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Theorem mndpsuppss 46437
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.)
Hypothesis
Ref Expression
mndpsuppss.r 𝑅 = (Base‘𝑀)
Assertion
Ref Expression
mndpsuppss (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → ((𝐴f (+g𝑀)𝐵) supp (0g𝑀)) ⊆ ((𝐴 supp (0g𝑀)) ∪ (𝐵 supp (0g𝑀))))

Proof of Theorem mndpsuppss
Dummy variables 𝑣 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ioran 982 . . . . . 6 (¬ ((𝐴𝑥) ≠ (0g𝑀) ∨ (𝐵𝑥) ≠ (0g𝑀)) ↔ (¬ (𝐴𝑥) ≠ (0g𝑀) ∧ ¬ (𝐵𝑥) ≠ (0g𝑀)))
2 nne 2947 . . . . . . 7 (¬ (𝐴𝑥) ≠ (0g𝑀) ↔ (𝐴𝑥) = (0g𝑀))
3 nne 2947 . . . . . . 7 (¬ (𝐵𝑥) ≠ (0g𝑀) ↔ (𝐵𝑥) = (0g𝑀))
42, 3anbi12i 627 . . . . . 6 ((¬ (𝐴𝑥) ≠ (0g𝑀) ∧ ¬ (𝐵𝑥) ≠ (0g𝑀)) ↔ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀)))
51, 4bitri 274 . . . . 5 (¬ ((𝐴𝑥) ≠ (0g𝑀) ∨ (𝐵𝑥) ≠ (0g𝑀)) ↔ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀)))
6 elmapfn 8803 . . . . . . . . . . . 12 (𝐴 ∈ (𝑅m 𝑉) → 𝐴 Fn 𝑉)
76ad2antrl 726 . . . . . . . . . . 11 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → 𝐴 Fn 𝑉)
87adantr 481 . . . . . . . . . 10 ((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) → 𝐴 Fn 𝑉)
9 elmapfn 8803 . . . . . . . . . . . 12 (𝐵 ∈ (𝑅m 𝑉) → 𝐵 Fn 𝑉)
109ad2antll 727 . . . . . . . . . . 11 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → 𝐵 Fn 𝑉)
1110adantr 481 . . . . . . . . . 10 ((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) → 𝐵 Fn 𝑉)
12 simplr 767 . . . . . . . . . . 11 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → 𝑉𝑋)
1312adantr 481 . . . . . . . . . 10 ((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) → 𝑉𝑋)
14 inidm 4178 . . . . . . . . . 10 (𝑉𝑉) = 𝑉
15 simplrl 775 . . . . . . . . . 10 (((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) ∧ 𝑥𝑉) → (𝐴𝑥) = (0g𝑀))
16 simplrr 776 . . . . . . . . . 10 (((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) ∧ 𝑥𝑉) → (𝐵𝑥) = (0g𝑀))
178, 11, 13, 13, 14, 15, 16ofval 7628 . . . . . . . . 9 (((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) ∧ 𝑥𝑉) → ((𝐴f (+g𝑀)𝐵)‘𝑥) = ((0g𝑀)(+g𝑀)(0g𝑀)))
1817an32s 650 . . . . . . . 8 (((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑥𝑉) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) → ((𝐴f (+g𝑀)𝐵)‘𝑥) = ((0g𝑀)(+g𝑀)(0g𝑀)))
19 eqid 2736 . . . . . . . . . . . 12 (Base‘𝑀) = (Base‘𝑀)
20 eqid 2736 . . . . . . . . . . . 12 (0g𝑀) = (0g𝑀)
2119, 20mndidcl 18571 . . . . . . . . . . 11 (𝑀 ∈ Mnd → (0g𝑀) ∈ (Base‘𝑀))
2221ancli 549 . . . . . . . . . 10 (𝑀 ∈ Mnd → (𝑀 ∈ Mnd ∧ (0g𝑀) ∈ (Base‘𝑀)))
2322ad4antr 730 . . . . . . . . 9 (((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑥𝑉) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) → (𝑀 ∈ Mnd ∧ (0g𝑀) ∈ (Base‘𝑀)))
24 eqid 2736 . . . . . . . . . 10 (+g𝑀) = (+g𝑀)
2519, 24, 20mndlid 18576 . . . . . . . . 9 ((𝑀 ∈ Mnd ∧ (0g𝑀) ∈ (Base‘𝑀)) → ((0g𝑀)(+g𝑀)(0g𝑀)) = (0g𝑀))
2623, 25syl 17 . . . . . . . 8 (((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑥𝑉) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) → ((0g𝑀)(+g𝑀)(0g𝑀)) = (0g𝑀))
2718, 26eqtrd 2776 . . . . . . 7 (((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑥𝑉) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) → ((𝐴f (+g𝑀)𝐵)‘𝑥) = (0g𝑀))
28 nne 2947 . . . . . . 7 (¬ ((𝐴f (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀) ↔ ((𝐴f (+g𝑀)𝐵)‘𝑥) = (0g𝑀))
2927, 28sylibr 233 . . . . . 6 (((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑥𝑉) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) → ¬ ((𝐴f (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀))
3029ex 413 . . . . 5 ((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑥𝑉) → (((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀)) → ¬ ((𝐴f (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀)))
315, 30biimtrid 241 . . . 4 ((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑥𝑉) → (¬ ((𝐴𝑥) ≠ (0g𝑀) ∨ (𝐵𝑥) ≠ (0g𝑀)) → ¬ ((𝐴f (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀)))
3231con4d 115 . . 3 ((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑥𝑉) → (((𝐴f (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀) → ((𝐴𝑥) ≠ (0g𝑀) ∨ (𝐵𝑥) ≠ (0g𝑀))))
3332ss2rabdv 4033 . 2 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → {𝑥𝑉 ∣ ((𝐴f (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀)} ⊆ {𝑥𝑉 ∣ ((𝐴𝑥) ≠ (0g𝑀) ∨ (𝐵𝑥) ≠ (0g𝑀))})
347, 10, 12, 12offun 7631 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → Fun (𝐴f (+g𝑀)𝐵))
35 ovexd 7392 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → (𝐴f (+g𝑀)𝐵) ∈ V)
36 fvexd 6857 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → (0g𝑀) ∈ V)
37 suppval1 8098 . . . 4 ((Fun (𝐴f (+g𝑀)𝐵) ∧ (𝐴f (+g𝑀)𝐵) ∈ V ∧ (0g𝑀) ∈ V) → ((𝐴f (+g𝑀)𝐵) supp (0g𝑀)) = {𝑥 ∈ dom (𝐴f (+g𝑀)𝐵) ∣ ((𝐴f (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀)})
3834, 35, 36, 37syl3anc 1371 . . 3 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → ((𝐴f (+g𝑀)𝐵) supp (0g𝑀)) = {𝑥 ∈ dom (𝐴f (+g𝑀)𝐵) ∣ ((𝐴f (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀)})
3912, 7, 10offvalfv 46408 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → (𝐴f (+g𝑀)𝐵) = (𝑣𝑉 ↦ ((𝐴𝑣)(+g𝑀)(𝐵𝑣))))
4039dmeqd 5861 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → dom (𝐴f (+g𝑀)𝐵) = dom (𝑣𝑉 ↦ ((𝐴𝑣)(+g𝑀)(𝐵𝑣))))
41 ovex 7390 . . . . . 6 ((𝐴𝑣)(+g𝑀)(𝐵𝑣)) ∈ V
42 eqid 2736 . . . . . 6 (𝑣𝑉 ↦ ((𝐴𝑣)(+g𝑀)(𝐵𝑣))) = (𝑣𝑉 ↦ ((𝐴𝑣)(+g𝑀)(𝐵𝑣)))
4341, 42dmmpti 6645 . . . . 5 dom (𝑣𝑉 ↦ ((𝐴𝑣)(+g𝑀)(𝐵𝑣))) = 𝑉
4440, 43eqtrdi 2792 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → dom (𝐴f (+g𝑀)𝐵) = 𝑉)
4544rabeqdv 3422 . . 3 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → {𝑥 ∈ dom (𝐴f (+g𝑀)𝐵) ∣ ((𝐴f (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀)} = {𝑥𝑉 ∣ ((𝐴f (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀)})
4638, 45eqtrd 2776 . 2 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → ((𝐴f (+g𝑀)𝐵) supp (0g𝑀)) = {𝑥𝑉 ∣ ((𝐴f (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀)})
47 elmapfun 8804 . . . . . . 7 (𝐴 ∈ (𝑅m 𝑉) → Fun 𝐴)
48 id 22 . . . . . . 7 (𝐴 ∈ (𝑅m 𝑉) → 𝐴 ∈ (𝑅m 𝑉))
49 fvexd 6857 . . . . . . 7 (𝐴 ∈ (𝑅m 𝑉) → (0g𝑀) ∈ V)
50 suppval1 8098 . . . . . . 7 ((Fun 𝐴𝐴 ∈ (𝑅m 𝑉) ∧ (0g𝑀) ∈ V) → (𝐴 supp (0g𝑀)) = {𝑥 ∈ dom 𝐴 ∣ (𝐴𝑥) ≠ (0g𝑀)})
5147, 48, 49, 50syl3anc 1371 . . . . . 6 (𝐴 ∈ (𝑅m 𝑉) → (𝐴 supp (0g𝑀)) = {𝑥 ∈ dom 𝐴 ∣ (𝐴𝑥) ≠ (0g𝑀)})
52 elmapi 8787 . . . . . . 7 (𝐴 ∈ (𝑅m 𝑉) → 𝐴:𝑉𝑅)
53 fdm 6677 . . . . . . 7 (𝐴:𝑉𝑅 → dom 𝐴 = 𝑉)
54 rabeq 3421 . . . . . . 7 (dom 𝐴 = 𝑉 → {𝑥 ∈ dom 𝐴 ∣ (𝐴𝑥) ≠ (0g𝑀)} = {𝑥𝑉 ∣ (𝐴𝑥) ≠ (0g𝑀)})
5552, 53, 543syl 18 . . . . . 6 (𝐴 ∈ (𝑅m 𝑉) → {𝑥 ∈ dom 𝐴 ∣ (𝐴𝑥) ≠ (0g𝑀)} = {𝑥𝑉 ∣ (𝐴𝑥) ≠ (0g𝑀)})
5651, 55eqtrd 2776 . . . . 5 (𝐴 ∈ (𝑅m 𝑉) → (𝐴 supp (0g𝑀)) = {𝑥𝑉 ∣ (𝐴𝑥) ≠ (0g𝑀)})
5756ad2antrl 726 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → (𝐴 supp (0g𝑀)) = {𝑥𝑉 ∣ (𝐴𝑥) ≠ (0g𝑀)})
58 elmapfun 8804 . . . . . . 7 (𝐵 ∈ (𝑅m 𝑉) → Fun 𝐵)
5958ad2antll 727 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → Fun 𝐵)
60 simprr 771 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → 𝐵 ∈ (𝑅m 𝑉))
61 suppval1 8098 . . . . . 6 ((Fun 𝐵𝐵 ∈ (𝑅m 𝑉) ∧ (0g𝑀) ∈ V) → (𝐵 supp (0g𝑀)) = {𝑥 ∈ dom 𝐵 ∣ (𝐵𝑥) ≠ (0g𝑀)})
6259, 60, 36, 61syl3anc 1371 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → (𝐵 supp (0g𝑀)) = {𝑥 ∈ dom 𝐵 ∣ (𝐵𝑥) ≠ (0g𝑀)})
63 elmapi 8787 . . . . . . . 8 (𝐵 ∈ (𝑅m 𝑉) → 𝐵:𝑉𝑅)
6463fdmd 6679 . . . . . . 7 (𝐵 ∈ (𝑅m 𝑉) → dom 𝐵 = 𝑉)
6564ad2antll 727 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → dom 𝐵 = 𝑉)
6665rabeqdv 3422 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → {𝑥 ∈ dom 𝐵 ∣ (𝐵𝑥) ≠ (0g𝑀)} = {𝑥𝑉 ∣ (𝐵𝑥) ≠ (0g𝑀)})
6762, 66eqtrd 2776 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → (𝐵 supp (0g𝑀)) = {𝑥𝑉 ∣ (𝐵𝑥) ≠ (0g𝑀)})
6857, 67uneq12d 4124 . . 3 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → ((𝐴 supp (0g𝑀)) ∪ (𝐵 supp (0g𝑀))) = ({𝑥𝑉 ∣ (𝐴𝑥) ≠ (0g𝑀)} ∪ {𝑥𝑉 ∣ (𝐵𝑥) ≠ (0g𝑀)}))
69 unrab 4265 . . 3 ({𝑥𝑉 ∣ (𝐴𝑥) ≠ (0g𝑀)} ∪ {𝑥𝑉 ∣ (𝐵𝑥) ≠ (0g𝑀)}) = {𝑥𝑉 ∣ ((𝐴𝑥) ≠ (0g𝑀) ∨ (𝐵𝑥) ≠ (0g𝑀))}
7068, 69eqtrdi 2792 . 2 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → ((𝐴 supp (0g𝑀)) ∪ (𝐵 supp (0g𝑀))) = {𝑥𝑉 ∣ ((𝐴𝑥) ≠ (0g𝑀) ∨ (𝐵𝑥) ≠ (0g𝑀))})
7133, 46, 703sstr4d 3991 1 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → ((𝐴f (+g𝑀)𝐵) supp (0g𝑀)) ⊆ ((𝐴 supp (0g𝑀)) ∪ (𝐵 supp (0g𝑀))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 845   = wceq 1541  wcel 2106  wne 2943  {crab 3407  Vcvv 3445  cun 3908  wss 3910  cmpt 5188  dom cdm 5633  Fun wfun 6490   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  f cof 7615   supp csupp 8092  m cmap 8765  Basecbs 17083  +gcplusg 17133  0gc0g 17321  Mndcmnd 18556
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-of 7617  df-1st 7921  df-2nd 7922  df-supp 8093  df-map 8767  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557
This theorem is referenced by:  mndpsuppfi  46441
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