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Theorem scmsuppss 42478
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 7921 . . . . 5 (𝐴 ∈ (𝑅𝑚 𝑉) → 𝐴:𝑉𝑅)
2 fdm 6089 . . . . . 6 (𝐴:𝑉𝑅 → dom 𝐴 = 𝑉)
3 eqidd 2652 . . . . . . . . . . . 12 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) = (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)))
4 fveq2 6229 . . . . . . . . . . . . . 14 (𝑣 = 𝑥 → (𝐴𝑣) = (𝐴𝑥))
5 id 22 . . . . . . . . . . . . . 14 (𝑣 = 𝑥𝑣 = 𝑥)
64, 5oveq12d 6708 . . . . . . . . . . . . 13 (𝑣 = 𝑥 → ((𝐴𝑣)( ·𝑠𝑀)𝑣) = ((𝐴𝑥)( ·𝑠𝑀)𝑥))
76adantl 481 . . . . . . . . . . . 12 (((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) ∧ 𝑣 = 𝑥) → ((𝐴𝑣)( ·𝑠𝑀)𝑣) = ((𝐴𝑥)( ·𝑠𝑀)𝑥))
8 simpr 476 . . . . . . . . . . . 12 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → 𝑥𝑉)
9 ovex 6718 . . . . . . . . . . . . 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 6327 . . . . . . . . . . 11 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → ((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))‘𝑥) = ((𝐴𝑥)( ·𝑠𝑀)𝑥))
1211neeq1d 2882 . . . . . . . . . 10 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → (((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))‘𝑥) ≠ (0g𝑀) ↔ ((𝐴𝑥)( ·𝑠𝑀)𝑥) ≠ (0g𝑀)))
13 oveq1 6697 . . . . . . . . . . . . 13 ((𝐴𝑥) = (0g𝑆) → ((𝐴𝑥)( ·𝑠𝑀)𝑥) = ((0g𝑆)( ·𝑠𝑀)𝑥))
14 simplrr 818 . . . . . . . . . . . . . 14 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → 𝑀 ∈ LMod)
15 elelpwi 4204 . . . . . . . . . . . . . . . . . 18 ((𝑥𝑉𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑥 ∈ (Base‘𝑀))
1615expcom 450 . . . . . . . . . . . . . . . . 17 (𝑉 ∈ 𝒫 (Base‘𝑀) → (𝑥𝑉𝑥 ∈ (Base‘𝑀)))
1716adantr 480 . . . . . . . . . . . . . . . 16 ((𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod) → (𝑥𝑉𝑥 ∈ (Base‘𝑀)))
1817adantl 481 . . . . . . . . . . . . . . 15 (((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) → (𝑥𝑉𝑥 ∈ (Base‘𝑀)))
1918imp 444 . . . . . . . . . . . . . 14 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → 𝑥 ∈ (Base‘𝑀))
20 eqid 2651 . . . . . . . . . . . . . . 15 (Base‘𝑀) = (Base‘𝑀)
21 scmsuppss.s . . . . . . . . . . . . . . 15 𝑆 = (Scalar‘𝑀)
22 eqid 2651 . . . . . . . . . . . . . . 15 ( ·𝑠𝑀) = ( ·𝑠𝑀)
23 eqid 2651 . . . . . . . . . . . . . . 15 (0g𝑆) = (0g𝑆)
24 eqid 2651 . . . . . . . . . . . . . . 15 (0g𝑀) = (0g𝑀)
2520, 21, 22, 23, 24lmod0vs 18944 . . . . . . . . . . . . . 14 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑀)) → ((0g𝑆)( ·𝑠𝑀)𝑥) = (0g𝑀))
2614, 19, 25syl2anc 694 . . . . . . . . . . . . 13 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → ((0g𝑆)( ·𝑠𝑀)𝑥) = (0g𝑀))
2713, 26sylan9eqr 2707 . . . . . . . . . . . 12 (((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) ∧ (𝐴𝑥) = (0g𝑆)) → ((𝐴𝑥)( ·𝑠𝑀)𝑥) = (0g𝑀))
2827ex 449 . . . . . . . . . . 11 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → ((𝐴𝑥) = (0g𝑆) → ((𝐴𝑥)( ·𝑠𝑀)𝑥) = (0g𝑀)))
2928necon3d 2844 . . . . . . . . . 10 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → (((𝐴𝑥)( ·𝑠𝑀)𝑥) ≠ (0g𝑀) → (𝐴𝑥) ≠ (0g𝑆)))
3012, 29sylbid 230 . . . . . . . . 9 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → (((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))‘𝑥) ≠ (0g𝑀) → (𝐴𝑥) ≠ (0g𝑆)))
3130ss2rabdv 3716 . . . . . . . 8 (((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) → {𝑥𝑉 ∣ ((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))‘𝑥) ≠ (0g𝑀)} ⊆ {𝑥𝑉 ∣ (𝐴𝑥) ≠ (0g𝑆)})
32 ovex 6718 . . . . . . . . . . . . 13 ((𝐴𝑣)( ·𝑠𝑀)𝑣) ∈ V
33 eqid 2651 . . . . . . . . . . . . 13 (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) = (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))
3432, 33dmmpti 6061 . . . . . . . . . . . 12 dom (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) = 𝑉
35 rabeq 3223 . . . . . . . . . . . 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 3223 . . . . . . . . . . 11 (dom 𝐴 = 𝑉 → {𝑥 ∈ dom 𝐴 ∣ (𝐴𝑥) ≠ (0g𝑆)} = {𝑥𝑉 ∣ (𝐴𝑥) ≠ (0g𝑆)})
3836, 37sseq12d 3667 . . . . . . . . . 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 480 . . . . . . . . 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 480 . . . . . . . 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 247 . . . . . . 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 639 . . . . . 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 38 . . . . 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 88 . . 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 1275 . 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 5964 . . . 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 6525 . . . 4 (𝑉 ∈ 𝒫 (Base‘𝑀) → (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) ∈ V)
50493ad2ant2 1103 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅𝑚 𝑉)) → (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) ∈ V)
51 fvexd 6241 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅𝑚 𝑉)) → (0g𝑀) ∈ V)
52 suppval1 7346 . . 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)})
5348, 50, 51, 52syl3anc 1366 . 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)})
54 elmapfun 7923 . . . 4 (𝐴 ∈ (𝑅𝑚 𝑉) → Fun 𝐴)
55543ad2ant3 1104 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅𝑚 𝑉)) → Fun 𝐴)
56 simp3 1083 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅𝑚 𝑉)) → 𝐴 ∈ (𝑅𝑚 𝑉))
57 fvexd 6241 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅𝑚 𝑉)) → (0g𝑆) ∈ V)
58 suppval1 7346 . . 3 ((Fun 𝐴𝐴 ∈ (𝑅𝑚 𝑉) ∧ (0g𝑆) ∈ V) → (𝐴 supp (0g𝑆)) = {𝑥 ∈ dom 𝐴 ∣ (𝐴𝑥) ≠ (0g𝑆)})
5955, 56, 57, 58syl3anc 1366 . 2 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅𝑚 𝑉)) → (𝐴 supp (0g𝑆)) = {𝑥 ∈ dom 𝐴 ∣ (𝐴𝑥) ≠ (0g𝑆)})
6046, 53, 593sstr4d 3681 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 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  {crab 2945  Vcvv 3231  wss 3607  𝒫 cpw 4191  cmpt 4762  dom cdm 5143  Fun wfun 5920  wf 5922  cfv 5926  (class class class)co 6690   supp csupp 7340  𝑚 cmap 7899  Basecbs 15904  Scalarcsca 15991   ·𝑠 cvsca 15992  0gc0g 16147  LModclmod 18911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-supp 7341  df-map 7901  df-0g 16149  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-ring 18595  df-lmod 18913
This theorem is referenced by:  scmsuppfi  42483
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