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Theorem scmsuppss 41946
Description: The support of a mapping of a scalar multiplication with a function of scalars is a subset of the support of the function of scalars. (Contributed by AV, 5-Apr-2019.)
Hypotheses
Ref Expression
scmsuppss.s 𝑆 = (Scalar‘𝑀)
scmsuppss.r 𝑅 = (Base‘𝑆)
Assertion
Ref Expression
scmsuppss ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅𝑚 𝑉)) → ((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) supp (0g𝑀)) ⊆ (𝐴 supp (0g𝑆)))
Distinct variable groups:   𝑣,𝐴   𝑣,𝑀   𝑣,𝑅   𝑣,𝑉
Allowed substitution hint:   𝑆(𝑣)

Proof of Theorem scmsuppss
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elmapi 7739 . . . . 5 (𝐴 ∈ (𝑅𝑚 𝑉) → 𝐴:𝑉𝑅)
2 fdm 5947 . . . . . 6 (𝐴:𝑉𝑅 → dom 𝐴 = 𝑉)
3 eqidd 2607 . . . . . . . . . . . 12 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) = (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)))
4 fveq2 6085 . . . . . . . . . . . . . 14 (𝑣 = 𝑥 → (𝐴𝑣) = (𝐴𝑥))
5 id 22 . . . . . . . . . . . . . 14 (𝑣 = 𝑥𝑣 = 𝑥)
64, 5oveq12d 6542 . . . . . . . . . . . . 13 (𝑣 = 𝑥 → ((𝐴𝑣)( ·𝑠𝑀)𝑣) = ((𝐴𝑥)( ·𝑠𝑀)𝑥))
76adantl 480 . . . . . . . . . . . 12 (((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) ∧ 𝑣 = 𝑥) → ((𝐴𝑣)( ·𝑠𝑀)𝑣) = ((𝐴𝑥)( ·𝑠𝑀)𝑥))
8 simpr 475 . . . . . . . . . . . 12 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → 𝑥𝑉)
9 ovex 6552 . . . . . . . . . . . . 13 ((𝐴𝑥)( ·𝑠𝑀)𝑥) ∈ V
109a1i 11 . . . . . . . . . . . 12 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → ((𝐴𝑥)( ·𝑠𝑀)𝑥) ∈ V)
113, 7, 8, 10fvmptd 6179 . . . . . . . . . . 11 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → ((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))‘𝑥) = ((𝐴𝑥)( ·𝑠𝑀)𝑥))
1211neeq1d 2837 . . . . . . . . . 10 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → (((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))‘𝑥) ≠ (0g𝑀) ↔ ((𝐴𝑥)( ·𝑠𝑀)𝑥) ≠ (0g𝑀)))
13 oveq1 6531 . . . . . . . . . . . . 13 ((𝐴𝑥) = (0g𝑆) → ((𝐴𝑥)( ·𝑠𝑀)𝑥) = ((0g𝑆)( ·𝑠𝑀)𝑥))
14 simplrr 796 . . . . . . . . . . . . . 14 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → 𝑀 ∈ LMod)
15 elelpwi 4115 . . . . . . . . . . . . . . . . . 18 ((𝑥𝑉𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑥 ∈ (Base‘𝑀))
1615expcom 449 . . . . . . . . . . . . . . . . 17 (𝑉 ∈ 𝒫 (Base‘𝑀) → (𝑥𝑉𝑥 ∈ (Base‘𝑀)))
1716adantr 479 . . . . . . . . . . . . . . . 16 ((𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod) → (𝑥𝑉𝑥 ∈ (Base‘𝑀)))
1817adantl 480 . . . . . . . . . . . . . . 15 (((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) → (𝑥𝑉𝑥 ∈ (Base‘𝑀)))
1918imp 443 . . . . . . . . . . . . . 14 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → 𝑥 ∈ (Base‘𝑀))
20 eqid 2606 . . . . . . . . . . . . . . 15 (Base‘𝑀) = (Base‘𝑀)
21 scmsuppss.s . . . . . . . . . . . . . . 15 𝑆 = (Scalar‘𝑀)
22 eqid 2606 . . . . . . . . . . . . . . 15 ( ·𝑠𝑀) = ( ·𝑠𝑀)
23 eqid 2606 . . . . . . . . . . . . . . 15 (0g𝑆) = (0g𝑆)
24 eqid 2606 . . . . . . . . . . . . . . 15 (0g𝑀) = (0g𝑀)
2520, 21, 22, 23, 24lmod0vs 18662 . . . . . . . . . . . . . 14 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑀)) → ((0g𝑆)( ·𝑠𝑀)𝑥) = (0g𝑀))
2614, 19, 25syl2anc 690 . . . . . . . . . . . . 13 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → ((0g𝑆)( ·𝑠𝑀)𝑥) = (0g𝑀))
2713, 26sylan9eqr 2662 . . . . . . . . . . . 12 (((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) ∧ (𝐴𝑥) = (0g𝑆)) → ((𝐴𝑥)( ·𝑠𝑀)𝑥) = (0g𝑀))
2827ex 448 . . . . . . . . . . 11 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → ((𝐴𝑥) = (0g𝑆) → ((𝐴𝑥)( ·𝑠𝑀)𝑥) = (0g𝑀)))
2928necon3d 2799 . . . . . . . . . 10 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → (((𝐴𝑥)( ·𝑠𝑀)𝑥) ≠ (0g𝑀) → (𝐴𝑥) ≠ (0g𝑆)))
3012, 29sylbid 228 . . . . . . . . 9 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → (((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))‘𝑥) ≠ (0g𝑀) → (𝐴𝑥) ≠ (0g𝑆)))
3130ss2rabdv 3642 . . . . . . . 8 (((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) → {𝑥𝑉 ∣ ((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))‘𝑥) ≠ (0g𝑀)} ⊆ {𝑥𝑉 ∣ (𝐴𝑥) ≠ (0g𝑆)})
32 ovex 6552 . . . . . . . . . . . . 13 ((𝐴𝑣)( ·𝑠𝑀)𝑣) ∈ V
33 eqid 2606 . . . . . . . . . . . . 13 (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) = (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))
3432, 33dmmpti 5919 . . . . . . . . . . . 12 dom (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) = 𝑉
35 rabeq 3162 . . . . . . . . . . . 12 (dom (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) = 𝑉 → {𝑥 ∈ dom (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) ∣ ((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))‘𝑥) ≠ (0g𝑀)} = {𝑥𝑉 ∣ ((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))‘𝑥) ≠ (0g𝑀)})
3634, 35mp1i 13 . . . . . . . . . . 11 (dom 𝐴 = 𝑉 → {𝑥 ∈ dom (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) ∣ ((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))‘𝑥) ≠ (0g𝑀)} = {𝑥𝑉 ∣ ((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))‘𝑥) ≠ (0g𝑀)})
37 rabeq 3162 . . . . . . . . . . 11 (dom 𝐴 = 𝑉 → {𝑥 ∈ dom 𝐴 ∣ (𝐴𝑥) ≠ (0g𝑆)} = {𝑥𝑉 ∣ (𝐴𝑥) ≠ (0g𝑆)})
3836, 37sseq12d 3593 . . . . . . . . . 10 (dom 𝐴 = 𝑉 → ({𝑥 ∈ dom (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) ∣ ((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))‘𝑥) ≠ (0g𝑀)} ⊆ {𝑥 ∈ dom 𝐴 ∣ (𝐴𝑥) ≠ (0g𝑆)} ↔ {𝑥𝑉 ∣ ((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))‘𝑥) ≠ (0g𝑀)} ⊆ {𝑥𝑉 ∣ (𝐴𝑥) ≠ (0g𝑆)}))
3938adantr 479 . . . . . . . . 9 ((dom 𝐴 = 𝑉𝐴:𝑉𝑅) → ({𝑥 ∈ dom (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) ∣ ((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))‘𝑥) ≠ (0g𝑀)} ⊆ {𝑥 ∈ dom 𝐴 ∣ (𝐴𝑥) ≠ (0g𝑆)} ↔ {𝑥𝑉 ∣ ((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))‘𝑥) ≠ (0g𝑀)} ⊆ {𝑥𝑉 ∣ (𝐴𝑥) ≠ (0g𝑆)}))
4039adantr 479 . . . . . . . 8 (((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) → ({𝑥 ∈ dom (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) ∣ ((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))‘𝑥) ≠ (0g𝑀)} ⊆ {𝑥 ∈ dom 𝐴 ∣ (𝐴𝑥) ≠ (0g𝑆)} ↔ {𝑥𝑉 ∣ ((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))‘𝑥) ≠ (0g𝑀)} ⊆ {𝑥𝑉 ∣ (𝐴𝑥) ≠ (0g𝑆)}))
4131, 40mpbird 245 . . . . . . 7 (((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) → {𝑥 ∈ dom (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) ∣ ((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))‘𝑥) ≠ (0g𝑀)} ⊆ {𝑥 ∈ dom 𝐴 ∣ (𝐴𝑥) ≠ (0g𝑆)})
4241exp43 637 . . . . . 6 (dom 𝐴 = 𝑉 → (𝐴:𝑉𝑅 → (𝑉 ∈ 𝒫 (Base‘𝑀) → (𝑀 ∈ LMod → {𝑥 ∈ dom (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) ∣ ((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))‘𝑥) ≠ (0g𝑀)} ⊆ {𝑥 ∈ dom 𝐴 ∣ (𝐴𝑥) ≠ (0g𝑆)}))))
432, 42mpcom 37 . . . . 5 (𝐴:𝑉𝑅 → (𝑉 ∈ 𝒫 (Base‘𝑀) → (𝑀 ∈ LMod → {𝑥 ∈ dom (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) ∣ ((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))‘𝑥) ≠ (0g𝑀)} ⊆ {𝑥 ∈ dom 𝐴 ∣ (𝐴𝑥) ≠ (0g𝑆)})))
441, 43syl 17 . . . 4 (𝐴 ∈ (𝑅𝑚 𝑉) → (𝑉 ∈ 𝒫 (Base‘𝑀) → (𝑀 ∈ LMod → {𝑥 ∈ dom (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) ∣ ((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))‘𝑥) ≠ (0g𝑀)} ⊆ {𝑥 ∈ dom 𝐴 ∣ (𝐴𝑥) ≠ (0g𝑆)})))
4544com13 85 . . 3 (𝑀 ∈ LMod → (𝑉 ∈ 𝒫 (Base‘𝑀) → (𝐴 ∈ (𝑅𝑚 𝑉) → {𝑥 ∈ dom (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) ∣ ((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))‘𝑥) ≠ (0g𝑀)} ⊆ {𝑥 ∈ dom 𝐴 ∣ (𝐴𝑥) ≠ (0g𝑆)})))
46453imp 1248 . 2 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅𝑚 𝑉)) → {𝑥 ∈ dom (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) ∣ ((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))‘𝑥) ≠ (0g𝑀)} ⊆ {𝑥 ∈ dom 𝐴 ∣ (𝐴𝑥) ≠ (0g𝑆)})
47 funmpt 5823 . . . 4 Fun (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))
4847a1i 11 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅𝑚 𝑉)) → Fun (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)))
49 mptexg 6364 . . . 4 (𝑉 ∈ 𝒫 (Base‘𝑀) → (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) ∈ V)
50493ad2ant2 1075 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅𝑚 𝑉)) → (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) ∈ V)
51 fvex 6095 . . . 4 (0g𝑀) ∈ V
5251a1i 11 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅𝑚 𝑉)) → (0g𝑀) ∈ V)
53 suppval1 7162 . . 3 ((Fun (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) ∧ (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) ∈ V ∧ (0g𝑀) ∈ V) → ((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) supp (0g𝑀)) = {𝑥 ∈ dom (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) ∣ ((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))‘𝑥) ≠ (0g𝑀)})
5448, 50, 52, 53syl3anc 1317 . 2 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅𝑚 𝑉)) → ((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) supp (0g𝑀)) = {𝑥 ∈ dom (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) ∣ ((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))‘𝑥) ≠ (0g𝑀)})
55 elmapfun 7741 . . . 4 (𝐴 ∈ (𝑅𝑚 𝑉) → Fun 𝐴)
56553ad2ant3 1076 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅𝑚 𝑉)) → Fun 𝐴)
57 simp3 1055 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅𝑚 𝑉)) → 𝐴 ∈ (𝑅𝑚 𝑉))
58 fvex 6095 . . . 4 (0g𝑆) ∈ V
5958a1i 11 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅𝑚 𝑉)) → (0g𝑆) ∈ V)
60 suppval1 7162 . . 3 ((Fun 𝐴𝐴 ∈ (𝑅𝑚 𝑉) ∧ (0g𝑆) ∈ V) → (𝐴 supp (0g𝑆)) = {𝑥 ∈ dom 𝐴 ∣ (𝐴𝑥) ≠ (0g𝑆)})
6156, 57, 59, 60syl3anc 1317 . 2 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅𝑚 𝑉)) → (𝐴 supp (0g𝑆)) = {𝑥 ∈ dom 𝐴 ∣ (𝐴𝑥) ≠ (0g𝑆)})
6246, 54, 613sstr4d 3607 1 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅𝑚 𝑉)) → ((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) supp (0g𝑀)) ⊆ (𝐴 supp (0g𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wne 2776  {crab 2896  Vcvv 3169  wss 3536  𝒫 cpw 4104  cmpt 4634  dom cdm 5025  Fun wfun 5781  wf 5783  cfv 5787  (class class class)co 6524   supp csupp 7156  𝑚 cmap 7718  Basecbs 15638  Scalarcsca 15714   ·𝑠 cvsca 15715  0gc0g 15866  LModclmod 18629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-ral 2897  df-rex 2898  df-reu 2899  df-rmo 2900  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-op 4128  df-uni 4364  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-id 4940  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-riota 6486  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-1st 7033  df-2nd 7034  df-supp 7157  df-map 7720  df-0g 15868  df-mgm 17008  df-sgrp 17050  df-mnd 17061  df-grp 17191  df-ring 18315  df-lmod 18631
This theorem is referenced by:  scmsuppfi  41951
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