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Theorem lincsum 46500
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 + = (+g𝑀)
lincsum.x 𝑋 = (𝐴( linC ‘𝑀)𝑉)
lincsum.y 𝑌 = (𝐵( linC ‘𝑀)𝑉)
lincsum.s 𝑆 = (Scalar‘𝑀)
lincsum.r 𝑅 = (Base‘𝑆)
lincsum.b = (+g𝑆)
Assertion
Ref Expression
lincsum (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) → (𝑋 + 𝑌) = ((𝐴f 𝐵)( linC ‘𝑀)𝑉))

Proof of Theorem lincsum
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2736 . . 3 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2736 . . 3 (0g𝑀) = (0g𝑀)
3 lincsum.p . . 3 + = (+g𝑀)
4 lmodcmn 20370 . . . . 5 (𝑀 ∈ LMod → 𝑀 ∈ CMnd)
54adantr 481 . . . 4 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑀 ∈ CMnd)
653ad2ant1 1133 . . 3 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) → 𝑀 ∈ CMnd)
7 simpr 485 . . . 4 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑉 ∈ 𝒫 (Base‘𝑀))
873ad2ant1 1133 . . 3 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) → 𝑉 ∈ 𝒫 (Base‘𝑀))
9 simpl 483 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑀 ∈ LMod)
1093ad2ant1 1133 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) → 𝑀 ∈ LMod)
1110adantr 481 . . . 4 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) ∧ 𝑥𝑉) → 𝑀 ∈ LMod)
12 elmapi 8787 . . . . . . . 8 (𝐴 ∈ (𝑅m 𝑉) → 𝐴:𝑉𝑅)
13 ffvelcdm 7032 . . . . . . . . 9 ((𝐴:𝑉𝑅𝑥𝑉) → (𝐴𝑥) ∈ 𝑅)
1413ex 413 . . . . . . . 8 (𝐴:𝑉𝑅 → (𝑥𝑉 → (𝐴𝑥) ∈ 𝑅))
1512, 14syl 17 . . . . . . 7 (𝐴 ∈ (𝑅m 𝑉) → (𝑥𝑉 → (𝐴𝑥) ∈ 𝑅))
1615adantr 481 . . . . . 6 ((𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) → (𝑥𝑉 → (𝐴𝑥) ∈ 𝑅))
17163ad2ant2 1134 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) → (𝑥𝑉 → (𝐴𝑥) ∈ 𝑅))
1817imp 407 . . . 4 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) ∧ 𝑥𝑉) → (𝐴𝑥) ∈ 𝑅)
19 elelpwi 4570 . . . . . . . 8 ((𝑥𝑉𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑥 ∈ (Base‘𝑀))
2019expcom 414 . . . . . . 7 (𝑉 ∈ 𝒫 (Base‘𝑀) → (𝑥𝑉𝑥 ∈ (Base‘𝑀)))
2120adantl 482 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑥𝑉𝑥 ∈ (Base‘𝑀)))
22213ad2ant1 1133 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) → (𝑥𝑉𝑥 ∈ (Base‘𝑀)))
2322imp 407 . . . 4 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) ∧ 𝑥𝑉) → 𝑥 ∈ (Base‘𝑀))
24 lincsum.s . . . . 5 𝑆 = (Scalar‘𝑀)
25 eqid 2736 . . . . 5 ( ·𝑠𝑀) = ( ·𝑠𝑀)
26 lincsum.r . . . . 5 𝑅 = (Base‘𝑆)
271, 24, 25, 26lmodvscl 20339 . . . 4 ((𝑀 ∈ LMod ∧ (𝐴𝑥) ∈ 𝑅𝑥 ∈ (Base‘𝑀)) → ((𝐴𝑥)( ·𝑠𝑀)𝑥) ∈ (Base‘𝑀))
2811, 18, 23, 27syl3anc 1371 . . 3 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) ∧ 𝑥𝑉) → ((𝐴𝑥)( ·𝑠𝑀)𝑥) ∈ (Base‘𝑀))
29 elmapi 8787 . . . . . . . 8 (𝐵 ∈ (𝑅m 𝑉) → 𝐵:𝑉𝑅)
30 ffvelcdm 7032 . . . . . . . . 9 ((𝐵:𝑉𝑅𝑥𝑉) → (𝐵𝑥) ∈ 𝑅)
3130ex 413 . . . . . . . 8 (𝐵:𝑉𝑅 → (𝑥𝑉 → (𝐵𝑥) ∈ 𝑅))
3229, 31syl 17 . . . . . . 7 (𝐵 ∈ (𝑅m 𝑉) → (𝑥𝑉 → (𝐵𝑥) ∈ 𝑅))
3332adantl 482 . . . . . 6 ((𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) → (𝑥𝑉 → (𝐵𝑥) ∈ 𝑅))
34333ad2ant2 1134 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) → (𝑥𝑉 → (𝐵𝑥) ∈ 𝑅))
3534imp 407 . . . 4 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) ∧ 𝑥𝑉) → (𝐵𝑥) ∈ 𝑅)
361, 24, 25, 26lmodvscl 20339 . . . 4 ((𝑀 ∈ LMod ∧ (𝐵𝑥) ∈ 𝑅𝑥 ∈ (Base‘𝑀)) → ((𝐵𝑥)( ·𝑠𝑀)𝑥) ∈ (Base‘𝑀))
3711, 35, 23, 36syl3anc 1371 . . 3 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) ∧ 𝑥𝑉) → ((𝐵𝑥)( ·𝑠𝑀)𝑥) ∈ (Base‘𝑀))
38 eqidd 2737 . . 3 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) → (𝑥𝑉 ↦ ((𝐴𝑥)( ·𝑠𝑀)𝑥)) = (𝑥𝑉 ↦ ((𝐴𝑥)( ·𝑠𝑀)𝑥)))
39 eqidd 2737 . . 3 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) → (𝑥𝑉 ↦ ((𝐵𝑥)( ·𝑠𝑀)𝑥)) = (𝑥𝑉 ↦ ((𝐵𝑥)( ·𝑠𝑀)𝑥)))
40 id 22 . . . 4 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)))
41 simpl 483 . . . 4 ((𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) → 𝐴 ∈ (𝑅m 𝑉))
42 simpl 483 . . . 4 ((𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆)) → 𝐴 finSupp (0g𝑆))
4324, 26scmfsupp 46444 . . . 4 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅m 𝑉) ∧ 𝐴 finSupp (0g𝑆)) → (𝑥𝑉 ↦ ((𝐴𝑥)( ·𝑠𝑀)𝑥)) finSupp (0g𝑀))
4440, 41, 42, 43syl3an 1160 . . 3 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) → (𝑥𝑉 ↦ ((𝐴𝑥)( ·𝑠𝑀)𝑥)) finSupp (0g𝑀))
45 simpr 485 . . . 4 ((𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) → 𝐵 ∈ (𝑅m 𝑉))
46 simpr 485 . . . 4 ((𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆)) → 𝐵 finSupp (0g𝑆))
4724, 26scmfsupp 46444 . . . 4 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐵 ∈ (𝑅m 𝑉) ∧ 𝐵 finSupp (0g𝑆)) → (𝑥𝑉 ↦ ((𝐵𝑥)( ·𝑠𝑀)𝑥)) finSupp (0g𝑀))
4840, 45, 46, 47syl3an 1160 . . 3 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) → (𝑥𝑉 ↦ ((𝐵𝑥)( ·𝑠𝑀)𝑥)) finSupp (0g𝑀))
491, 2, 3, 6, 8, 28, 37, 38, 39, 44, 48gsummptfsadd 19701 . 2 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) → (𝑀 Σg (𝑥𝑉 ↦ (((𝐴𝑥)( ·𝑠𝑀)𝑥) + ((𝐵𝑥)( ·𝑠𝑀)𝑥)))) = ((𝑀 Σg (𝑥𝑉 ↦ ((𝐴𝑥)( ·𝑠𝑀)𝑥))) + (𝑀 Σg (𝑥𝑉 ↦ ((𝐵𝑥)( ·𝑠𝑀)𝑥)))))
507adantr 481 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → 𝑉 ∈ 𝒫 (Base‘𝑀))
51 elmapfn 8803 . . . . . . . 8 (𝐴 ∈ (𝑅m 𝑉) → 𝐴 Fn 𝑉)
5251ad2antrl 726 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → 𝐴 Fn 𝑉)
53 elmapfn 8803 . . . . . . . 8 (𝐵 ∈ (𝑅m 𝑉) → 𝐵 Fn 𝑉)
5453ad2antll 727 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → 𝐵 Fn 𝑉)
5550, 52, 54offvalfv 46408 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → (𝐴f 𝐵) = (𝑦𝑉 ↦ ((𝐴𝑦) (𝐵𝑦))))
56553adant3 1132 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) → (𝐴f 𝐵) = (𝑦𝑉 ↦ ((𝐴𝑦) (𝐵𝑦))))
5724lmodfgrp 20331 . . . . . . . . . . 11 (𝑀 ∈ LMod → 𝑆 ∈ Grp)
5857grpmndd 18760 . . . . . . . . . 10 (𝑀 ∈ LMod → 𝑆 ∈ Mnd)
5958ad3antrrr 728 . . . . . . . . 9 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑦𝑉) → 𝑆 ∈ Mnd)
60 ffvelcdm 7032 . . . . . . . . . . . . . 14 ((𝐴:𝑉𝑅𝑦𝑉) → (𝐴𝑦) ∈ 𝑅)
6160ex 413 . . . . . . . . . . . . 13 (𝐴:𝑉𝑅 → (𝑦𝑉 → (𝐴𝑦) ∈ 𝑅))
6212, 61syl 17 . . . . . . . . . . . 12 (𝐴 ∈ (𝑅m 𝑉) → (𝑦𝑉 → (𝐴𝑦) ∈ 𝑅))
6362ad2antrl 726 . . . . . . . . . . 11 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → (𝑦𝑉 → (𝐴𝑦) ∈ 𝑅))
6463imp 407 . . . . . . . . . 10 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑦𝑉) → (𝐴𝑦) ∈ 𝑅)
6524fveq2i 6845 . . . . . . . . . . 11 (Base‘𝑆) = (Base‘(Scalar‘𝑀))
6626, 65eqtri 2764 . . . . . . . . . 10 𝑅 = (Base‘(Scalar‘𝑀))
6764, 66eleqtrdi 2848 . . . . . . . . 9 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑦𝑉) → (𝐴𝑦) ∈ (Base‘(Scalar‘𝑀)))
68 ffvelcdm 7032 . . . . . . . . . . . . . 14 ((𝐵:𝑉𝑅𝑦𝑉) → (𝐵𝑦) ∈ 𝑅)
6968, 66eleqtrdi 2848 . . . . . . . . . . . . 13 ((𝐵:𝑉𝑅𝑦𝑉) → (𝐵𝑦) ∈ (Base‘(Scalar‘𝑀)))
7069ex 413 . . . . . . . . . . . 12 (𝐵:𝑉𝑅 → (𝑦𝑉 → (𝐵𝑦) ∈ (Base‘(Scalar‘𝑀))))
7129, 70syl 17 . . . . . . . . . . 11 (𝐵 ∈ (𝑅m 𝑉) → (𝑦𝑉 → (𝐵𝑦) ∈ (Base‘(Scalar‘𝑀))))
7271ad2antll 727 . . . . . . . . . 10 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → (𝑦𝑉 → (𝐵𝑦) ∈ (Base‘(Scalar‘𝑀))))
7372imp 407 . . . . . . . . 9 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑦𝑉) → (𝐵𝑦) ∈ (Base‘(Scalar‘𝑀)))
7424eqcomi 2745 . . . . . . . . . . 11 (Scalar‘𝑀) = 𝑆
7574fveq2i 6845 . . . . . . . . . 10 (Base‘(Scalar‘𝑀)) = (Base‘𝑆)
76 lincsum.b . . . . . . . . . 10 = (+g𝑆)
7775, 76mndcl 18564 . . . . . . . . 9 ((𝑆 ∈ Mnd ∧ (𝐴𝑦) ∈ (Base‘(Scalar‘𝑀)) ∧ (𝐵𝑦) ∈ (Base‘(Scalar‘𝑀))) → ((𝐴𝑦) (𝐵𝑦)) ∈ (Base‘(Scalar‘𝑀)))
7859, 67, 73, 77syl3anc 1371 . . . . . . . 8 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑦𝑉) → ((𝐴𝑦) (𝐵𝑦)) ∈ (Base‘(Scalar‘𝑀)))
7978fmpttd 7063 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → (𝑦𝑉 ↦ ((𝐴𝑦) (𝐵𝑦))):𝑉⟶(Base‘(Scalar‘𝑀)))
80 fvex 6855 . . . . . . . 8 (Base‘(Scalar‘𝑀)) ∈ V
81 elmapg 8778 . . . . . . . 8 (((Base‘(Scalar‘𝑀)) ∈ V ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ((𝑦𝑉 ↦ ((𝐴𝑦) (𝐵𝑦))) ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ↔ (𝑦𝑉 ↦ ((𝐴𝑦) (𝐵𝑦))):𝑉⟶(Base‘(Scalar‘𝑀))))
8280, 50, 81sylancr 587 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → ((𝑦𝑉 ↦ ((𝐴𝑦) (𝐵𝑦))) ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ↔ (𝑦𝑉 ↦ ((𝐴𝑦) (𝐵𝑦))):𝑉⟶(Base‘(Scalar‘𝑀))))
8379, 82mpbird 256 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → (𝑦𝑉 ↦ ((𝐴𝑦) (𝐵𝑦))) ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
84833adant3 1132 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) → (𝑦𝑉 ↦ ((𝐴𝑦) (𝐵𝑦))) ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
8556, 84eqeltrd 2838 . . . 4 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) → (𝐴f 𝐵) ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
86 lincval 46480 . . . 4 ((𝑀 ∈ LMod ∧ (𝐴f 𝐵) ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ((𝐴f 𝐵)( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑥𝑉 ↦ (((𝐴f 𝐵)‘𝑥)( ·𝑠𝑀)𝑥))))
8710, 85, 8, 86syl3anc 1371 . . 3 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) → ((𝐴f 𝐵)( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑥𝑉 ↦ (((𝐴f 𝐵)‘𝑥)( ·𝑠𝑀)𝑥))))
8851, 53anim12i 613 . . . . . . . . . . . 12 ((𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) → (𝐴 Fn 𝑉𝐵 Fn 𝑉))
8988adantl 482 . . . . . . . . . . 11 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → (𝐴 Fn 𝑉𝐵 Fn 𝑉))
9089adantr 481 . . . . . . . . . 10 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑥𝑉) → (𝐴 Fn 𝑉𝐵 Fn 𝑉))
9150anim1i 615 . . . . . . . . . 10 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑥𝑉) → (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑥𝑉))
92 fnfvof 7634 . . . . . . . . . 10 (((𝐴 Fn 𝑉𝐵 Fn 𝑉) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑥𝑉)) → ((𝐴f 𝐵)‘𝑥) = ((𝐴𝑥) (𝐵𝑥)))
9390, 91, 92syl2anc 584 . . . . . . . . 9 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑥𝑉) → ((𝐴f 𝐵)‘𝑥) = ((𝐴𝑥) (𝐵𝑥)))
9476a1i 11 . . . . . . . . . 10 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑥𝑉) → = (+g𝑆))
9594oveqd 7374 . . . . . . . . 9 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑥𝑉) → ((𝐴𝑥) (𝐵𝑥)) = ((𝐴𝑥)(+g𝑆)(𝐵𝑥)))
9693, 95eqtrd 2776 . . . . . . . 8 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑥𝑉) → ((𝐴f 𝐵)‘𝑥) = ((𝐴𝑥)(+g𝑆)(𝐵𝑥)))
9796oveq1d 7372 . . . . . . 7 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑥𝑉) → (((𝐴f 𝐵)‘𝑥)( ·𝑠𝑀)𝑥) = (((𝐴𝑥)(+g𝑆)(𝐵𝑥))( ·𝑠𝑀)𝑥))
989adantr 481 . . . . . . . . 9 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → 𝑀 ∈ LMod)
9998adantr 481 . . . . . . . 8 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑥𝑉) → 𝑀 ∈ LMod)
10015ad2antrl 726 . . . . . . . . 9 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → (𝑥𝑉 → (𝐴𝑥) ∈ 𝑅))
101100imp 407 . . . . . . . 8 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑥𝑉) → (𝐴𝑥) ∈ 𝑅)
10232ad2antll 727 . . . . . . . . 9 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → (𝑥𝑉 → (𝐵𝑥) ∈ 𝑅))
103102imp 407 . . . . . . . 8 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑥𝑉) → (𝐵𝑥) ∈ 𝑅)
10421adantr 481 . . . . . . . . 9 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → (𝑥𝑉𝑥 ∈ (Base‘𝑀)))
105104imp 407 . . . . . . . 8 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑥𝑉) → 𝑥 ∈ (Base‘𝑀))
106 eqid 2736 . . . . . . . . 9 (Scalar‘𝑀) = (Scalar‘𝑀)
10724fveq2i 6845 . . . . . . . . 9 (+g𝑆) = (+g‘(Scalar‘𝑀))
1081, 3, 106, 25, 66, 107lmodvsdir 20346 . . . . . . . 8 ((𝑀 ∈ LMod ∧ ((𝐴𝑥) ∈ 𝑅 ∧ (𝐵𝑥) ∈ 𝑅𝑥 ∈ (Base‘𝑀))) → (((𝐴𝑥)(+g𝑆)(𝐵𝑥))( ·𝑠𝑀)𝑥) = (((𝐴𝑥)( ·𝑠𝑀)𝑥) + ((𝐵𝑥)( ·𝑠𝑀)𝑥)))
10999, 101, 103, 105, 108syl13anc 1372 . . . . . . 7 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑥𝑉) → (((𝐴𝑥)(+g𝑆)(𝐵𝑥))( ·𝑠𝑀)𝑥) = (((𝐴𝑥)( ·𝑠𝑀)𝑥) + ((𝐵𝑥)( ·𝑠𝑀)𝑥)))
11097, 109eqtrd 2776 . . . . . 6 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑥𝑉) → (((𝐴f 𝐵)‘𝑥)( ·𝑠𝑀)𝑥) = (((𝐴𝑥)( ·𝑠𝑀)𝑥) + ((𝐵𝑥)( ·𝑠𝑀)𝑥)))
111110mpteq2dva 5205 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → (𝑥𝑉 ↦ (((𝐴f 𝐵)‘𝑥)( ·𝑠𝑀)𝑥)) = (𝑥𝑉 ↦ (((𝐴𝑥)( ·𝑠𝑀)𝑥) + ((𝐵𝑥)( ·𝑠𝑀)𝑥))))
112111oveq2d 7373 . . . 4 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → (𝑀 Σg (𝑥𝑉 ↦ (((𝐴f 𝐵)‘𝑥)( ·𝑠𝑀)𝑥))) = (𝑀 Σg (𝑥𝑉 ↦ (((𝐴𝑥)( ·𝑠𝑀)𝑥) + ((𝐵𝑥)( ·𝑠𝑀)𝑥)))))
1131123adant3 1132 . . 3 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) → (𝑀 Σg (𝑥𝑉 ↦ (((𝐴f 𝐵)‘𝑥)( ·𝑠𝑀)𝑥))) = (𝑀 Σg (𝑥𝑉 ↦ (((𝐴𝑥)( ·𝑠𝑀)𝑥) + ((𝐵𝑥)( ·𝑠𝑀)𝑥)))))
11487, 113eqtrd 2776 . 2 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) → ((𝐴f 𝐵)( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑥𝑉 ↦ (((𝐴𝑥)( ·𝑠𝑀)𝑥) + ((𝐵𝑥)( ·𝑠𝑀)𝑥)))))
115 lincsum.x . . . 4 𝑋 = (𝐴( linC ‘𝑀)𝑉)
116 lincsum.y . . . 4 𝑌 = (𝐵( linC ‘𝑀)𝑉)
117115, 116oveq12i 7369 . . 3 (𝑋 + 𝑌) = ((𝐴( linC ‘𝑀)𝑉) + (𝐵( linC ‘𝑀)𝑉))
11866oveq1i 7367 . . . . . . . . 9 (𝑅m 𝑉) = ((Base‘(Scalar‘𝑀)) ↑m 𝑉)
119118eleq2i 2829 . . . . . . . 8 (𝐴 ∈ (𝑅m 𝑉) ↔ 𝐴 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
120119biimpi 215 . . . . . . 7 (𝐴 ∈ (𝑅m 𝑉) → 𝐴 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
121120ad2antrl 726 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → 𝐴 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
122 lincval 46480 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝐴 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝐴( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑥𝑉 ↦ ((𝐴𝑥)( ·𝑠𝑀)𝑥))))
12398, 121, 50, 122syl3anc 1371 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → (𝐴( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑥𝑉 ↦ ((𝐴𝑥)( ·𝑠𝑀)𝑥))))
124118eleq2i 2829 . . . . . . . 8 (𝐵 ∈ (𝑅m 𝑉) ↔ 𝐵 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
125124biimpi 215 . . . . . . 7 (𝐵 ∈ (𝑅m 𝑉) → 𝐵 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
126125ad2antll 727 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → 𝐵 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
127 lincval 46480 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝐵 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝐵( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑥𝑉 ↦ ((𝐵𝑥)( ·𝑠𝑀)𝑥))))
12898, 126, 50, 127syl3anc 1371 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → (𝐵( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑥𝑉 ↦ ((𝐵𝑥)( ·𝑠𝑀)𝑥))))
129123, 128oveq12d 7375 . . . 4 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → ((𝐴( linC ‘𝑀)𝑉) + (𝐵( linC ‘𝑀)𝑉)) = ((𝑀 Σg (𝑥𝑉 ↦ ((𝐴𝑥)( ·𝑠𝑀)𝑥))) + (𝑀 Σg (𝑥𝑉 ↦ ((𝐵𝑥)( ·𝑠𝑀)𝑥)))))
1301293adant3 1132 . . 3 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) → ((𝐴( linC ‘𝑀)𝑉) + (𝐵( linC ‘𝑀)𝑉)) = ((𝑀 Σg (𝑥𝑉 ↦ ((𝐴𝑥)( ·𝑠𝑀)𝑥))) + (𝑀 Σg (𝑥𝑉 ↦ ((𝐵𝑥)( ·𝑠𝑀)𝑥)))))
131117, 130eqtrid 2788 . 2 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) → (𝑋 + 𝑌) = ((𝑀 Σg (𝑥𝑉 ↦ ((𝐴𝑥)( ·𝑠𝑀)𝑥))) + (𝑀 Σg (𝑥𝑉 ↦ ((𝐵𝑥)( ·𝑠𝑀)𝑥)))))
13249, 114, 1313eqtr4rd 2787 1 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) → (𝑋 + 𝑌) = ((𝐴f 𝐵)( linC ‘𝑀)𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  Vcvv 3445  𝒫 cpw 4560   class class class wbr 5105  cmpt 5188   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  f cof 7615  m cmap 8765   finSupp cfsupp 9305  Basecbs 17083  +gcplusg 17133  Scalarcsca 17136   ·𝑠 cvsca 17137  0gc0g 17321   Σg cgsu 17322  Mndcmnd 18556  CMndccmn 19562  LModclmod 20322   linC clinc 46475
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-of 7617  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-ress 17113  df-plusg 17146  df-0g 17323  df-gsum 17324  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-grp 18751  df-minusg 18752  df-cntz 19097  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-lmod 20324  df-linc 46477
This theorem is referenced by:  lincsumcl  46502
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