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Theorem lincsum 43066
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 2826 . . 3 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2826 . . 3 (0g𝑀) = (0g𝑀)
3 lincsum.p . . 3 + = (+g𝑀)
4 lmodcmn 19268 . . . . 5 (𝑀 ∈ LMod → 𝑀 ∈ CMnd)
54adantr 474 . . . 4 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑀 ∈ CMnd)
653ad2ant1 1169 . . 3 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) → 𝑀 ∈ CMnd)
7 simpr 479 . . . 4 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑉 ∈ 𝒫 (Base‘𝑀))
873ad2ant1 1169 . . 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 476 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑀 ∈ LMod)
1093ad2ant1 1169 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) → 𝑀 ∈ LMod)
1110adantr 474 . . . 4 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) ∧ 𝑥𝑉) → 𝑀 ∈ LMod)
12 elmapi 8145 . . . . . . . 8 (𝐴 ∈ (𝑅𝑚 𝑉) → 𝐴:𝑉𝑅)
13 ffvelrn 6607 . . . . . . . . 9 ((𝐴:𝑉𝑅𝑥𝑉) → (𝐴𝑥) ∈ 𝑅)
1413ex 403 . . . . . . . 8 (𝐴:𝑉𝑅 → (𝑥𝑉 → (𝐴𝑥) ∈ 𝑅))
1512, 14syl 17 . . . . . . 7 (𝐴 ∈ (𝑅𝑚 𝑉) → (𝑥𝑉 → (𝐴𝑥) ∈ 𝑅))
1615adantr 474 . . . . . 6 ((𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉)) → (𝑥𝑉 → (𝐴𝑥) ∈ 𝑅))
17163ad2ant2 1170 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) → (𝑥𝑉 → (𝐴𝑥) ∈ 𝑅))
1817imp 397 . . . 4 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) ∧ 𝑥𝑉) → (𝐴𝑥) ∈ 𝑅)
19 elelpwi 4392 . . . . . . . 8 ((𝑥𝑉𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑥 ∈ (Base‘𝑀))
2019expcom 404 . . . . . . 7 (𝑉 ∈ 𝒫 (Base‘𝑀) → (𝑥𝑉𝑥 ∈ (Base‘𝑀)))
2120adantl 475 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑥𝑉𝑥 ∈ (Base‘𝑀)))
22213ad2ant1 1169 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) → (𝑥𝑉𝑥 ∈ (Base‘𝑀)))
2322imp 397 . . . 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 2826 . . . . 5 ( ·𝑠𝑀) = ( ·𝑠𝑀)
26 lincsum.r . . . . 5 𝑅 = (Base‘𝑆)
271, 24, 25, 26lmodvscl 19237 . . . 4 ((𝑀 ∈ LMod ∧ (𝐴𝑥) ∈ 𝑅𝑥 ∈ (Base‘𝑀)) → ((𝐴𝑥)( ·𝑠𝑀)𝑥) ∈ (Base‘𝑀))
2811, 18, 23, 27syl3anc 1496 . . 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 8145 . . . . . . . 8 (𝐵 ∈ (𝑅𝑚 𝑉) → 𝐵:𝑉𝑅)
30 ffvelrn 6607 . . . . . . . . 9 ((𝐵:𝑉𝑅𝑥𝑉) → (𝐵𝑥) ∈ 𝑅)
3130ex 403 . . . . . . . 8 (𝐵:𝑉𝑅 → (𝑥𝑉 → (𝐵𝑥) ∈ 𝑅))
3229, 31syl 17 . . . . . . 7 (𝐵 ∈ (𝑅𝑚 𝑉) → (𝑥𝑉 → (𝐵𝑥) ∈ 𝑅))
3332adantl 475 . . . . . 6 ((𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉)) → (𝑥𝑉 → (𝐵𝑥) ∈ 𝑅))
34333ad2ant2 1170 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) → (𝑥𝑉 → (𝐵𝑥) ∈ 𝑅))
3534imp 397 . . . 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 19237 . . . 4 ((𝑀 ∈ LMod ∧ (𝐵𝑥) ∈ 𝑅𝑥 ∈ (Base‘𝑀)) → ((𝐵𝑥)( ·𝑠𝑀)𝑥) ∈ (Base‘𝑀))
3711, 35, 23, 36syl3anc 1496 . . 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 2827 . . 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 2827 . . 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 476 . . . 4 ((𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉)) → 𝐴 ∈ (𝑅𝑚 𝑉))
42 simpl 476 . . . 4 ((𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆)) → 𝐴 finSupp (0g𝑆))
4324, 26scmfsupp 43007 . . . 4 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐴 finSupp (0g𝑆)) → (𝑥𝑉 ↦ ((𝐴𝑥)( ·𝑠𝑀)𝑥)) finSupp (0g𝑀))
4440, 41, 42, 43syl3an 1205 . . 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 479 . . . 4 ((𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉)) → 𝐵 ∈ (𝑅𝑚 𝑉))
46 simpr 479 . . . 4 ((𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆)) → 𝐵 finSupp (0g𝑆))
4724, 26scmfsupp 43007 . . . 4 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐵 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 finSupp (0g𝑆)) → (𝑥𝑉 ↦ ((𝐵𝑥)( ·𝑠𝑀)𝑥)) finSupp (0g𝑀))
4840, 45, 46, 47syl3an 1205 . . 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 18678 . 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 474 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → 𝑉 ∈ 𝒫 (Base‘𝑀))
51 elmapfn 8146 . . . . . . . 8 (𝐴 ∈ (𝑅𝑚 𝑉) → 𝐴 Fn 𝑉)
5251ad2antrl 721 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → 𝐴 Fn 𝑉)
53 elmapfn 8146 . . . . . . . 8 (𝐵 ∈ (𝑅𝑚 𝑉) → 𝐵 Fn 𝑉)
5453ad2antll 722 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → 𝐵 Fn 𝑉)
5550, 52, 54offvalfv 42969 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝐴𝑓 𝐵) = (𝑦𝑉 ↦ ((𝐴𝑦) (𝐵𝑦))))
56553adant3 1168 . . . . 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 19229 . . . . . . . . . . 11 (𝑀 ∈ LMod → 𝑆 ∈ Grp)
58 grpmnd 17784 . . . . . . . . . . 11 (𝑆 ∈ Grp → 𝑆 ∈ Mnd)
5957, 58syl 17 . . . . . . . . . 10 (𝑀 ∈ LMod → 𝑆 ∈ Mnd)
6059ad3antrrr 723 . . . . . . . . 9 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑦𝑉) → 𝑆 ∈ Mnd)
61 ffvelrn 6607 . . . . . . . . . . . . . 14 ((𝐴:𝑉𝑅𝑦𝑉) → (𝐴𝑦) ∈ 𝑅)
6261ex 403 . . . . . . . . . . . . 13 (𝐴:𝑉𝑅 → (𝑦𝑉 → (𝐴𝑦) ∈ 𝑅))
6312, 62syl 17 . . . . . . . . . . . 12 (𝐴 ∈ (𝑅𝑚 𝑉) → (𝑦𝑉 → (𝐴𝑦) ∈ 𝑅))
6463ad2antrl 721 . . . . . . . . . . 11 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝑦𝑉 → (𝐴𝑦) ∈ 𝑅))
6564imp 397 . . . . . . . . . 10 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑦𝑉) → (𝐴𝑦) ∈ 𝑅)
6624fveq2i 6437 . . . . . . . . . . 11 (Base‘𝑆) = (Base‘(Scalar‘𝑀))
6726, 66eqtri 2850 . . . . . . . . . 10 𝑅 = (Base‘(Scalar‘𝑀))
6865, 67syl6eleq 2917 . . . . . . . . 9 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑦𝑉) → (𝐴𝑦) ∈ (Base‘(Scalar‘𝑀)))
69 ffvelrn 6607 . . . . . . . . . . . . . 14 ((𝐵:𝑉𝑅𝑦𝑉) → (𝐵𝑦) ∈ 𝑅)
7069, 67syl6eleq 2917 . . . . . . . . . . . . 13 ((𝐵:𝑉𝑅𝑦𝑉) → (𝐵𝑦) ∈ (Base‘(Scalar‘𝑀)))
7170ex 403 . . . . . . . . . . . 12 (𝐵:𝑉𝑅 → (𝑦𝑉 → (𝐵𝑦) ∈ (Base‘(Scalar‘𝑀))))
7229, 71syl 17 . . . . . . . . . . 11 (𝐵 ∈ (𝑅𝑚 𝑉) → (𝑦𝑉 → (𝐵𝑦) ∈ (Base‘(Scalar‘𝑀))))
7372ad2antll 722 . . . . . . . . . 10 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝑦𝑉 → (𝐵𝑦) ∈ (Base‘(Scalar‘𝑀))))
7473imp 397 . . . . . . . . 9 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑦𝑉) → (𝐵𝑦) ∈ (Base‘(Scalar‘𝑀)))
7524eqcomi 2835 . . . . . . . . . . 11 (Scalar‘𝑀) = 𝑆
7675fveq2i 6437 . . . . . . . . . 10 (Base‘(Scalar‘𝑀)) = (Base‘𝑆)
77 lincsum.b . . . . . . . . . 10 = (+g𝑆)
7876, 77mndcl 17655 . . . . . . . . 9 ((𝑆 ∈ Mnd ∧ (𝐴𝑦) ∈ (Base‘(Scalar‘𝑀)) ∧ (𝐵𝑦) ∈ (Base‘(Scalar‘𝑀))) → ((𝐴𝑦) (𝐵𝑦)) ∈ (Base‘(Scalar‘𝑀)))
7960, 68, 74, 78syl3anc 1496 . . . . . . . 8 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑦𝑉) → ((𝐴𝑦) (𝐵𝑦)) ∈ (Base‘(Scalar‘𝑀)))
8079fmpttd 6635 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝑦𝑉 ↦ ((𝐴𝑦) (𝐵𝑦))):𝑉⟶(Base‘(Scalar‘𝑀)))
81 fvex 6447 . . . . . . . 8 (Base‘(Scalar‘𝑀)) ∈ V
82 elmapg 8136 . . . . . . . 8 (((Base‘(Scalar‘𝑀)) ∈ V ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ((𝑦𝑉 ↦ ((𝐴𝑦) (𝐵𝑦))) ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ↔ (𝑦𝑉 ↦ ((𝐴𝑦) (𝐵𝑦))):𝑉⟶(Base‘(Scalar‘𝑀))))
8381, 50, 82sylancr 583 . . . . . . 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))))
8480, 83mpbird 249 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝑦𝑉 ↦ ((𝐴𝑦) (𝐵𝑦))) ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉))
85843adant3 1168 . . . . 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))
8656, 85eqeltrd 2907 . . . 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))
87 lincval 43046 . . . 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))))
8810, 86, 8, 87syl3anc 1496 . . 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))))
8951, 53anim12i 608 . . . . . . . . . . . 12 ((𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉)) → (𝐴 Fn 𝑉𝐵 Fn 𝑉))
9089adantl 475 . . . . . . . . . . 11 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝐴 Fn 𝑉𝐵 Fn 𝑉))
9190adantr 474 . . . . . . . . . 10 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑥𝑉) → (𝐴 Fn 𝑉𝐵 Fn 𝑉))
9250anim1i 610 . . . . . . . . . 10 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑥𝑉) → (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑥𝑉))
93 fnfvof 7172 . . . . . . . . . 10 (((𝐴 Fn 𝑉𝐵 Fn 𝑉) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑥𝑉)) → ((𝐴𝑓 𝐵)‘𝑥) = ((𝐴𝑥) (𝐵𝑥)))
9491, 92, 93syl2anc 581 . . . . . . . . 9 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑥𝑉) → ((𝐴𝑓 𝐵)‘𝑥) = ((𝐴𝑥) (𝐵𝑥)))
9577a1i 11 . . . . . . . . . 10 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑥𝑉) → = (+g𝑆))
9695oveqd 6923 . . . . . . . . 9 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑥𝑉) → ((𝐴𝑥) (𝐵𝑥)) = ((𝐴𝑥)(+g𝑆)(𝐵𝑥)))
9794, 96eqtrd 2862 . . . . . . . 8 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑥𝑉) → ((𝐴𝑓 𝐵)‘𝑥) = ((𝐴𝑥)(+g𝑆)(𝐵𝑥)))
9897oveq1d 6921 . . . . . . 7 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑥𝑉) → (((𝐴𝑓 𝐵)‘𝑥)( ·𝑠𝑀)𝑥) = (((𝐴𝑥)(+g𝑆)(𝐵𝑥))( ·𝑠𝑀)𝑥))
999adantr 474 . . . . . . . . 9 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → 𝑀 ∈ LMod)
10099adantr 474 . . . . . . . 8 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑥𝑉) → 𝑀 ∈ LMod)
10115ad2antrl 721 . . . . . . . . 9 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝑥𝑉 → (𝐴𝑥) ∈ 𝑅))
102101imp 397 . . . . . . . 8 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑥𝑉) → (𝐴𝑥) ∈ 𝑅)
10332ad2antll 722 . . . . . . . . 9 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝑥𝑉 → (𝐵𝑥) ∈ 𝑅))
104103imp 397 . . . . . . . 8 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑥𝑉) → (𝐵𝑥) ∈ 𝑅)
10521adantr 474 . . . . . . . . 9 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝑥𝑉𝑥 ∈ (Base‘𝑀)))
106105imp 397 . . . . . . . 8 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑥𝑉) → 𝑥 ∈ (Base‘𝑀))
107 eqid 2826 . . . . . . . . 9 (Scalar‘𝑀) = (Scalar‘𝑀)
10824fveq2i 6437 . . . . . . . . 9 (+g𝑆) = (+g‘(Scalar‘𝑀))
1091, 3, 107, 25, 67, 108lmodvsdir 19244 . . . . . . . 8 ((𝑀 ∈ LMod ∧ ((𝐴𝑥) ∈ 𝑅 ∧ (𝐵𝑥) ∈ 𝑅𝑥 ∈ (Base‘𝑀))) → (((𝐴𝑥)(+g𝑆)(𝐵𝑥))( ·𝑠𝑀)𝑥) = (((𝐴𝑥)( ·𝑠𝑀)𝑥) + ((𝐵𝑥)( ·𝑠𝑀)𝑥)))
110100, 102, 104, 106, 109syl13anc 1497 . . . . . . 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)))
11198, 110eqtrd 2862 . . . . . 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)))
112111mpteq2dva 4968 . . . . 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))))
113112oveq2d 6922 . . . 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)))))
1141133adant3 1168 . . 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)))))
11588, 114eqtrd 2862 . 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)))))
116 lincsum.x . . . 4 𝑋 = (𝐴( linC ‘𝑀)𝑉)
117 lincsum.y . . . 4 𝑌 = (𝐵( linC ‘𝑀)𝑉)
118116, 117oveq12i 6918 . . 3 (𝑋 + 𝑌) = ((𝐴( linC ‘𝑀)𝑉) + (𝐵( linC ‘𝑀)𝑉))
11967oveq1i 6916 . . . . . . . . 9 (𝑅𝑚 𝑉) = ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)
120119eleq2i 2899 . . . . . . . 8 (𝐴 ∈ (𝑅𝑚 𝑉) ↔ 𝐴 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉))
121120biimpi 208 . . . . . . 7 (𝐴 ∈ (𝑅𝑚 𝑉) → 𝐴 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉))
122121ad2antrl 721 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → 𝐴 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉))
123 lincval 43046 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝐴 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝐴( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑥𝑉 ↦ ((𝐴𝑥)( ·𝑠𝑀)𝑥))))
12499, 122, 50, 123syl3anc 1496 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝐴( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑥𝑉 ↦ ((𝐴𝑥)( ·𝑠𝑀)𝑥))))
125119eleq2i 2899 . . . . . . . 8 (𝐵 ∈ (𝑅𝑚 𝑉) ↔ 𝐵 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉))
126125biimpi 208 . . . . . . 7 (𝐵 ∈ (𝑅𝑚 𝑉) → 𝐵 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉))
127126ad2antll 722 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → 𝐵 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉))
128 lincval 43046 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝐵 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝐵( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑥𝑉 ↦ ((𝐵𝑥)( ·𝑠𝑀)𝑥))))
12999, 127, 50, 128syl3anc 1496 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝐵( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑥𝑉 ↦ ((𝐵𝑥)( ·𝑠𝑀)𝑥))))
130124, 129oveq12d 6924 . . . 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)))))
1311303adant3 1168 . . 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)))))
132118, 131syl5eq 2874 . 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)))))
13349, 115, 1323eqtr4rd 2873 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 198  wa 386  w3a 1113   = wceq 1658  wcel 2166  Vcvv 3415  𝒫 cpw 4379   class class class wbr 4874  cmpt 4953   Fn wfn 6119  wf 6120  cfv 6124  (class class class)co 6906  𝑓 cof 7156  𝑚 cmap 8123   finSupp cfsupp 8545  Basecbs 16223  +gcplusg 16306  Scalarcsca 16309   ·𝑠 cvsca 16310  0gc0g 16454   Σg cgsu 16455  Mndcmnd 17648  Grpcgrp 17777  CMndccmn 18547  LModclmod 19220   linC clinc 43041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210  ax-cnex 10309  ax-resscn 10310  ax-1cn 10311  ax-icn 10312  ax-addcl 10313  ax-addrcl 10314  ax-mulcl 10315  ax-mulrcl 10316  ax-mulcom 10317  ax-addass 10318  ax-mulass 10319  ax-distr 10320  ax-i2m1 10321  ax-1ne0 10322  ax-1rid 10323  ax-rnegex 10324  ax-rrecex 10325  ax-cnre 10326  ax-pre-lttri 10327  ax-pre-lttrn 10328  ax-pre-ltadd 10329  ax-pre-mulgt0 10330
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-nel 3104  df-ral 3123  df-rex 3124  df-reu 3125  df-rmo 3126  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-pss 3815  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4660  df-int 4699  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-tr 4977  df-id 5251  df-eprel 5256  df-po 5264  df-so 5265  df-fr 5302  df-se 5303  df-we 5304  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-pred 5921  df-ord 5967  df-on 5968  df-lim 5969  df-suc 5970  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-isom 6133  df-riota 6867  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-of 7158  df-om 7328  df-1st 7429  df-2nd 7430  df-supp 7561  df-wrecs 7673  df-recs 7735  df-rdg 7773  df-1o 7827  df-oadd 7831  df-er 8010  df-map 8125  df-en 8224  df-dom 8225  df-sdom 8226  df-fin 8227  df-fsupp 8546  df-oi 8685  df-card 9079  df-pnf 10394  df-mnf 10395  df-xr 10396  df-ltxr 10397  df-le 10398  df-sub 10588  df-neg 10589  df-nn 11352  df-2 11415  df-n0 11620  df-z 11706  df-uz 11970  df-fz 12621  df-fzo 12762  df-seq 13097  df-hash 13412  df-ndx 16226  df-slot 16227  df-base 16229  df-sets 16230  df-ress 16231  df-plusg 16319  df-0g 16456  df-gsum 16457  df-mgm 17596  df-sgrp 17638  df-mnd 17649  df-submnd 17690  df-grp 17780  df-minusg 17781  df-cntz 18101  df-cmn 18549  df-abl 18550  df-mgp 18845  df-ur 18857  df-ring 18904  df-lmod 19222  df-linc 43043
This theorem is referenced by:  lincsumcl  43068
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