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Theorem mndpsuppss 18791
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 985 . . . . . 6 (¬ ((𝐴𝑥) ≠ (0g𝑀) ∨ (𝐵𝑥) ≠ (0g𝑀)) ↔ (¬ (𝐴𝑥) ≠ (0g𝑀) ∧ ¬ (𝐵𝑥) ≠ (0g𝑀)))
2 nne 2942 . . . . . . 7 (¬ (𝐴𝑥) ≠ (0g𝑀) ↔ (𝐴𝑥) = (0g𝑀))
3 nne 2942 . . . . . . 7 (¬ (𝐵𝑥) ≠ (0g𝑀) ↔ (𝐵𝑥) = (0g𝑀))
42, 3anbi12i 628 . . . . . 6 ((¬ (𝐴𝑥) ≠ (0g𝑀) ∧ ¬ (𝐵𝑥) ≠ (0g𝑀)) ↔ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀)))
51, 4bitri 275 . . . . 5 (¬ ((𝐴𝑥) ≠ (0g𝑀) ∨ (𝐵𝑥) ≠ (0g𝑀)) ↔ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀)))
6 elmapfn 8904 . . . . . . . . . . . 12 (𝐴 ∈ (𝑅m 𝑉) → 𝐴 Fn 𝑉)
76ad2antrl 728 . . . . . . . . . . 11 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → 𝐴 Fn 𝑉)
87adantr 480 . . . . . . . . . 10 ((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) → 𝐴 Fn 𝑉)
9 elmapfn 8904 . . . . . . . . . . . 12 (𝐵 ∈ (𝑅m 𝑉) → 𝐵 Fn 𝑉)
109ad2antll 729 . . . . . . . . . . 11 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → 𝐵 Fn 𝑉)
1110adantr 480 . . . . . . . . . 10 ((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) → 𝐵 Fn 𝑉)
12 simplr 769 . . . . . . . . . . 11 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → 𝑉𝑋)
1312adantr 480 . . . . . . . . . 10 ((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) → 𝑉𝑋)
14 inidm 4235 . . . . . . . . . 10 (𝑉𝑉) = 𝑉
15 simplrl 777 . . . . . . . . . 10 (((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) ∧ 𝑥𝑉) → (𝐴𝑥) = (0g𝑀))
16 simplrr 778 . . . . . . . . . 10 (((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) ∧ 𝑥𝑉) → (𝐵𝑥) = (0g𝑀))
178, 11, 13, 13, 14, 15, 16ofval 7708 . . . . . . . . 9 (((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) ∧ 𝑥𝑉) → ((𝐴f (+g𝑀)𝐵)‘𝑥) = ((0g𝑀)(+g𝑀)(0g𝑀)))
1817an32s 652 . . . . . . . 8 (((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑥𝑉) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) → ((𝐴f (+g𝑀)𝐵)‘𝑥) = ((0g𝑀)(+g𝑀)(0g𝑀)))
19 eqid 2735 . . . . . . . . . . . 12 (Base‘𝑀) = (Base‘𝑀)
20 eqid 2735 . . . . . . . . . . . 12 (0g𝑀) = (0g𝑀)
2119, 20mndidcl 18775 . . . . . . . . . . 11 (𝑀 ∈ Mnd → (0g𝑀) ∈ (Base‘𝑀))
2221ancli 548 . . . . . . . . . 10 (𝑀 ∈ Mnd → (𝑀 ∈ Mnd ∧ (0g𝑀) ∈ (Base‘𝑀)))
2322ad4antr 732 . . . . . . . . 9 (((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑥𝑉) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) → (𝑀 ∈ Mnd ∧ (0g𝑀) ∈ (Base‘𝑀)))
24 eqid 2735 . . . . . . . . . 10 (+g𝑀) = (+g𝑀)
2519, 24, 20mndlid 18780 . . . . . . . . 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 2775 . . . . . . 7 (((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑥𝑉) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) → ((𝐴f (+g𝑀)𝐵)‘𝑥) = (0g𝑀))
28 nne 2942 . . . . . . 7 (¬ ((𝐴f (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀) ↔ ((𝐴f (+g𝑀)𝐵)‘𝑥) = (0g𝑀))
2927, 28sylibr 234 . . . . . 6 (((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑥𝑉) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) → ¬ ((𝐴f (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀))
3029ex 412 . . . . 5 ((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) ∧ 𝑥𝑉) → (((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀)) → ¬ ((𝐴f (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀)))
315, 30biimtrid 242 . . . 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 4086 . 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 7711 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → Fun (𝐴f (+g𝑀)𝐵))
35 ovexd 7466 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → (𝐴f (+g𝑀)𝐵) ∈ V)
36 fvexd 6922 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → (0g𝑀) ∈ V)
37 suppval1 8190 . . . 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 1370 . . 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 7719 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → (𝐴f (+g𝑀)𝐵) = (𝑣𝑉 ↦ ((𝐴𝑣)(+g𝑀)(𝐵𝑣))))
4039dmeqd 5919 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → dom (𝐴f (+g𝑀)𝐵) = dom (𝑣𝑉 ↦ ((𝐴𝑣)(+g𝑀)(𝐵𝑣))))
41 ovex 7464 . . . . . 6 ((𝐴𝑣)(+g𝑀)(𝐵𝑣)) ∈ V
42 eqid 2735 . . . . . 6 (𝑣𝑉 ↦ ((𝐴𝑣)(+g𝑀)(𝐵𝑣))) = (𝑣𝑉 ↦ ((𝐴𝑣)(+g𝑀)(𝐵𝑣)))
4341, 42dmmpti 6713 . . . . 5 dom (𝑣𝑉 ↦ ((𝐴𝑣)(+g𝑀)(𝐵𝑣))) = 𝑉
4440, 43eqtrdi 2791 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → dom (𝐴f (+g𝑀)𝐵) = 𝑉)
4544rabeqdv 3449 . . 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 2775 . 2 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → ((𝐴f (+g𝑀)𝐵) supp (0g𝑀)) = {𝑥𝑉 ∣ ((𝐴f (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀)})
47 elmapfun 8905 . . . . . . 7 (𝐴 ∈ (𝑅m 𝑉) → Fun 𝐴)
48 id 22 . . . . . . 7 (𝐴 ∈ (𝑅m 𝑉) → 𝐴 ∈ (𝑅m 𝑉))
49 fvexd 6922 . . . . . . 7 (𝐴 ∈ (𝑅m 𝑉) → (0g𝑀) ∈ V)
50 suppval1 8190 . . . . . . 7 ((Fun 𝐴𝐴 ∈ (𝑅m 𝑉) ∧ (0g𝑀) ∈ V) → (𝐴 supp (0g𝑀)) = {𝑥 ∈ dom 𝐴 ∣ (𝐴𝑥) ≠ (0g𝑀)})
5147, 48, 49, 50syl3anc 1370 . . . . . 6 (𝐴 ∈ (𝑅m 𝑉) → (𝐴 supp (0g𝑀)) = {𝑥 ∈ dom 𝐴 ∣ (𝐴𝑥) ≠ (0g𝑀)})
52 elmapi 8888 . . . . . . 7 (𝐴 ∈ (𝑅m 𝑉) → 𝐴:𝑉𝑅)
53 fdm 6746 . . . . . . 7 (𝐴:𝑉𝑅 → dom 𝐴 = 𝑉)
54 rabeq 3448 . . . . . . 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 2775 . . . . 5 (𝐴 ∈ (𝑅m 𝑉) → (𝐴 supp (0g𝑀)) = {𝑥𝑉 ∣ (𝐴𝑥) ≠ (0g𝑀)})
5756ad2antrl 728 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → (𝐴 supp (0g𝑀)) = {𝑥𝑉 ∣ (𝐴𝑥) ≠ (0g𝑀)})
58 elmapfun 8905 . . . . . . 7 (𝐵 ∈ (𝑅m 𝑉) → Fun 𝐵)
5958ad2antll 729 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → Fun 𝐵)
60 simprr 773 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → 𝐵 ∈ (𝑅m 𝑉))
61 suppval1 8190 . . . . . 6 ((Fun 𝐵𝐵 ∈ (𝑅m 𝑉) ∧ (0g𝑀) ∈ V) → (𝐵 supp (0g𝑀)) = {𝑥 ∈ dom 𝐵 ∣ (𝐵𝑥) ≠ (0g𝑀)})
6259, 60, 36, 61syl3anc 1370 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → (𝐵 supp (0g𝑀)) = {𝑥 ∈ dom 𝐵 ∣ (𝐵𝑥) ≠ (0g𝑀)})
63 elmapi 8888 . . . . . . . 8 (𝐵 ∈ (𝑅m 𝑉) → 𝐵:𝑉𝑅)
6463fdmd 6747 . . . . . . 7 (𝐵 ∈ (𝑅m 𝑉) → dom 𝐵 = 𝑉)
6564ad2antll 729 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → dom 𝐵 = 𝑉)
6665rabeqdv 3449 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → {𝑥 ∈ dom 𝐵 ∣ (𝐵𝑥) ≠ (0g𝑀)} = {𝑥𝑉 ∣ (𝐵𝑥) ≠ (0g𝑀)})
6762, 66eqtrd 2775 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → (𝐵 supp (0g𝑀)) = {𝑥𝑉 ∣ (𝐵𝑥) ≠ (0g𝑀)})
6857, 67uneq12d 4179 . . 3 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → ((𝐴 supp (0g𝑀)) ∪ (𝐵 supp (0g𝑀))) = ({𝑥𝑉 ∣ (𝐴𝑥) ≠ (0g𝑀)} ∪ {𝑥𝑉 ∣ (𝐵𝑥) ≠ (0g𝑀)}))
69 unrab 4321 . . 3 ({𝑥𝑉 ∣ (𝐴𝑥) ≠ (0g𝑀)} ∪ {𝑥𝑉 ∣ (𝐵𝑥) ≠ (0g𝑀)}) = {𝑥𝑉 ∣ ((𝐴𝑥) ≠ (0g𝑀) ∨ (𝐵𝑥) ≠ (0g𝑀))}
7068, 69eqtrdi 2791 . 2 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → ((𝐴 supp (0g𝑀)) ∪ (𝐵 supp (0g𝑀))) = {𝑥𝑉 ∣ ((𝐴𝑥) ≠ (0g𝑀) ∨ (𝐵𝑥) ≠ (0g𝑀))})
7133, 46, 703sstr4d 4043 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 395  wo 847   = wceq 1537  wcel 2106  wne 2938  {crab 3433  Vcvv 3478  cun 3961  wss 3963  cmpt 5231  dom cdm 5689  Fun wfun 6557   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  f cof 7695   supp csupp 8184  m cmap 8865  Basecbs 17245  +gcplusg 17298  0gc0g 17486  Mndcmnd 18760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-1st 8013  df-2nd 8014  df-supp 8185  df-map 8867  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761
This theorem is referenced by:  mndpsuppfi  18792
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