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Theorem scmsuppss 46438
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 8787 . . . . 5 (𝐴 ∈ (𝑅m 𝑉) → 𝐴:𝑉𝑅)
2 fdm 6677 . . . . . 6 (𝐴:𝑉𝑅 → dom 𝐴 = 𝑉)
3 eqidd 2737 . . . . . . . . . . . 12 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) = (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)))
4 fveq2 6842 . . . . . . . . . . . . . 14 (𝑣 = 𝑥 → (𝐴𝑣) = (𝐴𝑥))
5 id 22 . . . . . . . . . . . . . 14 (𝑣 = 𝑥𝑣 = 𝑥)
64, 5oveq12d 7375 . . . . . . . . . . . . 13 (𝑣 = 𝑥 → ((𝐴𝑣)( ·𝑠𝑀)𝑣) = ((𝐴𝑥)( ·𝑠𝑀)𝑥))
76adantl 482 . . . . . . . . . . . 12 (((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) ∧ 𝑣 = 𝑥) → ((𝐴𝑣)( ·𝑠𝑀)𝑣) = ((𝐴𝑥)( ·𝑠𝑀)𝑥))
8 simpr 485 . . . . . . . . . . . 12 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → 𝑥𝑉)
9 ovex 7390 . . . . . . . . . . . . 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 6955 . . . . . . . . . . 11 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → ((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))‘𝑥) = ((𝐴𝑥)( ·𝑠𝑀)𝑥))
1211neeq1d 3003 . . . . . . . . . 10 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → (((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))‘𝑥) ≠ (0g𝑀) ↔ ((𝐴𝑥)( ·𝑠𝑀)𝑥) ≠ (0g𝑀)))
13 oveq1 7364 . . . . . . . . . . . . 13 ((𝐴𝑥) = (0g𝑆) → ((𝐴𝑥)( ·𝑠𝑀)𝑥) = ((0g𝑆)( ·𝑠𝑀)𝑥))
14 simplrr 776 . . . . . . . . . . . . . 14 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → 𝑀 ∈ LMod)
15 elelpwi 4570 . . . . . . . . . . . . . . . . . 18 ((𝑥𝑉𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑥 ∈ (Base‘𝑀))
1615expcom 414 . . . . . . . . . . . . . . . . 17 (𝑉 ∈ 𝒫 (Base‘𝑀) → (𝑥𝑉𝑥 ∈ (Base‘𝑀)))
1716adantr 481 . . . . . . . . . . . . . . . 16 ((𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod) → (𝑥𝑉𝑥 ∈ (Base‘𝑀)))
1817adantl 482 . . . . . . . . . . . . . . 15 (((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) → (𝑥𝑉𝑥 ∈ (Base‘𝑀)))
1918imp 407 . . . . . . . . . . . . . 14 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → 𝑥 ∈ (Base‘𝑀))
20 eqid 2736 . . . . . . . . . . . . . . 15 (Base‘𝑀) = (Base‘𝑀)
21 scmsuppss.s . . . . . . . . . . . . . . 15 𝑆 = (Scalar‘𝑀)
22 eqid 2736 . . . . . . . . . . . . . . 15 ( ·𝑠𝑀) = ( ·𝑠𝑀)
23 eqid 2736 . . . . . . . . . . . . . . 15 (0g𝑆) = (0g𝑆)
24 eqid 2736 . . . . . . . . . . . . . . 15 (0g𝑀) = (0g𝑀)
2520, 21, 22, 23, 24lmod0vs 20355 . . . . . . . . . . . . . 14 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑀)) → ((0g𝑆)( ·𝑠𝑀)𝑥) = (0g𝑀))
2614, 19, 25syl2anc 584 . . . . . . . . . . . . 13 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → ((0g𝑆)( ·𝑠𝑀)𝑥) = (0g𝑀))
2713, 26sylan9eqr 2798 . . . . . . . . . . . 12 (((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) ∧ (𝐴𝑥) = (0g𝑆)) → ((𝐴𝑥)( ·𝑠𝑀)𝑥) = (0g𝑀))
2827ex 413 . . . . . . . . . . 11 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → ((𝐴𝑥) = (0g𝑆) → ((𝐴𝑥)( ·𝑠𝑀)𝑥) = (0g𝑀)))
2928necon3d 2964 . . . . . . . . . 10 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → (((𝐴𝑥)( ·𝑠𝑀)𝑥) ≠ (0g𝑀) → (𝐴𝑥) ≠ (0g𝑆)))
3012, 29sylbid 239 . . . . . . . . 9 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → (((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))‘𝑥) ≠ (0g𝑀) → (𝐴𝑥) ≠ (0g𝑆)))
3130ss2rabdv 4033 . . . . . . . 8 (((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) → {𝑥𝑉 ∣ ((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))‘𝑥) ≠ (0g𝑀)} ⊆ {𝑥𝑉 ∣ (𝐴𝑥) ≠ (0g𝑆)})
32 ovex 7390 . . . . . . . . . . . . 13 ((𝐴𝑣)( ·𝑠𝑀)𝑣) ∈ V
33 eqid 2736 . . . . . . . . . . . . 13 (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) = (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))
3432, 33dmmpti 6645 . . . . . . . . . . . 12 dom (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) = 𝑉
35 rabeq 3421 . . . . . . . . . . . 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 3421 . . . . . . . . . . 11 (dom 𝐴 = 𝑉 → {𝑥 ∈ dom 𝐴 ∣ (𝐴𝑥) ≠ (0g𝑆)} = {𝑥𝑉 ∣ (𝐴𝑥) ≠ (0g𝑆)})
3836, 37sseq12d 3977 . . . . . . . . . 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 481 . . . . . . . . 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 481 . . . . . . . 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 256 . . . . . . 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 437 . . . . . 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 1111 . 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 6539 . . . 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 7171 . . . 4 (𝑉 ∈ 𝒫 (Base‘𝑀) → (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) ∈ V)
50493ad2ant2 1134 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅m 𝑉)) → (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) ∈ V)
51 fvexd 6857 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅m 𝑉)) → (0g𝑀) ∈ V)
52 suppval1 8098 . . 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 1371 . 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 8804 . . . 4 (𝐴 ∈ (𝑅m 𝑉) → Fun 𝐴)
55543ad2ant3 1135 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅m 𝑉)) → Fun 𝐴)
56 simp3 1138 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅m 𝑉)) → 𝐴 ∈ (𝑅m 𝑉))
57 fvexd 6857 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅m 𝑉)) → (0g𝑆) ∈ V)
58 suppval1 8098 . . 3 ((Fun 𝐴𝐴 ∈ (𝑅m 𝑉) ∧ (0g𝑆) ∈ V) → (𝐴 supp (0g𝑆)) = {𝑥 ∈ dom 𝐴 ∣ (𝐴𝑥) ≠ (0g𝑆)})
5955, 56, 57, 58syl3anc 1371 . 2 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅m 𝑉)) → (𝐴 supp (0g𝑆)) = {𝑥 ∈ dom 𝐴 ∣ (𝐴𝑥) ≠ (0g𝑆)})
6046, 53, 593sstr4d 3991 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 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  {crab 3407  Vcvv 3445  wss 3910  𝒫 cpw 4560  cmpt 5188  dom cdm 5633  Fun wfun 6490  wf 6492  cfv 6496  (class class class)co 7357   supp csupp 8092  m cmap 8765  Basecbs 17083  Scalarcsca 17136   ·𝑠 cvsca 17137  0gc0g 17321  LModclmod 20322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-supp 8093  df-map 8767  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-ring 19966  df-lmod 20324
This theorem is referenced by:  scmsuppfi  46443
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