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Theorem lincsum 42747
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 2761 . . 3 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2761 . . 3 (0g𝑀) = (0g𝑀)
3 lincsum.p . . 3 + = (+g𝑀)
4 lmodcmn 19134 . . . . 5 (𝑀 ∈ LMod → 𝑀 ∈ CMnd)
54adantr 472 . . . 4 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑀 ∈ CMnd)
653ad2ant1 1128 . . 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 1128 . . 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 474 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑀 ∈ LMod)
1093ad2ant1 1128 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) → 𝑀 ∈ LMod)
1110adantr 472 . . . 4 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) ∧ 𝑥𝑉) → 𝑀 ∈ LMod)
12 elmapi 8048 . . . . . . . 8 (𝐴 ∈ (𝑅𝑚 𝑉) → 𝐴:𝑉𝑅)
13 ffvelrn 6522 . . . . . . . . 9 ((𝐴:𝑉𝑅𝑥𝑉) → (𝐴𝑥) ∈ 𝑅)
1413ex 449 . . . . . . . 8 (𝐴:𝑉𝑅 → (𝑥𝑉 → (𝐴𝑥) ∈ 𝑅))
1512, 14syl 17 . . . . . . 7 (𝐴 ∈ (𝑅𝑚 𝑉) → (𝑥𝑉 → (𝐴𝑥) ∈ 𝑅))
1615adantr 472 . . . . . 6 ((𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉)) → (𝑥𝑉 → (𝐴𝑥) ∈ 𝑅))
17163ad2ant2 1129 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) → (𝑥𝑉 → (𝐴𝑥) ∈ 𝑅))
1817imp 444 . . . 4 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) ∧ 𝑥𝑉) → (𝐴𝑥) ∈ 𝑅)
19 elelpwi 4316 . . . . . . . 8 ((𝑥𝑉𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑥 ∈ (Base‘𝑀))
2019expcom 450 . . . . . . 7 (𝑉 ∈ 𝒫 (Base‘𝑀) → (𝑥𝑉𝑥 ∈ (Base‘𝑀)))
2120adantl 473 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑥𝑉𝑥 ∈ (Base‘𝑀)))
22213ad2ant1 1128 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) → (𝑥𝑉𝑥 ∈ (Base‘𝑀)))
2322imp 444 . . . 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 2761 . . . . 5 ( ·𝑠𝑀) = ( ·𝑠𝑀)
26 lincsum.r . . . . 5 𝑅 = (Base‘𝑆)
271, 24, 25, 26lmodvscl 19103 . . . 4 ((𝑀 ∈ LMod ∧ (𝐴𝑥) ∈ 𝑅𝑥 ∈ (Base‘𝑀)) → ((𝐴𝑥)( ·𝑠𝑀)𝑥) ∈ (Base‘𝑀))
2811, 18, 23, 27syl3anc 1477 . . 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 8048 . . . . . . . 8 (𝐵 ∈ (𝑅𝑚 𝑉) → 𝐵:𝑉𝑅)
30 ffvelrn 6522 . . . . . . . . 9 ((𝐵:𝑉𝑅𝑥𝑉) → (𝐵𝑥) ∈ 𝑅)
3130ex 449 . . . . . . . 8 (𝐵:𝑉𝑅 → (𝑥𝑉 → (𝐵𝑥) ∈ 𝑅))
3229, 31syl 17 . . . . . . 7 (𝐵 ∈ (𝑅𝑚 𝑉) → (𝑥𝑉 → (𝐵𝑥) ∈ 𝑅))
3332adantl 473 . . . . . 6 ((𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉)) → (𝑥𝑉 → (𝐵𝑥) ∈ 𝑅))
34333ad2ant2 1129 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) → (𝑥𝑉 → (𝐵𝑥) ∈ 𝑅))
3534imp 444 . . . 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 19103 . . . 4 ((𝑀 ∈ LMod ∧ (𝐵𝑥) ∈ 𝑅𝑥 ∈ (Base‘𝑀)) → ((𝐵𝑥)( ·𝑠𝑀)𝑥) ∈ (Base‘𝑀))
3711, 35, 23, 36syl3anc 1477 . . 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 2762 . . 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 2762 . . 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 474 . . . 4 ((𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉)) → 𝐴 ∈ (𝑅𝑚 𝑉))
42 simpl 474 . . . 4 ((𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆)) → 𝐴 finSupp (0g𝑆))
4324, 26scmfsupp 42688 . . . 4 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐴 finSupp (0g𝑆)) → (𝑥𝑉 ↦ ((𝐴𝑥)( ·𝑠𝑀)𝑥)) finSupp (0g𝑀))
4440, 41, 42, 43syl3an 1164 . . 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 42688 . . . 4 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐵 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 finSupp (0g𝑆)) → (𝑥𝑉 ↦ ((𝐵𝑥)( ·𝑠𝑀)𝑥)) finSupp (0g𝑀))
4840, 45, 46, 47syl3an 1164 . . 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 18545 . 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 472 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → 𝑉 ∈ 𝒫 (Base‘𝑀))
51 elmapfn 8049 . . . . . . . 8 (𝐴 ∈ (𝑅𝑚 𝑉) → 𝐴 Fn 𝑉)
5251ad2antrl 766 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → 𝐴 Fn 𝑉)
53 elmapfn 8049 . . . . . . . 8 (𝐵 ∈ (𝑅𝑚 𝑉) → 𝐵 Fn 𝑉)
5453ad2antll 767 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → 𝐵 Fn 𝑉)
5550, 52, 54offvalfv 42650 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝐴𝑓 𝐵) = (𝑦𝑉 ↦ ((𝐴𝑦) (𝐵𝑦))))
56553adant3 1127 . . . . 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 19095 . . . . . . . . . . 11 (𝑀 ∈ LMod → 𝑆 ∈ Grp)
58 grpmnd 17651 . . . . . . . . . . 11 (𝑆 ∈ Grp → 𝑆 ∈ Mnd)
5957, 58syl 17 . . . . . . . . . 10 (𝑀 ∈ LMod → 𝑆 ∈ Mnd)
6059ad3antrrr 768 . . . . . . . . 9 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑦𝑉) → 𝑆 ∈ Mnd)
61 ffvelrn 6522 . . . . . . . . . . . . . 14 ((𝐴:𝑉𝑅𝑦𝑉) → (𝐴𝑦) ∈ 𝑅)
6261ex 449 . . . . . . . . . . . . 13 (𝐴:𝑉𝑅 → (𝑦𝑉 → (𝐴𝑦) ∈ 𝑅))
6312, 62syl 17 . . . . . . . . . . . 12 (𝐴 ∈ (𝑅𝑚 𝑉) → (𝑦𝑉 → (𝐴𝑦) ∈ 𝑅))
6463ad2antrl 766 . . . . . . . . . . 11 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝑦𝑉 → (𝐴𝑦) ∈ 𝑅))
6564imp 444 . . . . . . . . . 10 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑦𝑉) → (𝐴𝑦) ∈ 𝑅)
6624fveq2i 6357 . . . . . . . . . . 11 (Base‘𝑆) = (Base‘(Scalar‘𝑀))
6726, 66eqtri 2783 . . . . . . . . . 10 𝑅 = (Base‘(Scalar‘𝑀))
6865, 67syl6eleq 2850 . . . . . . . . 9 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑦𝑉) → (𝐴𝑦) ∈ (Base‘(Scalar‘𝑀)))
69 ffvelrn 6522 . . . . . . . . . . . . . 14 ((𝐵:𝑉𝑅𝑦𝑉) → (𝐵𝑦) ∈ 𝑅)
7069, 67syl6eleq 2850 . . . . . . . . . . . . 13 ((𝐵:𝑉𝑅𝑦𝑉) → (𝐵𝑦) ∈ (Base‘(Scalar‘𝑀)))
7170ex 449 . . . . . . . . . . . 12 (𝐵:𝑉𝑅 → (𝑦𝑉 → (𝐵𝑦) ∈ (Base‘(Scalar‘𝑀))))
7229, 71syl 17 . . . . . . . . . . 11 (𝐵 ∈ (𝑅𝑚 𝑉) → (𝑦𝑉 → (𝐵𝑦) ∈ (Base‘(Scalar‘𝑀))))
7372ad2antll 767 . . . . . . . . . 10 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝑦𝑉 → (𝐵𝑦) ∈ (Base‘(Scalar‘𝑀))))
7473imp 444 . . . . . . . . 9 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑦𝑉) → (𝐵𝑦) ∈ (Base‘(Scalar‘𝑀)))
7524eqcomi 2770 . . . . . . . . . . 11 (Scalar‘𝑀) = 𝑆
7675fveq2i 6357 . . . . . . . . . 10 (Base‘(Scalar‘𝑀)) = (Base‘𝑆)
77 lincsum.b . . . . . . . . . 10 = (+g𝑆)
7876, 77mndcl 17523 . . . . . . . . 9 ((𝑆 ∈ Mnd ∧ (𝐴𝑦) ∈ (Base‘(Scalar‘𝑀)) ∧ (𝐵𝑦) ∈ (Base‘(Scalar‘𝑀))) → ((𝐴𝑦) (𝐵𝑦)) ∈ (Base‘(Scalar‘𝑀)))
7960, 68, 74, 78syl3anc 1477 . . . . . . . 8 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑦𝑉) → ((𝐴𝑦) (𝐵𝑦)) ∈ (Base‘(Scalar‘𝑀)))
80 eqid 2761 . . . . . . . 8 (𝑦𝑉 ↦ ((𝐴𝑦) (𝐵𝑦))) = (𝑦𝑉 ↦ ((𝐴𝑦) (𝐵𝑦)))
8179, 80fmptd 6550 . . . . . . 7 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝑦𝑉 ↦ ((𝐴𝑦) (𝐵𝑦))):𝑉⟶(Base‘(Scalar‘𝑀)))
82 fvex 6364 . . . . . . . 8 (Base‘(Scalar‘𝑀)) ∈ V
83 elmapg 8039 . . . . . . . 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 698 . . . . . . 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 1127 . . . . 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 2840 . . . 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 42727 . . . 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 1477 . . 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 591 . . . . . . . . . . . 12 ((𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉)) → (𝐴 Fn 𝑉𝐵 Fn 𝑉))
9190adantl 473 . . . . . . . . . . 11 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝐴 Fn 𝑉𝐵 Fn 𝑉))
9291adantr 472 . . . . . . . . . 10 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑥𝑉) → (𝐴 Fn 𝑉𝐵 Fn 𝑉))
9350anim1i 593 . . . . . . . . . 10 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑥𝑉) → (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑥𝑉))
94 fnfvof 7078 . . . . . . . . . 10 (((𝐴 Fn 𝑉𝐵 Fn 𝑉) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑥𝑉)) → ((𝐴𝑓 𝐵)‘𝑥) = ((𝐴𝑥) (𝐵𝑥)))
9592, 93, 94syl2anc 696 . . . . . . . . 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 6832 . . . . . . . . 9 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑥𝑉) → ((𝐴𝑥) (𝐵𝑥)) = ((𝐴𝑥)(+g𝑆)(𝐵𝑥)))
9895, 97eqtrd 2795 . . . . . . . 8 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑥𝑉) → ((𝐴𝑓 𝐵)‘𝑥) = ((𝐴𝑥)(+g𝑆)(𝐵𝑥)))
9998oveq1d 6830 . . . . . . 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 472 . . . . . . . . 9 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → 𝑀 ∈ LMod)
101100adantr 472 . . . . . . . 8 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑥𝑉) → 𝑀 ∈ LMod)
10215ad2antrl 766 . . . . . . . . 9 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝑥𝑉 → (𝐴𝑥) ∈ 𝑅))
103102imp 444 . . . . . . . 8 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑥𝑉) → (𝐴𝑥) ∈ 𝑅)
10432ad2antll 767 . . . . . . . . 9 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝑥𝑉 → (𝐵𝑥) ∈ 𝑅))
105104imp 444 . . . . . . . 8 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑥𝑉) → (𝐵𝑥) ∈ 𝑅)
10621adantr 472 . . . . . . . . 9 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝑥𝑉𝑥 ∈ (Base‘𝑀)))
107106imp 444 . . . . . . . 8 ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑥𝑉) → 𝑥 ∈ (Base‘𝑀))
108 eqid 2761 . . . . . . . . 9 (Scalar‘𝑀) = (Scalar‘𝑀)
10924fveq2i 6357 . . . . . . . . 9 (+g𝑆) = (+g‘(Scalar‘𝑀))
1101, 3, 108, 25, 67, 109lmodvsdir 19110 . . . . . . . 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 1479 . . . . . . 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 2795 . . . . . 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 4897 . . . . 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 6831 . . . 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 1127 . . 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 2795 . 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 6827 . . 3 (𝑋 + 𝑌) = ((𝐴( linC ‘𝑀)𝑉) + (𝐵( linC ‘𝑀)𝑉))
12067oveq1i 6825 . . . . . . . . 9 (𝑅𝑚 𝑉) = ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)
121120eleq2i 2832 . . . . . . . 8 (𝐴 ∈ (𝑅𝑚 𝑉) ↔ 𝐴 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉))
122121biimpi 206 . . . . . . 7 (𝐴 ∈ (𝑅𝑚 𝑉) → 𝐴 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉))
123122ad2antrl 766 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → 𝐴 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉))
124 lincval 42727 . . . . . 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 1477 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝐴( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑥𝑉 ↦ ((𝐴𝑥)( ·𝑠𝑀)𝑥))))
126120eleq2i 2832 . . . . . . . 8 (𝐵 ∈ (𝑅𝑚 𝑉) ↔ 𝐵 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉))
127126biimpi 206 . . . . . . 7 (𝐵 ∈ (𝑅𝑚 𝑉) → 𝐵 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉))
128127ad2antll 767 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → 𝐵 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉))
129 lincval 42727 . . . . . 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 1477 . . . . 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 6833 . . . 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 1127 . . 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 2807 . 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 2806 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 383  w3a 1072   = wceq 1632  wcel 2140  Vcvv 3341  𝒫 cpw 4303   class class class wbr 4805  cmpt 4882   Fn wfn 6045  wf 6046  cfv 6050  (class class class)co 6815  𝑓 cof 7062  𝑚 cmap 8026   finSupp cfsupp 8443  Basecbs 16080  +gcplusg 16164  Scalarcsca 16167   ·𝑠 cvsca 16168  0gc0g 16323   Σg cgsu 16324  Mndcmnd 17516  Grpcgrp 17644  CMndccmn 18414  LModclmod 19086   linC clinc 42722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116  ax-cnex 10205  ax-resscn 10206  ax-1cn 10207  ax-icn 10208  ax-addcl 10209  ax-addrcl 10210  ax-mulcl 10211  ax-mulrcl 10212  ax-mulcom 10213  ax-addass 10214  ax-mulass 10215  ax-distr 10216  ax-i2m1 10217  ax-1ne0 10218  ax-1rid 10219  ax-rnegex 10220  ax-rrecex 10221  ax-cnre 10222  ax-pre-lttri 10223  ax-pre-lttrn 10224  ax-pre-ltadd 10225  ax-pre-mulgt0 10226
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-nel 3037  df-ral 3056  df-rex 3057  df-reu 3058  df-rmo 3059  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-pss 3732  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-tp 4327  df-op 4329  df-uni 4590  df-int 4629  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-tr 4906  df-id 5175  df-eprel 5180  df-po 5188  df-so 5189  df-fr 5226  df-se 5227  df-we 5228  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-pred 5842  df-ord 5888  df-on 5889  df-lim 5890  df-suc 5891  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-isom 6059  df-riota 6776  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-of 7064  df-om 7233  df-1st 7335  df-2nd 7336  df-supp 7466  df-wrecs 7578  df-recs 7639  df-rdg 7677  df-1o 7731  df-oadd 7735  df-er 7914  df-map 8028  df-en 8125  df-dom 8126  df-sdom 8127  df-fin 8128  df-fsupp 8444  df-oi 8583  df-card 8976  df-pnf 10289  df-mnf 10290  df-xr 10291  df-ltxr 10292  df-le 10293  df-sub 10481  df-neg 10482  df-nn 11234  df-2 11292  df-n0 11506  df-z 11591  df-uz 11901  df-fz 12541  df-fzo 12681  df-seq 13017  df-hash 13333  df-ndx 16083  df-slot 16084  df-base 16086  df-sets 16087  df-ress 16088  df-plusg 16177  df-0g 16325  df-gsum 16326  df-mgm 17464  df-sgrp 17506  df-mnd 17517  df-submnd 17558  df-grp 17647  df-minusg 17648  df-cntz 17971  df-cmn 18416  df-abl 18417  df-mgp 18711  df-ur 18723  df-ring 18770  df-lmod 19088  df-linc 42724
This theorem is referenced by:  lincsumcl  42749
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