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 41471
 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 7839 . . . . 5 (𝐴 ∈ (𝑅𝑚 𝑉) → 𝐴:𝑉𝑅)
2 fdm 6018 . . . . . 6 (𝐴:𝑉𝑅 → dom 𝐴 = 𝑉)
3 eqidd 2622 . . . . . . . . . . . 12 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) = (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)))
4 fveq2 6158 . . . . . . . . . . . . . 14 (𝑣 = 𝑥 → (𝐴𝑣) = (𝐴𝑥))
5 id 22 . . . . . . . . . . . . . 14 (𝑣 = 𝑥𝑣 = 𝑥)
64, 5oveq12d 6633 . . . . . . . . . . . . 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 477 . . . . . . . . . . . 12 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → 𝑥𝑉)
9 ovex 6643 . . . . . . . . . . . . 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 6255 . . . . . . . . . . 11 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → ((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))‘𝑥) = ((𝐴𝑥)( ·𝑠𝑀)𝑥))
1211neeq1d 2849 . . . . . . . . . 10 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → (((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))‘𝑥) ≠ (0g𝑀) ↔ ((𝐴𝑥)( ·𝑠𝑀)𝑥) ≠ (0g𝑀)))
13 oveq1 6622 . . . . . . . . . . . . 13 ((𝐴𝑥) = (0g𝑆) → ((𝐴𝑥)( ·𝑠𝑀)𝑥) = ((0g𝑆)( ·𝑠𝑀)𝑥))
14 simplrr 800 . . . . . . . . . . . . . 14 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → 𝑀 ∈ LMod)
15 elelpwi 4149 . . . . . . . . . . . . . . . . . 18 ((𝑥𝑉𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑥 ∈ (Base‘𝑀))
1615expcom 451 . . . . . . . . . . . . . . . . 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 445 . . . . . . . . . . . . . 14 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → 𝑥 ∈ (Base‘𝑀))
20 eqid 2621 . . . . . . . . . . . . . . 15 (Base‘𝑀) = (Base‘𝑀)
21 scmsuppss.s . . . . . . . . . . . . . . 15 𝑆 = (Scalar‘𝑀)
22 eqid 2621 . . . . . . . . . . . . . . 15 ( ·𝑠𝑀) = ( ·𝑠𝑀)
23 eqid 2621 . . . . . . . . . . . . . . 15 (0g𝑆) = (0g𝑆)
24 eqid 2621 . . . . . . . . . . . . . . 15 (0g𝑀) = (0g𝑀)
2520, 21, 22, 23, 24lmod0vs 18836 . . . . . . . . . . . . . 14 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑀)) → ((0g𝑆)( ·𝑠𝑀)𝑥) = (0g𝑀))
2614, 19, 25syl2anc 692 . . . . . . . . . . . . 13 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → ((0g𝑆)( ·𝑠𝑀)𝑥) = (0g𝑀))
2713, 26sylan9eqr 2677 . . . . . . . . . . . 12 (((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) ∧ (𝐴𝑥) = (0g𝑆)) → ((𝐴𝑥)( ·𝑠𝑀)𝑥) = (0g𝑀))
2827ex 450 . . . . . . . . . . 11 ((((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥𝑉) → ((𝐴𝑥) = (0g𝑆) → ((𝐴𝑥)( ·𝑠𝑀)𝑥) = (0g𝑀)))
2928necon3d 2811 . . . . . . . . . 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 3668 . . . . . . . 8 (((dom 𝐴 = 𝑉𝐴:𝑉𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) → {𝑥𝑉 ∣ ((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))‘𝑥) ≠ (0g𝑀)} ⊆ {𝑥𝑉 ∣ (𝐴𝑥) ≠ (0g𝑆)})
32 ovex 6643 . . . . . . . . . . . . 13 ((𝐴𝑣)( ·𝑠𝑀)𝑣) ∈ V
33 eqid 2621 . . . . . . . . . . . . 13 (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) = (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣))
3432, 33dmmpti 5990 . . . . . . . . . . . 12 dom (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) = 𝑉
35 rabeq 3183 . . . . . . . . . . . 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 3183 . . . . . . . . . . 11 (dom 𝐴 = 𝑉 → {𝑥 ∈ dom 𝐴 ∣ (𝐴𝑥) ≠ (0g𝑆)} = {𝑥𝑉 ∣ (𝐴𝑥) ≠ (0g𝑆)})
3836, 37sseq12d 3619 . . . . . . . . . 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 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 1254 . 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 5894 . . . 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 6449 . . . 4 (𝑉 ∈ 𝒫 (Base‘𝑀) → (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) ∈ V)
50493ad2ant2 1081 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅𝑚 𝑉)) → (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) ∈ V)
51 fvexd 6170 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅𝑚 𝑉)) → (0g𝑀) ∈ V)
52 suppval1 7261 . . 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 1323 . 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 7841 . . . 4 (𝐴 ∈ (𝑅𝑚 𝑉) → Fun 𝐴)
55543ad2ant3 1082 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅𝑚 𝑉)) → Fun 𝐴)
56 simp3 1061 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅𝑚 𝑉)) → 𝐴 ∈ (𝑅𝑚 𝑉))
57 fvexd 6170 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅𝑚 𝑉)) → (0g𝑆) ∈ V)
58 suppval1 7261 . . 3 ((Fun 𝐴𝐴 ∈ (𝑅𝑚 𝑉) ∧ (0g𝑆) ∈ V) → (𝐴 supp (0g𝑆)) = {𝑥 ∈ dom 𝐴 ∣ (𝐴𝑥) ≠ (0g𝑆)})
5955, 56, 57, 58syl3anc 1323 . 2 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅𝑚 𝑉)) → (𝐴 supp (0g𝑆)) = {𝑥 ∈ dom 𝐴 ∣ (𝐴𝑥) ≠ (0g𝑆)})
6046, 53, 593sstr4d 3633 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 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987   ≠ wne 2790  {crab 2912  Vcvv 3190   ⊆ wss 3560  𝒫 cpw 4136   ↦ cmpt 4683  dom cdm 5084  Fun wfun 5851  ⟶wf 5853  ‘cfv 5857  (class class class)co 6615   supp csupp 7255   ↑𝑚 cmap 7817  Basecbs 15800  Scalarcsca 15884   ·𝑠 cvsca 15885  0gc0g 16040  LModclmod 18803 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-1st 7128  df-2nd 7129  df-supp 7256  df-map 7819  df-0g 16042  df-mgm 17182  df-sgrp 17224  df-mnd 17235  df-grp 17365  df-ring 18489  df-lmod 18805 This theorem is referenced by:  scmsuppfi  41476
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