Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  scmsuppss Structured version   Visualization version   GIF version

Theorem scmsuppss 44761
 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 8415 . . . . 5 (𝐴 ∈ (𝑅m 𝑉) → 𝐴:𝑉𝑅)
2 fdm 6499 . . . . . 6 (𝐴:𝑉𝑅 → dom 𝐴 = 𝑉)
3 eqidd 2802 . . . . . . . . . . . 12 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) = (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)))
4 fveq2 6649 . . . . . . . . . . . . . 14 (𝑣 = 𝑥 → (𝐴𝑣) = (𝐴𝑥))
5 id 22 . . . . . . . . . . . . . 14 (𝑣 = 𝑥𝑣 = 𝑥)
64, 5oveq12d 7157 . . . . . . . . . . . . 13 (𝑣 = 𝑥 → ((𝐴𝑣)( ·𝑠𝑀)𝑣) = ((𝐴𝑥)( ·𝑠𝑀)𝑥))
76adantl 485 . . . . . . . . . . . 12 (((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) ∧ 𝑣 = 𝑥) → ((𝐴𝑣)( ·𝑠𝑀)𝑣) = ((𝐴𝑥)( ·𝑠𝑀)𝑥))
8 simpr 488 . . . . . . . . . . . 12 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → 𝑥𝑉)
9 ovex 7172 . . . . . . . . . . . . 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 6756 . . . . . . . . . . 11 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → ((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))‘𝑥) = ((𝐴𝑥)( ·𝑠𝑀)𝑥))
1211neeq1d 3049 . . . . . . . . . 10 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → (((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))‘𝑥) ≠ (0g𝑀) ↔ ((𝐴𝑥)( ·𝑠𝑀)𝑥) ≠ (0g𝑀)))
13 oveq1 7146 . . . . . . . . . . . . 13 ((𝐴𝑥) = (0g𝑆) → ((𝐴𝑥)( ·𝑠𝑀)𝑥) = ((0g𝑆)( ·𝑠𝑀)𝑥))
14 simplrr 777 . . . . . . . . . . . . . 14 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → 𝑀 ∈ LMod)
15 elelpwi 4512 . . . . . . . . . . . . . . . . . 18 ((𝑥𝑉𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑥 ∈ (Base‘𝑀))
1615expcom 417 . . . . . . . . . . . . . . . . 17 (𝑉 ∈ 𝒫 (Base‘𝑀) → (𝑥𝑉𝑥 ∈ (Base‘𝑀)))
1716adantr 484 . . . . . . . . . . . . . . . 16 ((𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod) → (𝑥𝑉𝑥 ∈ (Base‘𝑀)))
1817adantl 485 . . . . . . . . . . . . . . 15 (((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) → (𝑥𝑉𝑥 ∈ (Base‘𝑀)))
1918imp 410 . . . . . . . . . . . . . 14 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → 𝑥 ∈ (Base‘𝑀))
20 eqid 2801 . . . . . . . . . . . . . . 15 (Base‘𝑀) = (Base‘𝑀)
21 scmsuppss.s . . . . . . . . . . . . . . 15 𝑆 = (Scalar‘𝑀)
22 eqid 2801 . . . . . . . . . . . . . . 15 ( ·𝑠𝑀) = ( ·𝑠𝑀)
23 eqid 2801 . . . . . . . . . . . . . . 15 (0g𝑆) = (0g𝑆)
24 eqid 2801 . . . . . . . . . . . . . . 15 (0g𝑀) = (0g𝑀)
2520, 21, 22, 23, 24lmod0vs 19663 . . . . . . . . . . . . . 14 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑀)) → ((0g𝑆)( ·𝑠𝑀)𝑥) = (0g𝑀))
2614, 19, 25syl2anc 587 . . . . . . . . . . . . 13 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → ((0g𝑆)( ·𝑠𝑀)𝑥) = (0g𝑀))
2713, 26sylan9eqr 2858 . . . . . . . . . . . 12 (((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) ∧ (𝐴𝑥) = (0g𝑆)) → ((𝐴𝑥)( ·𝑠𝑀)𝑥) = (0g𝑀))
2827ex 416 . . . . . . . . . . 11 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → ((𝐴𝑥) = (0g𝑆) → ((𝐴𝑥)( ·𝑠𝑀)𝑥) = (0g𝑀)))
2928necon3d 3011 . . . . . . . . . 10 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → (((𝐴𝑥)( ·𝑠𝑀)𝑥) ≠ (0g𝑀) → (𝐴𝑥) ≠ (0g𝑆)))
3012, 29sylbid 243 . . . . . . . . 9 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → (((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))‘𝑥) ≠ (0g𝑀) → (𝐴𝑥) ≠ (0g𝑆)))
3130ss2rabdv 4006 . . . . . . . 8 (((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) → {𝑥𝑉 ∣ ((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))‘𝑥) ≠ (0g𝑀)} ⊆ {𝑥𝑉 ∣ (𝐴𝑥) ≠ (0g𝑆)})
32 ovex 7172 . . . . . . . . . . . . 13 ((𝐴𝑣)( ·𝑠𝑀)𝑣) ∈ V
33 eqid 2801 . . . . . . . . . . . . 13 (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) = (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))
3432, 33dmmpti 6468 . . . . . . . . . . . 12 dom (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) = 𝑉
35 rabeq 3434 . . . . . . . . . . . 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 3434 . . . . . . . . . . 11 (dom 𝐴 = 𝑉 → {𝑥 ∈ dom 𝐴 ∣ (𝐴𝑥) ≠ (0g𝑆)} = {𝑥𝑉 ∣ (𝐴𝑥) ≠ (0g𝑆)})
3836, 37sseq12d 3951 . . . . . . . . . 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 484 . . . . . . . . 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 484 . . . . . . . 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 260 . . . . . . 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 440 . . . . . 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 1108 . 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 6366 . . . 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 6965 . . . 4 (𝑉 ∈ 𝒫 (Base‘𝑀) → (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) ∈ V)
50493ad2ant2 1131 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅m 𝑉)) → (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) ∈ V)
51 fvexd 6664 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅m 𝑉)) → (0g𝑀) ∈ V)
52 suppval1 7823 . . 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 1368 . 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 8417 . . . 4 (𝐴 ∈ (𝑅m 𝑉) → Fun 𝐴)
55543ad2ant3 1132 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅m 𝑉)) → Fun 𝐴)
56 simp3 1135 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅m 𝑉)) → 𝐴 ∈ (𝑅m 𝑉))
57 fvexd 6664 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅m 𝑉)) → (0g𝑆) ∈ V)
58 suppval1 7823 . . 3 ((Fun 𝐴𝐴 ∈ (𝑅m 𝑉) ∧ (0g𝑆) ∈ V) → (𝐴 supp (0g𝑆)) = {𝑥 ∈ dom 𝐴 ∣ (𝐴𝑥) ≠ (0g𝑆)})
5955, 56, 57, 58syl3anc 1368 . 2 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅m 𝑉)) → (𝐴 supp (0g𝑆)) = {𝑥 ∈ dom 𝐴 ∣ (𝐴𝑥) ≠ (0g𝑆)})
6046, 53, 593sstr4d 3965 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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2112   ≠ wne 2990  {crab 3113  Vcvv 3444   ⊆ wss 3884  𝒫 cpw 4500   ↦ cmpt 5113  dom cdm 5523  Fun wfun 6322  ⟶wf 6324  ‘cfv 6328  (class class class)co 7139   supp csupp 7817   ↑m cmap 8393  Basecbs 16478  Scalarcsca 16563   ·𝑠 cvsca 16564  0gc0g 16708  LModclmod 19630 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-supp 7818  df-map 8395  df-0g 16710  df-mgm 17847  df-sgrp 17896  df-mnd 17907  df-grp 18101  df-ring 19295  df-lmod 19632 This theorem is referenced by:  scmsuppfi  44766
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