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Theorem lincsum 45472
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 2738 . . 3 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2738 . . 3 (0g𝑀) = (0g𝑀)
3 lincsum.p . . 3 + = (+g𝑀)
4 lmodcmn 19972 . . . . 5 (𝑀 ∈ LMod → 𝑀 ∈ CMnd)
54adantr 484 . . . 4 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑀 ∈ CMnd)
653ad2ant1 1135 . . 3 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) → 𝑀 ∈ CMnd)
7 simpr 488 . . . 4 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑉 ∈ 𝒫 (Base‘𝑀))
873ad2ant1 1135 . . 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 486 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑀 ∈ LMod)
1093ad2ant1 1135 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) → 𝑀 ∈ LMod)
1110adantr 484 . . . 4 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) ∧ 𝑥𝑉) → 𝑀 ∈ LMod)
12 elmapi 8551 . . . . . . . 8 (𝐴 ∈ (𝑅m 𝑉) → 𝐴:𝑉𝑅)
13 ffvelrn 6921 . . . . . . . . 9 ((𝐴:𝑉𝑅𝑥𝑉) → (𝐴𝑥) ∈ 𝑅)
1413ex 416 . . . . . . . 8 (𝐴:𝑉𝑅 → (𝑥𝑉 → (𝐴𝑥) ∈ 𝑅))
1512, 14syl 17 . . . . . . 7 (𝐴 ∈ (𝑅m 𝑉) → (𝑥𝑉 → (𝐴𝑥) ∈ 𝑅))
1615adantr 484 . . . . . 6 ((𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) → (𝑥𝑉 → (𝐴𝑥) ∈ 𝑅))
17163ad2ant2 1136 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) → (𝑥𝑉 → (𝐴𝑥) ∈ 𝑅))
1817imp 410 . . . 4 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) ∧ 𝑥𝑉) → (𝐴𝑥) ∈ 𝑅)
19 elelpwi 4540 . . . . . . . 8 ((𝑥𝑉𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑥 ∈ (Base‘𝑀))
2019expcom 417 . . . . . . 7 (𝑉 ∈ 𝒫 (Base‘𝑀) → (𝑥𝑉𝑥 ∈ (Base‘𝑀)))
2120adantl 485 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑥𝑉𝑥 ∈ (Base‘𝑀)))
22213ad2ant1 1135 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) → (𝑥𝑉𝑥 ∈ (Base‘𝑀)))
2322imp 410 . . . 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 2738 . . . . 5 ( ·𝑠𝑀) = ( ·𝑠𝑀)
26 lincsum.r . . . . 5 𝑅 = (Base‘𝑆)
271, 24, 25, 26lmodvscl 19941 . . . 4 ((𝑀 ∈ LMod ∧ (𝐴𝑥) ∈ 𝑅𝑥 ∈ (Base‘𝑀)) → ((𝐴𝑥)( ·𝑠𝑀)𝑥) ∈ (Base‘𝑀))
2811, 18, 23, 27syl3anc 1373 . . 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 8551 . . . . . . . 8 (𝐵 ∈ (𝑅m 𝑉) → 𝐵:𝑉𝑅)
30 ffvelrn 6921 . . . . . . . . 9 ((𝐵:𝑉𝑅𝑥𝑉) → (𝐵𝑥) ∈ 𝑅)
3130ex 416 . . . . . . . 8 (𝐵:𝑉𝑅 → (𝑥𝑉 → (𝐵𝑥) ∈ 𝑅))
3229, 31syl 17 . . . . . . 7 (𝐵 ∈ (𝑅m 𝑉) → (𝑥𝑉 → (𝐵𝑥) ∈ 𝑅))
3332adantl 485 . . . . . 6 ((𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) → (𝑥𝑉 → (𝐵𝑥) ∈ 𝑅))
34333ad2ant2 1136 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) → (𝑥𝑉 → (𝐵𝑥) ∈ 𝑅))
3534imp 410 . . . 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 19941 . . . 4 ((𝑀 ∈ LMod ∧ (𝐵𝑥) ∈ 𝑅𝑥 ∈ (Base‘𝑀)) → ((𝐵𝑥)( ·𝑠𝑀)𝑥) ∈ (Base‘𝑀))
3711, 35, 23, 36syl3anc 1373 . . 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 2739 . . 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 2739 . . 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 486 . . . 4 ((𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) → 𝐴 ∈ (𝑅m 𝑉))
42 simpl 486 . . . 4 ((𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆)) → 𝐴 finSupp (0g𝑆))
4324, 26scmfsupp 45416 . . . 4 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅m 𝑉) ∧ 𝐴 finSupp (0g𝑆)) → (𝑥𝑉 ↦ ((𝐴𝑥)( ·𝑠𝑀)𝑥)) finSupp (0g𝑀))
4440, 41, 42, 43syl3an 1162 . . 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 488 . . . 4 ((𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) → 𝐵 ∈ (𝑅m 𝑉))
46 simpr 488 . . . 4 ((𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆)) → 𝐵 finSupp (0g𝑆))
4724, 26scmfsupp 45416 . . . 4 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐵 ∈ (𝑅m 𝑉) ∧ 𝐵 finSupp (0g𝑆)) → (𝑥𝑉 ↦ ((𝐵𝑥)( ·𝑠𝑀)𝑥)) finSupp (0g𝑀))
4840, 45, 46, 47syl3an 1162 . . 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 19334 . 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 484 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → 𝑉 ∈ 𝒫 (Base‘𝑀))
51 elmapfn 8567 . . . . . . . 8 (𝐴 ∈ (𝑅m 𝑉) → 𝐴 Fn 𝑉)
5251ad2antrl 728 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → 𝐴 Fn 𝑉)
53 elmapfn 8567 . . . . . . . 8 (𝐵 ∈ (𝑅m 𝑉) → 𝐵 Fn 𝑉)
5453ad2antll 729 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → 𝐵 Fn 𝑉)
5550, 52, 54offvalfv 45380 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → (𝐴f 𝐵) = (𝑦𝑉 ↦ ((𝐴𝑦) (𝐵𝑦))))
56553adant3 1134 . . . . 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 19933 . . . . . . . . . . 11 (𝑀 ∈ LMod → 𝑆 ∈ Grp)
5857grpmndd 18402 . . . . . . . . . 10 (𝑀 ∈ LMod → 𝑆 ∈ Mnd)
5958ad3antrrr 730 . . . . . . . . 9 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑦𝑉) → 𝑆 ∈ Mnd)
60 ffvelrn 6921 . . . . . . . . . . . . . 14 ((𝐴:𝑉𝑅𝑦𝑉) → (𝐴𝑦) ∈ 𝑅)
6160ex 416 . . . . . . . . . . . . 13 (𝐴:𝑉𝑅 → (𝑦𝑉 → (𝐴𝑦) ∈ 𝑅))
6212, 61syl 17 . . . . . . . . . . . 12 (𝐴 ∈ (𝑅m 𝑉) → (𝑦𝑉 → (𝐴𝑦) ∈ 𝑅))
6362ad2antrl 728 . . . . . . . . . . 11 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → (𝑦𝑉 → (𝐴𝑦) ∈ 𝑅))
6463imp 410 . . . . . . . . . 10 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑦𝑉) → (𝐴𝑦) ∈ 𝑅)
6524fveq2i 6739 . . . . . . . . . . 11 (Base‘𝑆) = (Base‘(Scalar‘𝑀))
6626, 65eqtri 2766 . . . . . . . . . 10 𝑅 = (Base‘(Scalar‘𝑀))
6764, 66eleqtrdi 2849 . . . . . . . . 9 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑦𝑉) → (𝐴𝑦) ∈ (Base‘(Scalar‘𝑀)))
68 ffvelrn 6921 . . . . . . . . . . . . . 14 ((𝐵:𝑉𝑅𝑦𝑉) → (𝐵𝑦) ∈ 𝑅)
6968, 66eleqtrdi 2849 . . . . . . . . . . . . 13 ((𝐵:𝑉𝑅𝑦𝑉) → (𝐵𝑦) ∈ (Base‘(Scalar‘𝑀)))
7069ex 416 . . . . . . . . . . . 12 (𝐵:𝑉𝑅 → (𝑦𝑉 → (𝐵𝑦) ∈ (Base‘(Scalar‘𝑀))))
7129, 70syl 17 . . . . . . . . . . 11 (𝐵 ∈ (𝑅m 𝑉) → (𝑦𝑉 → (𝐵𝑦) ∈ (Base‘(Scalar‘𝑀))))
7271ad2antll 729 . . . . . . . . . 10 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → (𝑦𝑉 → (𝐵𝑦) ∈ (Base‘(Scalar‘𝑀))))
7372imp 410 . . . . . . . . 9 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑦𝑉) → (𝐵𝑦) ∈ (Base‘(Scalar‘𝑀)))
7424eqcomi 2747 . . . . . . . . . . 11 (Scalar‘𝑀) = 𝑆
7574fveq2i 6739 . . . . . . . . . 10 (Base‘(Scalar‘𝑀)) = (Base‘𝑆)
76 lincsum.b . . . . . . . . . 10 = (+g𝑆)
7775, 76mndcl 18206 . . . . . . . . 9 ((𝑆 ∈ Mnd ∧ (𝐴𝑦) ∈ (Base‘(Scalar‘𝑀)) ∧ (𝐵𝑦) ∈ (Base‘(Scalar‘𝑀))) → ((𝐴𝑦) (𝐵𝑦)) ∈ (Base‘(Scalar‘𝑀)))
7859, 67, 73, 77syl3anc 1373 . . . . . . . 8 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑦𝑉) → ((𝐴𝑦) (𝐵𝑦)) ∈ (Base‘(Scalar‘𝑀)))
7978fmpttd 6951 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → (𝑦𝑉 ↦ ((𝐴𝑦) (𝐵𝑦))):𝑉⟶(Base‘(Scalar‘𝑀)))
80 fvex 6749 . . . . . . . 8 (Base‘(Scalar‘𝑀)) ∈ V
81 elmapg 8542 . . . . . . . 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 590 . . . . . . 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 260 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → (𝑦𝑉 ↦ ((𝐴𝑦) (𝐵𝑦))) ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
84833adant3 1134 . . . . 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 2839 . . . 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 45452 . . . 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 1373 . . 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 616 . . . . . . . . . . . 12 ((𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) → (𝐴 Fn 𝑉𝐵 Fn 𝑉))
8988adantl 485 . . . . . . . . . . 11 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → (𝐴 Fn 𝑉𝐵 Fn 𝑉))
9089adantr 484 . . . . . . . . . 10 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑥𝑉) → (𝐴 Fn 𝑉𝐵 Fn 𝑉))
9150anim1i 618 . . . . . . . . . 10 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑥𝑉) → (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑥𝑉))
92 fnfvof 7504 . . . . . . . . . 10 (((𝐴 Fn 𝑉𝐵 Fn 𝑉) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑥𝑉)) → ((𝐴f 𝐵)‘𝑥) = ((𝐴𝑥) (𝐵𝑥)))
9390, 91, 92syl2anc 587 . . . . . . . . 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 7249 . . . . . . . . 9 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑥𝑉) → ((𝐴𝑥) (𝐵𝑥)) = ((𝐴𝑥)(+g𝑆)(𝐵𝑥)))
9693, 95eqtrd 2778 . . . . . . . 8 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑥𝑉) → ((𝐴f 𝐵)‘𝑥) = ((𝐴𝑥)(+g𝑆)(𝐵𝑥)))
9796oveq1d 7247 . . . . . . 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 484 . . . . . . . . 9 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → 𝑀 ∈ LMod)
9998adantr 484 . . . . . . . 8 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑥𝑉) → 𝑀 ∈ LMod)
10015ad2antrl 728 . . . . . . . . 9 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → (𝑥𝑉 → (𝐴𝑥) ∈ 𝑅))
101100imp 410 . . . . . . . 8 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑥𝑉) → (𝐴𝑥) ∈ 𝑅)
10232ad2antll 729 . . . . . . . . 9 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → (𝑥𝑉 → (𝐵𝑥) ∈ 𝑅))
103102imp 410 . . . . . . . 8 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑥𝑉) → (𝐵𝑥) ∈ 𝑅)
10421adantr 484 . . . . . . . . 9 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → (𝑥𝑉𝑥 ∈ (Base‘𝑀)))
105104imp 410 . . . . . . . 8 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑥𝑉) → 𝑥 ∈ (Base‘𝑀))
106 eqid 2738 . . . . . . . . 9 (Scalar‘𝑀) = (Scalar‘𝑀)
10724fveq2i 6739 . . . . . . . . 9 (+g𝑆) = (+g‘(Scalar‘𝑀))
1081, 3, 106, 25, 66, 107lmodvsdir 19948 . . . . . . . 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 1374 . . . . . . 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 2778 . . . . . 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 5165 . . . . 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 7248 . . . 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 1134 . . 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 2778 . 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 7244 . . 3 (𝑋 + 𝑌) = ((𝐴( linC ‘𝑀)𝑉) + (𝐵( linC ‘𝑀)𝑉))
11866oveq1i 7242 . . . . . . . . 9 (𝑅m 𝑉) = ((Base‘(Scalar‘𝑀)) ↑m 𝑉)
119118eleq2i 2830 . . . . . . . 8 (𝐴 ∈ (𝑅m 𝑉) ↔ 𝐴 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
120119biimpi 219 . . . . . . 7 (𝐴 ∈ (𝑅m 𝑉) → 𝐴 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
121120ad2antrl 728 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → 𝐴 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
122 lincval 45452 . . . . . 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 1373 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → (𝐴( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑥𝑉 ↦ ((𝐴𝑥)( ·𝑠𝑀)𝑥))))
124118eleq2i 2830 . . . . . . . 8 (𝐵 ∈ (𝑅m 𝑉) ↔ 𝐵 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
125124biimpi 219 . . . . . . 7 (𝐵 ∈ (𝑅m 𝑉) → 𝐵 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
126125ad2antll 729 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → 𝐵 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
127 lincval 45452 . . . . . 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 1373 . . . . 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 7250 . . . 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 1134 . . 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, 130syl5eq 2791 . 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 2789 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 209  wa 399  w3a 1089   = wceq 1543  wcel 2111  Vcvv 3421  𝒫 cpw 4528   class class class wbr 5068  cmpt 5150   Fn wfn 6393  wf 6394  cfv 6398  (class class class)co 7232  f cof 7486  m cmap 8529   finSupp cfsupp 9010  Basecbs 16785  +gcplusg 16827  Scalarcsca 16830   ·𝑠 cvsca 16831  0gc0g 16969   Σg cgsu 16970  Mndcmnd 18198  CMndccmn 19195  LModclmod 19924   linC clinc 45447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-rep 5194  ax-sep 5207  ax-nul 5214  ax-pow 5273  ax-pr 5337  ax-un 7542  ax-cnex 10810  ax-resscn 10811  ax-1cn 10812  ax-icn 10813  ax-addcl 10814  ax-addrcl 10815  ax-mulcl 10816  ax-mulrcl 10817  ax-mulcom 10818  ax-addass 10819  ax-mulass 10820  ax-distr 10821  ax-i2m1 10822  ax-1ne0 10823  ax-1rid 10824  ax-rnegex 10825  ax-rrecex 10826  ax-cnre 10827  ax-pre-lttri 10828  ax-pre-lttrn 10829  ax-pre-ltadd 10830  ax-pre-mulgt0 10831
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3067  df-rex 3068  df-reu 3069  df-rmo 3070  df-rab 3071  df-v 3423  df-sbc 3710  df-csb 3827  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4253  df-if 4455  df-pw 4530  df-sn 4557  df-pr 4559  df-tp 4561  df-op 4563  df-uni 4835  df-int 4875  df-iun 4921  df-br 5069  df-opab 5131  df-mpt 5151  df-tr 5177  df-id 5470  df-eprel 5475  df-po 5483  df-so 5484  df-fr 5524  df-se 5525  df-we 5526  df-xp 5572  df-rel 5573  df-cnv 5574  df-co 5575  df-dm 5576  df-rn 5577  df-res 5578  df-ima 5579  df-pred 6176  df-ord 6234  df-on 6235  df-lim 6236  df-suc 6237  df-iota 6356  df-fun 6400  df-fn 6401  df-f 6402  df-f1 6403  df-fo 6404  df-f1o 6405  df-fv 6406  df-isom 6407  df-riota 7189  df-ov 7235  df-oprab 7236  df-mpo 7237  df-of 7488  df-om 7664  df-1st 7780  df-2nd 7781  df-supp 7925  df-wrecs 8068  df-recs 8129  df-rdg 8167  df-1o 8223  df-er 8412  df-map 8531  df-en 8648  df-dom 8649  df-sdom 8650  df-fin 8651  df-fsupp 9011  df-oi 9151  df-card 9580  df-pnf 10894  df-mnf 10895  df-xr 10896  df-ltxr 10897  df-le 10898  df-sub 11089  df-neg 11090  df-nn 11856  df-2 11918  df-n0 12116  df-z 12202  df-uz 12464  df-fz 13121  df-fzo 13264  df-seq 13600  df-hash 13922  df-sets 16742  df-slot 16760  df-ndx 16770  df-base 16786  df-ress 16810  df-plusg 16840  df-0g 16971  df-gsum 16972  df-mgm 18139  df-sgrp 18188  df-mnd 18199  df-submnd 18244  df-grp 18393  df-minusg 18394  df-cntz 18736  df-cmn 19197  df-abl 19198  df-mgp 19530  df-ur 19542  df-ring 19589  df-lmod 19926  df-linc 45449
This theorem is referenced by:  lincsumcl  45474
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