜π‘€)
26 lincsum.r . . . . 5 𝑅 = (Baseβ€˜π‘†)
271, 24, 25, 26lmodvscl 20355 . . . 4 ((𝑀 ∈ LMod ∧ (π΄β€˜π‘₯) ∈ 𝑅 ∧ π‘₯ ∈ (Baseβ€˜π‘€)) β†’ ((π΄β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯) ∈ (Baseβ€˜π‘€))
2811, 18, 23, 27syl3anc 1372 . . 3 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0gβ€˜π‘†) ∧ 𝐡 finSupp (0gβ€˜π‘†))) ∧ π‘₯ ∈ 𝑉) β†’ ((π΄β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯) ∈ (Baseβ€˜π‘€))
29 elmapi 8794 . . . . . . . 8 (𝐡 ∈ (𝑅 ↑m 𝑉) β†’ 𝐡:π‘‰βŸΆπ‘…)
30 ffvelcdm 7037 . . . . . . . . 9 ((𝐡:π‘‰βŸΆπ‘… ∧ π‘₯ ∈ 𝑉) β†’ (π΅β€˜π‘₯) ∈ 𝑅)
3130ex 414 . . . . . . . 8 (𝐡:π‘‰βŸΆπ‘… β†’ (π‘₯ ∈ 𝑉 β†’ (π΅β€˜π‘₯) ∈ 𝑅))
3229, 31syl 17 . . . . . . 7 (𝐡 ∈ (𝑅 ↑m 𝑉) β†’ (π‘₯ ∈ 𝑉 β†’ (π΅β€˜π‘₯) ∈ 𝑅))
3332adantl 483 . . . . . 6 ((𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉)) β†’ (π‘₯ ∈ 𝑉 β†’ (π΅β€˜π‘₯) ∈ 𝑅))
34333ad2ant2 1135 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0gβ€˜π‘†) ∧ 𝐡 finSupp (0gβ€˜π‘†))) β†’ (π‘₯ ∈ 𝑉 β†’ (π΅β€˜π‘₯) ∈ 𝑅))
3534imp 408 . . . 4 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0gβ€˜π‘†) ∧ 𝐡 finSupp (0gβ€˜π‘†))) ∧ π‘₯ ∈ 𝑉) β†’ (π΅β€˜π‘₯) ∈ 𝑅)
361, 24, 25, 26lmodvscl 20355 . . . 4 ((𝑀 ∈ LMod ∧ (π΅β€˜π‘₯) ∈ 𝑅 ∧ π‘₯ ∈ (Baseβ€˜π‘€)) β†’ ((π΅β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯) ∈ (Baseβ€˜π‘€))
3711, 35, 23, 36syl3anc 1372 . . 3 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0gβ€˜π‘†) ∧ 𝐡 finSupp (0gβ€˜π‘†))) ∧ π‘₯ ∈ 𝑉) β†’ ((π΅β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯) ∈ (Baseβ€˜π‘€))
38 eqidd 2738 . . 3 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0gβ€˜π‘†) ∧ 𝐡 finSupp (0gβ€˜π‘†))) β†’ (π‘₯ ∈ 𝑉 ↦ ((π΄β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯)) = (π‘₯ ∈ 𝑉 ↦ ((π΄β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯)))
39 eqidd 2738 . . 3 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0gβ€˜π‘†) ∧ 𝐡 finSupp (0gβ€˜π‘†))) β†’ (π‘₯ ∈ 𝑉 ↦ ((π΅β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯)) = (π‘₯ ∈ 𝑉 ↦ ((π΅β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯)))
40 id 22 . . . 4 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)))
41 simpl 484 . . . 4 ((𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉)) β†’ 𝐴 ∈ (𝑅 ↑m 𝑉))
42 simpl 484 . . . 4 ((𝐴 finSupp (0gβ€˜π‘†) ∧ 𝐡 finSupp (0gβ€˜π‘†)) β†’ 𝐴 finSupp (0gβ€˜π‘†))
4324, 26scmfsupp 46528 . . . 4 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐴 finSupp (0gβ€˜π‘†)) β†’ (π‘₯ ∈ 𝑉 ↦ ((π΄β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯)) finSupp (0gβ€˜π‘€))
4440, 41, 42, 43syl3an 1161 . . 3 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0gβ€˜π‘†) ∧ 𝐡 finSupp (0gβ€˜π‘†))) β†’ (π‘₯ ∈ 𝑉 ↦ ((π΄β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯)) finSupp (0gβ€˜π‘€))
45 simpr 486 . . . 4 ((𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉)) β†’ 𝐡 ∈ (𝑅 ↑m 𝑉))
46 simpr 486 . . . 4 ((𝐴 finSupp (0gβ€˜π‘†) ∧ 𝐡 finSupp (0gβ€˜π‘†)) β†’ 𝐡 finSupp (0gβ€˜π‘†))
4724, 26scmfsupp 46528 . . . 4 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 finSupp (0gβ€˜π‘†)) β†’ (π‘₯ ∈ 𝑉 ↦ ((π΅β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯)) finSupp (0gβ€˜π‘€))
4840, 45, 46, 47syl3an 1161 . . 3 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0gβ€˜π‘†) ∧ 𝐡 finSupp (0gβ€˜π‘†))) β†’ (π‘₯ ∈ 𝑉 ↦ ((π΅β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯)) finSupp (0gβ€˜π‘€))
491, 2, 3, 6, 8, 28, 37, 38, 39, 44, 48gsummptfsadd 19708 . 2 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0gβ€˜π‘†) ∧ 𝐡 finSupp (0gβ€˜π‘†))) β†’ (𝑀 Ξ£g (π‘₯ ∈ 𝑉 ↦ (((π΄β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯) + ((π΅β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯)))) = ((𝑀 Ξ£g (π‘₯ ∈ 𝑉 ↦ ((π΄β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯))) + (𝑀 Ξ£g (π‘₯ ∈ 𝑉 ↦ ((π΅β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯)))))
507adantr 482 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉))) β†’ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))
51 elmapfn 8810 . . . . . . . 8 (𝐴 ∈ (𝑅 ↑m 𝑉) β†’ 𝐴 Fn 𝑉)
5251ad2antrl 727 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉))) β†’ 𝐴 Fn 𝑉)
53 elmapfn 8810 . . . . . . . 8 (𝐡 ∈ (𝑅 ↑m 𝑉) β†’ 𝐡 Fn 𝑉)
5453ad2antll 728 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉))) β†’ 𝐡 Fn 𝑉)
5550, 52, 54offvalfv 46492 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉))) β†’ (𝐴 ∘f ✚ 𝐡) = (𝑦 ∈ 𝑉 ↦ ((π΄β€˜π‘¦) ✚ (π΅β€˜π‘¦))))
56553adant3 1133 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0gβ€˜π‘†) ∧ 𝐡 finSupp (0gβ€˜π‘†))) β†’ (𝐴 ∘f ✚ 𝐡) = (𝑦 ∈ 𝑉 ↦ ((π΄β€˜π‘¦) ✚ (π΅β€˜π‘¦))))
5724lmodfgrp 20347 . . . . . . . . . . 11 (𝑀 ∈ LMod β†’ 𝑆 ∈ Grp)
5857grpmndd 18767 . . . . . . . . . 10 (𝑀 ∈ LMod β†’ 𝑆 ∈ Mnd)
5958ad3antrrr 729 . . . . . . . . 9 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉))) ∧ 𝑦 ∈ 𝑉) β†’ 𝑆 ∈ Mnd)
60 ffvelcdm 7037 . . . . . . . . . . . . . 14 ((𝐴:π‘‰βŸΆπ‘… ∧ 𝑦 ∈ 𝑉) β†’ (π΄β€˜π‘¦) ∈ 𝑅)
6160ex 414 . . . . . . . . . . . . 13 (𝐴:π‘‰βŸΆπ‘… β†’ (𝑦 ∈ 𝑉 β†’ (π΄β€˜π‘¦) ∈ 𝑅))
6212, 61syl 17 . . . . . . . . . . . 12 (𝐴 ∈ (𝑅 ↑m 𝑉) β†’ (𝑦 ∈ 𝑉 β†’ (π΄β€˜π‘¦) ∈ 𝑅))
6362ad2antrl 727 . . . . . . . . . . 11 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉))) β†’ (𝑦 ∈ 𝑉 β†’ (π΄β€˜π‘¦) ∈ 𝑅))
6463imp 408 . . . . . . . . . 10 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉))) ∧ 𝑦 ∈ 𝑉) β†’ (π΄β€˜π‘¦) ∈ 𝑅)
6524fveq2i 6850 . . . . . . . . . . 11 (Baseβ€˜π‘†) = (Baseβ€˜(Scalarβ€˜π‘€))
6626, 65eqtri 2765 . . . . . . . . . 10 𝑅 = (Baseβ€˜(Scalarβ€˜π‘€))
6764, 66eleqtrdi 2848 . . . . . . . . 9 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉))) ∧ 𝑦 ∈ 𝑉) β†’ (π΄β€˜π‘¦) ∈ (Baseβ€˜(Scalarβ€˜π‘€)))
68 ffvelcdm 7037 . . . . . . . . . . . . . 14 ((𝐡:π‘‰βŸΆπ‘… ∧ 𝑦 ∈ 𝑉) β†’ (π΅β€˜π‘¦) ∈ 𝑅)
6968, 66eleqtrdi 2848 . . . . . . . . . . . . 13 ((𝐡:π‘‰βŸΆπ‘… ∧ 𝑦 ∈ 𝑉) β†’ (π΅β€˜π‘¦) ∈ (Baseβ€˜(Scalarβ€˜π‘€)))
7069ex 414 . . . . . . . . . . . 12 (𝐡:π‘‰βŸΆπ‘… β†’ (𝑦 ∈ 𝑉 β†’ (π΅β€˜π‘¦) ∈ (Baseβ€˜(Scalarβ€˜π‘€))))
7129, 70syl 17 . . . . . . . . . . 11 (𝐡 ∈ (𝑅 ↑m 𝑉) β†’ (𝑦 ∈ 𝑉 β†’ (π΅β€˜π‘¦) ∈ (Baseβ€˜(Scalarβ€˜π‘€))))
7271ad2antll 728 . . . . . . . . . 10 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉))) β†’ (𝑦 ∈ 𝑉 β†’ (π΅β€˜π‘¦) ∈ (Baseβ€˜(Scalarβ€˜π‘€))))
7372imp 408 . . . . . . . . 9 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉))) ∧ 𝑦 ∈ 𝑉) β†’ (π΅β€˜π‘¦) ∈ (Baseβ€˜(Scalarβ€˜π‘€)))
7424eqcomi 2746 . . . . . . . . . . 11 (Scalarβ€˜π‘€) = 𝑆
7574fveq2i 6850 . . . . . . . . . 10 (Baseβ€˜(Scalarβ€˜π‘€)) = (Baseβ€˜π‘†)
76 lincsum.b . . . . . . . . . 10 ✚ = (+gβ€˜π‘†)
7775, 76mndcl 18571 . . . . . . . . 9 ((𝑆 ∈ Mnd ∧ (π΄β€˜π‘¦) ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ (π΅β€˜π‘¦) ∈ (Baseβ€˜(Scalarβ€˜π‘€))) β†’ ((π΄β€˜π‘¦) ✚ (π΅β€˜π‘¦)) ∈ (Baseβ€˜(Scalarβ€˜π‘€)))
7859, 67, 73, 77syl3anc 1372 . . . . . . . 8 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉))) ∧ 𝑦 ∈ 𝑉) β†’ ((π΄β€˜π‘¦) ✚ (π΅β€˜π‘¦)) ∈ (Baseβ€˜(Scalarβ€˜π‘€)))
7978fmpttd 7068 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉))) β†’ (𝑦 ∈ 𝑉 ↦ ((π΄β€˜π‘¦) ✚ (π΅β€˜π‘¦))):π‘‰βŸΆ(Baseβ€˜(Scalarβ€˜π‘€)))
80 fvex 6860 . . . . . . . 8 (Baseβ€˜(Scalarβ€˜π‘€)) ∈ V
81 elmapg 8785 . . . . . . . 8 (((Baseβ€˜(Scalarβ€˜π‘€)) ∈ V ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ ((𝑦 ∈ 𝑉 ↦ ((π΄β€˜π‘¦) ✚ (π΅β€˜π‘¦))) ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ↔ (𝑦 ∈ 𝑉 ↦ ((π΄β€˜π‘¦) ✚ (π΅β€˜π‘¦))):π‘‰βŸΆ(Baseβ€˜(Scalarβ€˜π‘€))))
8280, 50, 81sylancr 588 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉))) β†’ ((𝑦 ∈ 𝑉 ↦ ((π΄β€˜π‘¦) ✚ (π΅β€˜π‘¦))) ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ↔ (𝑦 ∈ 𝑉 ↦ ((π΄β€˜π‘¦) ✚ (π΅β€˜π‘¦))):π‘‰βŸΆ(Baseβ€˜(Scalarβ€˜π‘€))))
8379, 82mpbird 257 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉))) β†’ (𝑦 ∈ 𝑉 ↦ ((π΄β€˜π‘¦) ✚ (π΅β€˜π‘¦))) ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉))
84833adant3 1133 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0gβ€˜π‘†) ∧ 𝐡 finSupp (0gβ€˜π‘†))) β†’ (𝑦 ∈ 𝑉 ↦ ((π΄β€˜π‘¦) ✚ (π΅β€˜π‘¦))) ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉))
8556, 84eqeltrd 2838 . . . 4 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0gβ€˜π‘†) ∧ 𝐡 finSupp (0gβ€˜π‘†))) β†’ (𝐴 ∘f ✚ 𝐡) ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉))
86 lincval 46564 . . . 4 ((𝑀 ∈ LMod ∧ (𝐴 ∘f ✚ 𝐡) ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ ((𝐴 ∘f ✚ 𝐡)( linC β€˜π‘€)𝑉) = (𝑀 Ξ£g (π‘₯ ∈ 𝑉 ↦ (((𝐴 ∘f ✚ 𝐡)β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯))))
8710, 85, 8, 86syl3anc 1372 . . 3 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0gβ€˜π‘†) ∧ 𝐡 finSupp (0gβ€˜π‘†))) β†’ ((𝐴 ∘f ✚ 𝐡)( linC β€˜π‘€)𝑉) = (𝑀 Ξ£g (π‘₯ ∈ 𝑉 ↦ (((𝐴 ∘f ✚ 𝐡)β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯))))
8851, 53anim12i 614 . . . . . . . . . . . 12 ((𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉)) β†’ (𝐴 Fn 𝑉 ∧ 𝐡 Fn 𝑉))
8988adantl 483 . . . . . . . . . . 11 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉))) β†’ (𝐴 Fn 𝑉 ∧ 𝐡 Fn 𝑉))
9089adantr 482 . . . . . . . . . 10 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉))) ∧ π‘₯ ∈ 𝑉) β†’ (𝐴 Fn 𝑉 ∧ 𝐡 Fn 𝑉))
9150anim1i 616 . . . . . . . . . 10 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉))) ∧ π‘₯ ∈ 𝑉) β†’ (𝑉 ∈ 𝒫 (Baseβ€˜π‘€) ∧ π‘₯ ∈ 𝑉))
92 fnfvof 7639 . . . . . . . . . 10 (((𝐴 Fn 𝑉 ∧ 𝐡 Fn 𝑉) ∧ (𝑉 ∈ 𝒫 (Baseβ€˜π‘€) ∧ π‘₯ ∈ 𝑉)) β†’ ((𝐴 ∘f ✚ 𝐡)β€˜π‘₯) = ((π΄β€˜π‘₯) ✚ (π΅β€˜π‘₯)))
9390, 91, 92syl2anc 585 . . . . . . . . 9 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉))) ∧ π‘₯ ∈ 𝑉) β†’ ((𝐴 ∘f ✚ 𝐡)β€˜π‘₯) = ((π΄β€˜π‘₯) ✚ (π΅β€˜π‘₯)))
9476a1i 11 . . . . . . . . . 10 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉))) ∧ π‘₯ ∈ 𝑉) β†’ ✚ = (+gβ€˜π‘†))
9594oveqd 7379 . . . . . . . . 9 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉))) ∧ π‘₯ ∈ 𝑉) β†’ ((π΄β€˜π‘₯) ✚ (π΅β€˜π‘₯)) = ((π΄β€˜π‘₯)(+gβ€˜π‘†)(π΅β€˜π‘₯)))
9693, 95eqtrd 2777 . . . . . . . 8 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉))) ∧ π‘₯ ∈ 𝑉) β†’ ((𝐴 ∘f ✚ 𝐡)β€˜π‘₯) = ((π΄β€˜π‘₯)(+gβ€˜π‘†)(π΅β€˜π‘₯)))
9796oveq1d 7377 . . . . . . 7 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉))) ∧ π‘₯ ∈ 𝑉) β†’ (((𝐴 ∘f ✚ 𝐡)β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯) = (((π΄β€˜π‘₯)(+gβ€˜π‘†)(π΅β€˜π‘₯))( ·𝑠 β€˜π‘€)π‘₯))
989adantr 482 . . . . . . . . 9 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉))) β†’ 𝑀 ∈ LMod)
9998adantr 482 . . . . . . . 8 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉))) ∧ π‘₯ ∈ 𝑉) β†’ 𝑀 ∈ LMod)
10015ad2antrl 727 . . . . . . . . 9 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉))) β†’ (π‘₯ ∈ 𝑉 β†’ (π΄β€˜π‘₯) ∈ 𝑅))
101100imp 408 . . . . . . . 8 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉))) ∧ π‘₯ ∈ 𝑉) β†’ (π΄β€˜π‘₯) ∈ 𝑅)
10232ad2antll 728 . . . . . . . . 9 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉))) β†’ (π‘₯ ∈ 𝑉 β†’ (π΅β€˜π‘₯) ∈ 𝑅))
103102imp 408 . . . . . . . 8 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉))) ∧ π‘₯ ∈ 𝑉) β†’ (π΅β€˜π‘₯) ∈ 𝑅)
10421adantr 482 . . . . . . . . 9 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉))) β†’ (π‘₯ ∈ 𝑉 β†’ π‘₯ ∈ (Baseβ€˜π‘€)))
105104imp 408 . . . . . . . 8 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉))) ∧ π‘₯ ∈ 𝑉) β†’ π‘₯ ∈ (Baseβ€˜π‘€))
106 eqid 2737 . . . . . . . . 9 (Scalarβ€˜π‘€) = (Scalarβ€˜π‘€)
10724fveq2i 6850 . . . . . . . . 9 (+gβ€˜π‘†) = (+gβ€˜(Scalarβ€˜π‘€))
1081, 3, 106, 25, 66, 107lmodvsdir 20362 . . . . . . . 8 ((𝑀 ∈ LMod ∧ ((π΄β€˜π‘₯) ∈ 𝑅 ∧ (π΅β€˜π‘₯) ∈ 𝑅 ∧ π‘₯ ∈ (Baseβ€˜π‘€))) β†’ (((π΄β€˜π‘₯)(+gβ€˜π‘†)(π΅β€˜π‘₯))( ·𝑠 β€˜π‘€)π‘₯) = (((π΄β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯) + ((π΅β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯)))
10999, 101, 103, 105, 108syl13anc 1373 . . . . . . 7 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉))) ∧ π‘₯ ∈ 𝑉) β†’ (((π΄β€˜π‘₯)(+gβ€˜π‘†)(π΅β€˜π‘₯))( ·𝑠 β€˜π‘€)π‘₯) = (((π΄β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯) + ((π΅β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯)))
11097, 109eqtrd 2777 . . . . . 6 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉))) ∧ π‘₯ ∈ 𝑉) β†’ (((𝐴 ∘f ✚ 𝐡)β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯) = (((π΄β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯) + ((π΅β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯)))
111110mpteq2dva 5210 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉))) β†’ (π‘₯ ∈ 𝑉 ↦ (((𝐴 ∘f ✚ 𝐡)β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯)) = (π‘₯ ∈ 𝑉 ↦ (((π΄β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯) + ((π΅β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯))))
112111oveq2d 7378 . . . 4 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉))) β†’ (𝑀 Ξ£g (π‘₯ ∈ 𝑉 ↦ (((𝐴 ∘f ✚ 𝐡)β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯))) = (𝑀 Ξ£g (π‘₯ ∈ 𝑉 ↦ (((π΄β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯) + ((π΅β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯)))))
1131123adant3 1133 . . 3 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0gβ€˜π‘†) ∧ 𝐡 finSupp (0gβ€˜π‘†))) β†’ (𝑀 Ξ£g (π‘₯ ∈ 𝑉 ↦ (((𝐴 ∘f ✚ 𝐡)β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯))) = (𝑀 Ξ£g (π‘₯ ∈ 𝑉 ↦ (((π΄β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯) + ((π΅β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯)))))
11487, 113eqtrd 2777 . 2 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0gβ€˜π‘†) ∧ 𝐡 finSupp (0gβ€˜π‘†))) β†’ ((𝐴 ∘f ✚ 𝐡)( linC β€˜π‘€)𝑉) = (𝑀 Ξ£g (π‘₯ ∈ 𝑉 ↦ (((π΄β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯) + ((π΅β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯)))))
115 lincsum.x . . . 4 𝑋 = (𝐴( linC β€˜π‘€)𝑉)
116 lincsum.y . . . 4 π‘Œ = (𝐡( linC β€˜π‘€)𝑉)
117115, 116oveq12i 7374 . . 3 (𝑋 + π‘Œ) = ((𝐴( linC β€˜π‘€)𝑉) + (𝐡( linC β€˜π‘€)𝑉))
11866oveq1i 7372 . . . . . . . . 9 (𝑅 ↑m 𝑉) = ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)
119118eleq2i 2830 . . . . . . . 8 (𝐴 ∈ (𝑅 ↑m 𝑉) ↔ 𝐴 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉))
120119biimpi 215 . . . . . . 7 (𝐴 ∈ (𝑅 ↑m 𝑉) β†’ 𝐴 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉))
121120ad2antrl 727 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉))) β†’ 𝐴 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉))
122 lincval 46564 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝐴 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ (𝐴( linC β€˜π‘€)𝑉) = (𝑀 Ξ£g (π‘₯ ∈ 𝑉 ↦ ((π΄β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯))))
12398, 121, 50, 122syl3anc 1372 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉))) β†’ (𝐴( linC β€˜π‘€)𝑉) = (𝑀 Ξ£g (π‘₯ ∈ 𝑉 ↦ ((π΄β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯))))
124118eleq2i 2830 . . . . . . . 8 (𝐡 ∈ (𝑅 ↑m 𝑉) ↔ 𝐡 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉))
125124biimpi 215 . . . . . . 7 (𝐡 ∈ (𝑅 ↑m 𝑉) β†’ 𝐡 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉))
126125ad2antll 728 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉))) β†’ 𝐡 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉))
127 lincval 46564 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝐡 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ (𝐡( linC β€˜π‘€)𝑉) = (𝑀 Ξ£g (π‘₯ ∈ 𝑉 ↦ ((π΅β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯))))
12898, 126, 50, 127syl3anc 1372 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉))) β†’ (𝐡( linC β€˜π‘€)𝑉) = (𝑀 Ξ£g (π‘₯ ∈ 𝑉 ↦ ((π΅β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯))))
129123, 128oveq12d 7380 . . . 4 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉))) β†’ ((𝐴( linC β€˜π‘€)𝑉) + (𝐡( linC β€˜π‘€)𝑉)) = ((𝑀 Ξ£g (π‘₯ ∈ 𝑉 ↦ ((π΄β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯))) + (𝑀 Ξ£g (π‘₯ ∈ 𝑉 ↦ ((π΅β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯)))))
1301293adant3 1133 . . 3 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0gβ€˜π‘†) ∧ 𝐡 finSupp (0gβ€˜π‘†))) β†’ ((𝐴( linC β€˜π‘€)𝑉) + (𝐡( linC β€˜π‘€)𝑉)) = ((𝑀 Ξ£g (π‘₯ ∈ 𝑉 ↦ ((π΄β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯))) + (𝑀 Ξ£g (π‘₯ ∈ 𝑉 ↦ ((π΅β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯)))))
131117, 130eqtrid 2789 . 2 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0gβ€˜π‘†) ∧ 𝐡 finSupp (0gβ€˜π‘†))) β†’ (𝑋 + π‘Œ) = ((𝑀 Ξ£g (π‘₯ ∈ 𝑉 ↦ ((π΄β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯))) + (𝑀 Ξ£g (π‘₯ ∈ 𝑉 ↦ ((π΅β€˜π‘₯)( ·𝑠 β€˜π‘€)π‘₯)))))
13249, 114, 1313eqtr4rd 2788 1 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0gβ€˜π‘†) ∧ 𝐡 finSupp (0gβ€˜π‘†))) β†’ (𝑋 + π‘Œ) = ((𝐴 ∘f ✚ 𝐡)( linC β€˜π‘€)𝑉))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  Vcvv 3448  π’« cpw 4565   class class class wbr 5110   ↦ cmpt 5193   Fn wfn 6496  βŸΆwf 6497  β€˜cfv 6501  (class class class)co 7362   ∘f cof 7620   ↑m cmap 8772   finSupp cfsupp 9312  Basecbs 17090  +gcplusg 17140  Scalarcsca 17143   ·𝑠 cvsca 17144  0gc0g 17328   Ξ£g cgsu 17329  Mndcmnd 18563  CMndccmn 19569  LModclmod 20338   linC clinc 46559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-of 7622  df-om 7808  df-1st 7926  df-2nd 7927  df-supp 8098  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-er 8655  df-map 8774  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-fsupp 9313  df-oi 9453  df-card 9882  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-nn 12161  df-2 12223  df-n0 12421  df-z 12507  df-uz 12771  df-fz 13432  df-fzo 13575  df-seq 13914  df-hash 14238  df-sets 17043  df-slot 17061  df-ndx 17073  df-base 17091  df-ress 17120  df-plusg 17153  df-0g 17330  df-gsum 17331  df-mgm 18504  df-sgrp 18553  df-mnd 18564  df-submnd 18609  df-grp 18758  df-minusg 18759  df-cntz 19104  df-cmn 19571  df-abl 19572  df-mgp 19904  df-ur 19921  df-ring 19973  df-lmod 20340  df-linc 46561
This theorem is referenced by:  lincsumcl  46586
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