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Theorem lincsum 41532
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 2621 . . 3 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2621 . . 3 (0g𝑀) = (0g𝑀)
3 lincsum.p . . 3 + = (+g𝑀)
4 lmodcmn 18843 . . . . 5 (𝑀 ∈ LMod → 𝑀 ∈ CMnd)
54adantr 481 . . . 4 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑀 ∈ CMnd)
653ad2ant1 1080 . . 3 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) → 𝑀 ∈ CMnd)
7 simpr 477 . . . 4 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑉 ∈ 𝒫 (Base‘𝑀))
873ad2ant1 1080 . . 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 473 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑀 ∈ LMod)
1093ad2ant1 1080 . . . . 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 7831 . . . . . . . 8 (𝐴 ∈ (𝑅𝑚 𝑉) → 𝐴:𝑉𝑅)
13 ffvelrn 6318 . . . . . . . . 9 ((𝐴:𝑉𝑅𝑥𝑉) → (𝐴𝑥) ∈ 𝑅)
1413ex 450 . . . . . . . 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 1081 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) → (𝑥𝑉 → (𝐴𝑥) ∈ 𝑅))
1817imp 445 . . . 4 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) ∧ 𝑥𝑉) → (𝐴𝑥) ∈ 𝑅)
19 elelpwi 4147 . . . . . . . 8 ((𝑥𝑉𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑥 ∈ (Base‘𝑀))
2019expcom 451 . . . . . . 7 (𝑉 ∈ 𝒫 (Base‘𝑀) → (𝑥𝑉𝑥 ∈ (Base‘𝑀)))
2120adantl 482 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑥𝑉𝑥 ∈ (Base‘𝑀)))
22213ad2ant1 1080 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) → (𝑥𝑉𝑥 ∈ (Base‘𝑀)))
2322imp 445 . . . 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 2621 . . . . 5 ( ·𝑠𝑀) = ( ·𝑠𝑀)
26 lincsum.r . . . . 5 𝑅 = (Base‘𝑆)
271, 24, 25, 26lmodvscl 18812 . . . 4 ((𝑀 ∈ LMod ∧ (𝐴𝑥) ∈ 𝑅𝑥 ∈ (Base‘𝑀)) → ((𝐴𝑥)( ·𝑠𝑀)𝑥) ∈ (Base‘𝑀))
2811, 18, 23, 27syl3anc 1323 . . 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 7831 . . . . . . . 8 (𝐵 ∈ (𝑅𝑚 𝑉) → 𝐵:𝑉𝑅)
30 ffvelrn 6318 . . . . . . . . 9 ((𝐵:𝑉𝑅𝑥𝑉) → (𝐵𝑥) ∈ 𝑅)
3130ex 450 . . . . . . . 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 1081 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) → (𝑥𝑉 → (𝐵𝑥) ∈ 𝑅))
3534imp 445 . . . 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 18812 . . . 4 ((𝑀 ∈ LMod ∧ (𝐵𝑥) ∈ 𝑅𝑥 ∈ (Base‘𝑀)) → ((𝐵𝑥)( ·𝑠𝑀)𝑥) ∈ (Base‘𝑀))
3711, 35, 23, 36syl3anc 1323 . . 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 2622 . . 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 2622 . . 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 473 . . . 4 ((𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉)) → 𝐴 ∈ (𝑅𝑚 𝑉))
42 simpl 473 . . . 4 ((𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆)) → 𝐴 finSupp (0g𝑆))
4324, 26scmfsupp 41473 . . . 4 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐴 finSupp (0g𝑆)) → (𝑥𝑉 ↦ ((𝐴𝑥)( ·𝑠𝑀)𝑥)) finSupp (0g𝑀))
4440, 41, 42, 43syl3an 1365 . . 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 477 . . . 4 ((𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉)) → 𝐵 ∈ (𝑅𝑚 𝑉))
46 simpr 477 . . . 4 ((𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆)) → 𝐵 finSupp (0g𝑆))
4724, 26scmfsupp 41473 . . . 4 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐵 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 finSupp (0g𝑆)) → (𝑥𝑉 ↦ ((𝐵𝑥)( ·𝑠𝑀)𝑥)) finSupp (0g𝑀))
4840, 45, 46, 47syl3an 1365 . . 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 18256 . 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 7832 . . . . . . . 8 (𝐴 ∈ (𝑅𝑚 𝑉) → 𝐴 Fn 𝑉)
5251ad2antrl 763 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → 𝐴 Fn 𝑉)
53 elmapfn 7832 . . . . . . . 8 (𝐵 ∈ (𝑅𝑚 𝑉) → 𝐵 Fn 𝑉)
5453ad2antll 764 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → 𝐵 Fn 𝑉)
5550, 52, 54offvalfv 41435 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝐴𝑓 𝐵) = (𝑦𝑉 ↦ ((𝐴𝑦) (𝐵𝑦))))
56553adant3 1079 . . . . 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 18804 . . . . . . . . . . 11 (𝑀 ∈ LMod → 𝑆 ∈ Grp)
58 grpmnd 17361 . . . . . . . . . . 11 (𝑆 ∈ Grp → 𝑆 ∈ Mnd)
5957, 58syl 17 . . . . . . . . . 10 (𝑀 ∈ LMod → 𝑆 ∈ Mnd)
6059ad3antrrr 765 . . . . . . . . 9 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑦𝑉) → 𝑆 ∈ Mnd)
61 ffvelrn 6318 . . . . . . . . . . . . . 14 ((𝐴:𝑉𝑅𝑦𝑉) → (𝐴𝑦) ∈ 𝑅)
6261ex 450 . . . . . . . . . . . . 13 (𝐴:𝑉𝑅 → (𝑦𝑉 → (𝐴𝑦) ∈ 𝑅))
6312, 62syl 17 . . . . . . . . . . . 12 (𝐴 ∈ (𝑅𝑚 𝑉) → (𝑦𝑉 → (𝐴𝑦) ∈ 𝑅))
6463ad2antrl 763 . . . . . . . . . . 11 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝑦𝑉 → (𝐴𝑦) ∈ 𝑅))
6564imp 445 . . . . . . . . . 10 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑦𝑉) → (𝐴𝑦) ∈ 𝑅)
6624fveq2i 6156 . . . . . . . . . . 11 (Base‘𝑆) = (Base‘(Scalar‘𝑀))
6726, 66eqtri 2643 . . . . . . . . . 10 𝑅 = (Base‘(Scalar‘𝑀))
6865, 67syl6eleq 2708 . . . . . . . . 9 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑦𝑉) → (𝐴𝑦) ∈ (Base‘(Scalar‘𝑀)))
69 ffvelrn 6318 . . . . . . . . . . . . . 14 ((𝐵:𝑉𝑅𝑦𝑉) → (𝐵𝑦) ∈ 𝑅)
7069, 67syl6eleq 2708 . . . . . . . . . . . . 13 ((𝐵:𝑉𝑅𝑦𝑉) → (𝐵𝑦) ∈ (Base‘(Scalar‘𝑀)))
7170ex 450 . . . . . . . . . . . 12 (𝐵:𝑉𝑅 → (𝑦𝑉 → (𝐵𝑦) ∈ (Base‘(Scalar‘𝑀))))
7229, 71syl 17 . . . . . . . . . . 11 (𝐵 ∈ (𝑅𝑚 𝑉) → (𝑦𝑉 → (𝐵𝑦) ∈ (Base‘(Scalar‘𝑀))))
7372ad2antll 764 . . . . . . . . . 10 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝑦𝑉 → (𝐵𝑦) ∈ (Base‘(Scalar‘𝑀))))
7473imp 445 . . . . . . . . 9 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑦𝑉) → (𝐵𝑦) ∈ (Base‘(Scalar‘𝑀)))
7524eqcomi 2630 . . . . . . . . . . 11 (Scalar‘𝑀) = 𝑆
7675fveq2i 6156 . . . . . . . . . 10 (Base‘(Scalar‘𝑀)) = (Base‘𝑆)
77 lincsum.b . . . . . . . . . 10 = (+g𝑆)
7876, 77mndcl 17233 . . . . . . . . 9 ((𝑆 ∈ Mnd ∧ (𝐴𝑦) ∈ (Base‘(Scalar‘𝑀)) ∧ (𝐵𝑦) ∈ (Base‘(Scalar‘𝑀))) → ((𝐴𝑦) (𝐵𝑦)) ∈ (Base‘(Scalar‘𝑀)))
7960, 68, 74, 78syl3anc 1323 . . . . . . . 8 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑦𝑉) → ((𝐴𝑦) (𝐵𝑦)) ∈ (Base‘(Scalar‘𝑀)))
80 eqid 2621 . . . . . . . 8 (𝑦𝑉 ↦ ((𝐴𝑦) (𝐵𝑦))) = (𝑦𝑉 ↦ ((𝐴𝑦) (𝐵𝑦)))
8179, 80fmptd 6346 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝑦𝑉 ↦ ((𝐴𝑦) (𝐵𝑦))):𝑉⟶(Base‘(Scalar‘𝑀)))
82 fvex 6163 . . . . . . . 8 (Base‘(Scalar‘𝑀)) ∈ V
83 elmapg 7822 . . . . . . . 8 (((Base‘(Scalar‘𝑀)) ∈ V ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ((𝑦𝑉 ↦ ((𝐴𝑦) (𝐵𝑦))) ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ↔ (𝑦𝑉 ↦ ((𝐴𝑦) (𝐵𝑦))):𝑉⟶(Base‘(Scalar‘𝑀))))
8482, 50, 83sylancr 694 . . . . . . 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))))
8581, 84mpbird 247 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝑦𝑉 ↦ ((𝐴𝑦) (𝐵𝑦))) ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉))
86853adant3 1079 . . . . 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))
8756, 86eqeltrd 2698 . . . 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))
88 lincval 41512 . . . 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))))
8910, 87, 8, 88syl3anc 1323 . . 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))))
9051, 53anim12i 589 . . . . . . . . . . . 12 ((𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉)) → (𝐴 Fn 𝑉𝐵 Fn 𝑉))
9190adantl 482 . . . . . . . . . . 11 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝐴 Fn 𝑉𝐵 Fn 𝑉))
9291adantr 481 . . . . . . . . . 10 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑥𝑉) → (𝐴 Fn 𝑉𝐵 Fn 𝑉))
9350anim1i 591 . . . . . . . . . 10 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑥𝑉) → (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑥𝑉))
94 fnfvof 6871 . . . . . . . . . 10 (((𝐴 Fn 𝑉𝐵 Fn 𝑉) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑥𝑉)) → ((𝐴𝑓 𝐵)‘𝑥) = ((𝐴𝑥) (𝐵𝑥)))
9592, 93, 94syl2anc 692 . . . . . . . . 9 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑥𝑉) → ((𝐴𝑓 𝐵)‘𝑥) = ((𝐴𝑥) (𝐵𝑥)))
9677a1i 11 . . . . . . . . . 10 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑥𝑉) → = (+g𝑆))
9796oveqd 6627 . . . . . . . . 9 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑥𝑉) → ((𝐴𝑥) (𝐵𝑥)) = ((𝐴𝑥)(+g𝑆)(𝐵𝑥)))
9895, 97eqtrd 2655 . . . . . . . 8 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑥𝑉) → ((𝐴𝑓 𝐵)‘𝑥) = ((𝐴𝑥)(+g𝑆)(𝐵𝑥)))
9998oveq1d 6625 . . . . . . 7 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑥𝑉) → (((𝐴𝑓 𝐵)‘𝑥)( ·𝑠𝑀)𝑥) = (((𝐴𝑥)(+g𝑆)(𝐵𝑥))( ·𝑠𝑀)𝑥))
1009adantr 481 . . . . . . . . 9 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → 𝑀 ∈ LMod)
101100adantr 481 . . . . . . . 8 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑥𝑉) → 𝑀 ∈ LMod)
10215ad2antrl 763 . . . . . . . . 9 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝑥𝑉 → (𝐴𝑥) ∈ 𝑅))
103102imp 445 . . . . . . . 8 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑥𝑉) → (𝐴𝑥) ∈ 𝑅)
10432ad2antll 764 . . . . . . . . 9 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝑥𝑉 → (𝐵𝑥) ∈ 𝑅))
105104imp 445 . . . . . . . 8 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑥𝑉) → (𝐵𝑥) ∈ 𝑅)
10621adantr 481 . . . . . . . . 9 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝑥𝑉𝑥 ∈ (Base‘𝑀)))
107106imp 445 . . . . . . . 8 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑥𝑉) → 𝑥 ∈ (Base‘𝑀))
108 eqid 2621 . . . . . . . . 9 (Scalar‘𝑀) = (Scalar‘𝑀)
10924fveq2i 6156 . . . . . . . . 9 (+g𝑆) = (+g‘(Scalar‘𝑀))
1101, 3, 108, 25, 67, 109lmodvsdir 18819 . . . . . . . 8 ((𝑀 ∈ LMod ∧ ((𝐴𝑥) ∈ 𝑅 ∧ (𝐵𝑥) ∈ 𝑅𝑥 ∈ (Base‘𝑀))) → (((𝐴𝑥)(+g𝑆)(𝐵𝑥))( ·𝑠𝑀)𝑥) = (((𝐴𝑥)( ·𝑠𝑀)𝑥) + ((𝐵𝑥)( ·𝑠𝑀)𝑥)))
111101, 103, 105, 107, 110syl13anc 1325 . . . . . . 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)))
11299, 111eqtrd 2655 . . . . . 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)))
113112mpteq2dva 4709 . . . . 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))))
114113oveq2d 6626 . . . 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)))))
1151143adant3 1079 . . 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)))))
11689, 115eqtrd 2655 . 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)))))
117 lincsum.x . . . 4 𝑋 = (𝐴( linC ‘𝑀)𝑉)
118 lincsum.y . . . 4 𝑌 = (𝐵( linC ‘𝑀)𝑉)
119117, 118oveq12i 6622 . . 3 (𝑋 + 𝑌) = ((𝐴( linC ‘𝑀)𝑉) + (𝐵( linC ‘𝑀)𝑉))
12067oveq1i 6620 . . . . . . . . 9 (𝑅𝑚 𝑉) = ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)
121120eleq2i 2690 . . . . . . . 8 (𝐴 ∈ (𝑅𝑚 𝑉) ↔ 𝐴 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉))
122121biimpi 206 . . . . . . 7 (𝐴 ∈ (𝑅𝑚 𝑉) → 𝐴 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉))
123122ad2antrl 763 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → 𝐴 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉))
124 lincval 41512 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝐴 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝐴( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑥𝑉 ↦ ((𝐴𝑥)( ·𝑠𝑀)𝑥))))
125100, 123, 50, 124syl3anc 1323 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝐴( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑥𝑉 ↦ ((𝐴𝑥)( ·𝑠𝑀)𝑥))))
126120eleq2i 2690 . . . . . . . 8 (𝐵 ∈ (𝑅𝑚 𝑉) ↔ 𝐵 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉))
127126biimpi 206 . . . . . . 7 (𝐵 ∈ (𝑅𝑚 𝑉) → 𝐵 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉))
128127ad2antll 764 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → 𝐵 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉))
129 lincval 41512 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝐵 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝐵( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑥𝑉 ↦ ((𝐵𝑥)( ·𝑠𝑀)𝑥))))
130100, 128, 50, 129syl3anc 1323 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝐵( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑥𝑉 ↦ ((𝐵𝑥)( ·𝑠𝑀)𝑥))))
131125, 130oveq12d 6628 . . . 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)))))
1321313adant3 1079 . . 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)))))
133119, 132syl5eq 2667 . 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)))))
13449, 116, 1333eqtr4rd 2666 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 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  Vcvv 3189  𝒫 cpw 4135   class class class wbr 4618  cmpt 4678   Fn wfn 5847  wf 5848  cfv 5852  (class class class)co 6610  𝑓 cof 6855  𝑚 cmap 7809   finSupp cfsupp 8227  Basecbs 15792  +gcplusg 15873  Scalarcsca 15876   ·𝑠 cvsca 15877  0gc0g 16032   Σg cgsu 16033  Mndcmnd 17226  Grpcgrp 17354  CMndccmn 18125  LModclmod 18795   linC clinc 41507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-of 6857  df-om 7020  df-1st 7120  df-2nd 7121  df-supp 7248  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-map 7811  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-fsupp 8228  df-oi 8367  df-card 8717  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-2 11031  df-n0 11245  df-z 11330  df-uz 11640  df-fz 12277  df-fzo 12415  df-seq 12750  df-hash 13066  df-ndx 15795  df-slot 15796  df-base 15797  df-sets 15798  df-ress 15799  df-plusg 15886  df-0g 16034  df-gsum 16035  df-mgm 17174  df-sgrp 17216  df-mnd 17227  df-submnd 17268  df-grp 17357  df-minusg 17358  df-cntz 17682  df-cmn 18127  df-abl 18128  df-mgp 18422  df-ur 18434  df-ring 18481  df-lmod 18797  df-linc 41509
This theorem is referenced by:  lincsumcl  41534
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